Cohomology of arithmetic groups and special values of automorphic L -functions on GL_{n}

Joachim Mahnkopf

July 26, 2000

Introduction

To any automorphic representation π of the adelic points of the group GL_{n} one can attach an L -function L(π, s) defined by an infinte product of Euler type, convergent for Re(s) sufficiently large but which can be holomorphically continued to an entire function if π is cuspidal. We want to investigate the properties of the values of these L -functions at critical points l of the representation π . More precisely we want to study the properties of the collection of values L(π ⊗ χ, l) , where χ runs over all idele class characters of finite order and conductor a power of the prime 𝔭 and l is any non-positive critical value for π . In particular, this fixes the type at infinity of χ to be the trivial character id or the signum character sgn . In view of the functional equation we may restrict to points l ≤ 0 . If n is even, 𝔸 denotes the ring of adeles of ℚ and π is a cuspidal representation of GL_{n}(𝔸) , which is of cohomological type with coefficients in some finite dimensional representation (ρ, V) and which satisfies a certain regularity condition (cf. Definition 4.3), we obtain the following result (cf. Theorem 6.7):

For all critical points l ≤ 0 there is a complex period Ω_{l}(π) such that for all χ as above we have

(0.1)  \frac{L(π ⊗ χ, l)}{Ω_{l}(π)} ∈ E_{π}(χ),

where E_{π} is the field of definition of π . Moreover there is a finite abelian extension K/E_{ρ} , where E_{ρ} ≤ E_{π} is the field of definition of ρ such that for all σ ∈ Aut(ℂ/K) we have

(0.2)  (G(χ)^{\frac{n}{2}} \frac{L(π ⊗ χ, l)}{Ω_{l}(π)})^{σ} = G(χ^{σ})^{\frac{n}{2}} \frac{L(π^{σ} ⊗ χ^{σ}, l)}{Ω_{l}(π^{σ})}.

The proof of this result is by an induction over the rank n of the group GL_{n} as follows. Let π be a cuspidal representation of GL_{n}(𝔸) for arbitrary n ≥ 4 of the above type. Depending on the component at infinity π_{∞} we define a representation π̂_{∞} of GL_{n−2}(ℝ) such that for all cuspidal representation π̂ of GL_{n−2}(𝔸) with infinity component π̂_{∞} the following holds (cf. Theorem 6.6)

There is a period Ω_{l}(π, π′) ∈ ℂ^{∗} and an integer l′ ≤ 0 depending on l such that

(0.3)  \frac{L(π ⊗ χ, l)}{L(π′ ⊗ χ, l′)} ∈ Ω_{l}(π, π′)E_{π}E_{π′}(χ)

and for all σ ∈ Aut(ℂ/K)

(0.4)  (\frac{G(χ)}{Ω_{l}(π, π′)} \frac{L(π ⊗ χ, l)}{L(π′ ⊗ χ, l′)})^{σ} = \frac{G(χ^{σ})}{Ω_{l}(π^{σ}, π′^{σ})} \frac{L(π^{σ} ⊗ χ^{σ}, l)}{L(π′^{σ} ⊗ χ^{σ}, l′)}.

If now n is even and π satisfies the regularity condition we find that the same is true for π̂ and l′ again is a critical point for π′ . Hence, we can inductively apply Eq. (0.3) ,(0.4) to reduce to the case of L -functions on GL_{2}(𝔸) . Thus Eq. (0.1) and (0.2) finally follow from the well known algebraicity properties of the values at critical integers of the L -functions L(f, s) attached to holomorphic elliptic modular forms.

The central point is the compuation of the quotient of quotient of L -functions appearing in (0.3). Since we look to negative points l , we have to make use of the analytic continuation of the Euler product defining L(π, s) to compute (0.3). In general there are two quite different approaches to the problem of analytic continuation, the first one being the theory of Mellin transforms of automorphic forms and second one being the theory of the constant term of Eisenstein series as developped by Langlands and generalized to generic Fourier coefficients by Shahidi. We will combine the theory of the Mellin transform as well as the Langlands-Shahidi theory to compute (0.3).

In more detail, we let P ≤ GL_{n−1} be the maximal parabolic subgroup of type (n − 2, 1) . For any character χ as above the product π̂[χ] = π̂ ⊗ χ defines a cuspidal representation of the Levi subgroup M_{P} (𝔸) and we consider the Mellin transform on GL_{n} ×GL_{n−1} of automorphic forms, which are contained in the space belonging to the representation

π × Ind_{P (𝔸)}^{GL_{n−1}(𝔸)}π̂ ⊗ χ.

Such a Mellin transform for forms on the group GL_{n} × GL_{n−1} is given by the Zeta integral belonging to the Rankin-Selberg convolutions introduced in [J-S]. We shall choose a special pair of automorphic forms ϕ and E_{π̂[χ]} corresponding to π resp. Ind_{P (𝔸)}^{GL_{n−1}(𝔸)}π̂ ⊗ χ for which we compute the Mellin transform. To this end we compute the local factors of the Rankin-Selberg integral at all places, which in particular implies that we have to compute the Whittaker function of the Eisenstein series E_{π̂[χ]} . Since the representation π̂[χ] ramifies at 𝔭 we have to extend the Langlands-Shahidi method to certain ramified induced representations as well. Of course this relies on a generalization of the Casselman-Shalika formulas for the spherical Whittaker function.

As a result we obtain a formula for the quotient of L -functions in (0.3) as an integral involving the forms ϕ_{π} and E_{π̂[χ]} .

Let now n be even. Under the assumption that π has non-vanishing cohomology this integral involving ϕ_{π} and E_{π′} can be computed in a purely topological manner: there is a class ω corresponding to ϕ_{π} and an Eisenstein class ω_{Eis,π′} in the cohomology of the symmetric space of GL_{n} and GL_{n−1} with coefficient system of highest weight m and λ respectively such that

(0.5)  \frac{L(π ⊗ χ, l)}{L(π′ ⊗ χ, l′)} = Lin. comb. ∫_{fundamental class} r_{u}^{∗}ω ((\begin{matrix}g \\ & 1 \end{matrix})) ∪ ω_{Eis,π′},

where u ∈ N_{n}(F_{𝔭}) runs over a finite set of matrices. Using the theory developped in [Ha 1] and [Sch] we are able to compute the behaviour of the analytically defined class ω_{Eis,π′} under the operation of Aut(ℂ/ℚ) (cf. Theorem 5.4). To establish (0.5) as well as the rationality of the Esienstein class it is decisive that the highest weight λ of the coefficient system satisfies certain properties (cf. (W.1) - (W.5) ).

Since the integral (0.5) is constructed by topological methods it commutes with the action of Aut(ℂ/ℚ) and we deduce the claims (0.3) and (0.4).

In view of a study of integrality properties, we proceed a little differently than described above. We use the result of the computation of the Mellin transform of ϕ_{π} and E_{π′} to define a p -adic distribution dP_{π,l} on ℤ_{p}^{∗} in section 3 such that its p -adic Mellin transform satisfies for all χ as above

(0.6)  ∫_{ℤ_{p}^{∗}} χ dP_{π,l} = \frac{L(π ⊗ χ, l)}{L(π′ ⊗ χ, l′)}

and we then compute dP_{π,l} via an integral like (0.5) (cf. Proposition 6.2). Quite analogous we obtain the rationality of the values dP(U) , U ≤ ℤ_{p}^{∗} , which via (0.6) implies (0.3) and (0.4). Since it seems to be very difficult to determine the denominators fo the Eisenstein classes we can not say anything about p -adic analytic properties of the vlues of L(π ⊗ χ, l) ; on the other hand the Eisenstein classes obviously being independent of π at least we see, that there is a univeral bound for the denominators of the congruences satisfied by the values of L(π ⊗ χ, l) .

To ensure that we are not constructing the zero distribution we have to make the

Assumption. A certain integral involving terms at infinity does not vanish

(cf. section 6.2). Finally we remark that instead of the maximal parabolic subgroup P it may be possible to apply the method to more general parabolic subgroups P ≤ GL_{n−1} of type (n_{1}, …, n_{t}, 1) to obtain different formulas for the critical values L(π⊗ χη, l) .

In section 1 we select the Eisenstein series E_{π̂[χ]} and compute its Whittaker function. In particular we generalize the Casselman-Shalika formula for the computationof values of Whittaker functions on elements of the split torus to the place 𝔭 , where the representation π̂[χ] ramifies. The proof, again, is by induction on the rank n . In section 2 we introduce the Mellin transform and we reduce its computation to the computation of the local Rankin-Selberg convolution at 𝔭 . The final formula then appers in Theorem 2.7. In section 3 we define the distribution dP_{π} and compute its p -adic Mellin transform (cf. Theorem 3.5). In section 4 explain the rationality properties of (regular) algebraic automorphic representations relying on [C]. Section 5 contains the (analytic) construction of the Eisenstein cohomology classes and also the proof of their rationality in the cases, which are relevant to us (cf. Theorem 5.4). Finally in section 6 we apply all this to the computation of (0.3) and in 6.3 we explain the induction step, which finally yields (0.1) and (0.2). To conclude the induction step we separate two different cases into which the representation π can fall (cf. Definition 4.3).

Notations.

We denote by B_{n} ≤ GL_{n} the Borel subgroup of upper triangular matrices and we write N_{n} resp. T_{n} for the unipotent radical of B_{n} resp. the maximal torus consisting of diagonal matrices. Let P ≤ GL_{n} be a parabolic subgroup. We denote by U_{P} the unipotent radical and by M_{P} = P/U_{P} the Levi component of P and we write A_{P} for the maximal central torus in M_{P} . By 𝔪 , ^{0}𝔪 , 𝔫 we understand the Lie algebras of M_{P} (ℝ) , ^{0}M_{P} (ℝ) , N_{P} (ℝ) . We set 𝔞_{P} = Lie(A_{P}) and 𝔟_{P} = 𝔥 ∩ ^{0}𝔪 , where 𝔥 is the Lie algebra of the standard torus T_{n−1} of diagonal matrices in GL_{n−1} . We let Φ_{n−1} = Φ(GL_{n−1}, T_{n−1}) be the set of roots of GL_{n−1} with respect to the standard torus. Our choice of the standard minimal parabolic subgroup B_{n−1} ≤ GL_{n−1} determines a choice of positive roots Φ^{+}(GL_{n−1}, T_{n−1}) of GL_{n−1} and we denote by Δ_{n−1} = Δ(GL_{n−1}, T_{n−1}) resp. Δ(M_{P} , T_{n−1}) the set of simple positive roots of GL_{n−1} resp. M_{P} . with respect to the standard torus in GL_{n−1} . Moreover by Δ(P, A_{P}) we denote the simple roots of P with respect to the split component. We set

ρ_{P} = 1/2 ∑_{α∈Δ(P,A_{P})} α

and also ρ̌ = ρ_{B_{n}} , i.e. ρ̌ is the half sum of positive roots in GL_{n} . Finally we write δ_{P} to denote the modulus of the parabolic subgroup P .

For any algebraic group G/ℚ we denote by X(G) the set of all ℚ -morphisms G → GL_{1} and we denote by

⟨ , ⟩ : X(A_{P}) × A_{P} → 𝔾_{m}

the natural pairing. We define

0G = ⋂_{χ∈X_{ℚ}(G)} ker χ^{2}.

For any parabolic subgroup P ≤ GL_{n} the connected component of the unit element of the ℝ rational points A_{P}^{0} = A_{P} (ℝ) is a split component for P and we have the Langlands decomposition

P (ℝ) = ^{0}M_{P} (ℝ)A_{P}^{0} U_{P} (ℝ).

We note that A_{P}^{0} is isomorphic to (ℝ_{+}^{∗})^{t} via

(a_{1}, ⋯, a_{t} ↦ diag(a_{1}, …, a_{1}, …, a_{t}, …, a_{t}).

We identify characters of A_{P}^{0} with characters of 𝔞_{P} via the exponential mapping

exp : 𝔞_{P} → A_{P}^{0} .

Let F be a number field. We denote by O resp. 𝔸 the ring of integers resp. ring of adeles 𝔸 of F and S_{∞} is the set of archimedean places of F . We write O_{v} for the completion of O at v and we set Ô = ∏_{v∉S_{∞}} O_{v} . For any finite place v we denote by q_{v} resp. ϖ_{v} the cardinality of the residue field resp. a local prime element at the place v .

More specific, let F/ℚ be a totally real number field. For any finite place v we define the maximal compact subgroup K_{n,v} = GL_{n}(O_{v}) and if v ∈ S_{∞} corresponds to an embedding F ↪ ℝ we set K_{n,v} = SO_{n}(ℝ)Z_{n}^{0}(ℝ) . We write K_{n,f} for the product of the K_{n,v} over all finite places and K_{n,∞} for the product of the K_{n,v} over all infinite places. Moreover, by

π_{0} = ∏_{v∈S_{∞}} O_{n}(ℝ)/SO_{n}(ℝ)

we understand the group of connected components of GL_{n}(∏_{v∈S_{∞}} F_{v}) . Obviously, π_{0} ≅ (ℤ/2ℤ)^{[F:ℚ]} .

If P ≤ GL_{n} is a parabolic subgroup and K ≤ GL_{n}(ℝ) a compact subgroup we define K^{M_{P}} to be the image of the intersection K ∩ P (ℝ) in M_{P} (ℝ) . In the same way we define K^{M_{P}} ≤ M_{P} (𝔸_{f}) for compact subgroups K ≤ GL_{n}(𝔸_{f}) .

By A_{F} (n) we understand the space of automorphic forms on GL_{n}(𝔸) and L_{0}^{2}(GL_{n}(F)\GL_{n}(𝔸)) is the space of all cusp forms.

We write 1_{G} to denote the unit element in G and we abbreviate 1_{n} = 1_{GL_{n}} . Finally we denote by [x] , x ∈ ℝ the largest integer equal to or smaller than x .

Acknowledgements. This work was begun while staying at Université de Paris Sud, Orsay and the main part of it was completed during a stay at Max-PLanck Institut für Mathematik, Bonn. It was written down while staying at University of Heidelberg. I am greatful to these institutions for their invitations.

1 Generic Fourier coefficients of Eisenstein series

Let F/ℚ be a finite extension and let 𝔸 denote its ring of adeles. To technically simplify the exposition we assume that F has class number equal to one. Let π be a cuspidal automorphic representation of GL_{n}(𝔸) , n ∈ ℕ . We fix a prime 𝔭 of O_{F} such that π_{𝔭} does not ramify and we let X_{𝔭} be the set of all idele class characters χ : F^{∗}\𝔸^{∗} → ℂ^{∗} of finite order with infinity component χ_{∞} = id and conductor f_{χ} = 𝔭^{e} a 𝔭 -power.

We want to describe a formula, which for a pair of cuspidal representations π resp. π̂ on GL_{n}(𝔸) resp. GL_{n−2}(𝔸) computes the behaviour of the ratio

(1.1)  \frac{L(π ⊗ χ, n)}{L(π̂ ⊗ χ, n)}

of principal L -function as the representations vary over twists with all characters χ in X_{𝔭} . To derive this formula we shall simultaneously apply the theory of Rankin-Selberg convolutions an the Langlands-Shahidi theory. In this section we explain the way we apply the Langlands-Shahidi method, which relies on the computation of generic Fourier coefficients of Eisenstein series. In particular we extend the method to include ramified primes. This will rely on a extension of the formulas for the values of calss-1 Whittaker functions to certain ramified representations.

We let π̂ be a unitary, cuspidal representation of GL_{n−2}(𝔸) and we denote by S′ the set of all places, where π̂ ramifies including the archimedean ones. We assume that 𝔭 ∉ S′ , i.e. π̂_{𝔭} is of class-1. We let P ≤ GL_{n−1} be the maximal parabolic subgroup of type (n − 2, 1) . In particular, the Levi subgroup of M_{P} of P is isomorphic to GL_{n−2} × GL_{1} and we obtain an embedding of GL_{n−2} into M_{P}

P  ⊂ GL_{n−1}
↑
GL_{n−2} ↪ GL_{n−2} × GL_{1}

For any character χ ∈ X_{𝔭} we define the cuspidal representation of the Levi subgroup M_{P}

π̂[χ] = π̂ ⊗ χ

and we denote by V_{π̂[χ]} be the space of cusp forms on which π̂[χ] can be realized. We note that π̂[χ] factorizes over A_{P}^{0} \GL_{n−1}(𝔸) , hence the restriction of the central character ω_{π̂[χ]} to A_{P}^{0} vanishes identically.

For any place v we define a local height function

H_{P,v} : M_{P} (F_{v}) → 𝔞_{P}

attached to the parabolic P : we set e_{v} = q_{v} if v is finite and e_{v} = exp if v is archimedean; H_{P,v} then is the unique function satisfying

|ν(m)|_{v} = e_{v}^{⟨ν,H_{P,v}(m)⟩}

for all rational characters ν ∈ X(M_{P})_{ℚ} and m ∈ M_{P} (F_{v}) . Using the decomposition GL_{n−1}(F_{v}) = M_{P} (F_{v})N_{P} (F_{v})K_{n−1,v} we may extend H_{P,v} to GL_{n−1}(F_{v}) . The (global) height function

H_{P} : GL_{n−1}(𝔸) → 𝔞_{P}

is defined via the equation

e^{⟨ν,H_{P} (m)⟩} = ∏_{v} e_{v}^{⟨ν,H_{P,v}(m_{v})⟩}

for all ν ∈ X(M_{P})_{ℚ} .

Let Λ ∈ 𝔞_{P,ℂ}^{∗} . With the above choice of data we then define the (unitarily) induced representation

I_{P,Λ}π̂[χ] = Ind_{P_{𝔸}}^{GL_{n−1}(𝔸)}e^{⟨Λ,H_{P} (·)⟩} ⊗ π̂[χ].

We immediately deduce I_{P,Λ}π̂[χ] = ⊗_{v}I_{P,Λ,v}π̂[χ] , where

I_{P,Λ,v}π̂[χ] = Ind_{P (F_{v})}^{GL_{n−1}(F_{v})}e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ π̂[χ].

We denote by f = ∏_{v} f_{v} the conductor of I_{P,Λ}π̂[χ] .

The theory of Eisenstein series yields an intertwining operator

Eis_{P,π̂[χ],Λ} : I_{P,Λ}π̂[χ] → A_{F} (n − 1).

and for the moment we shall assume that this map is well defined. More precisely, let f ∈ C^{∞}(N_{P} (𝔸)A_{P}^{0} P (F)\GL_{n−1}(𝔸)) such that for any g ∈ GL_{n−1}(𝔸) the map

{m ↦ f(mg)} ∈ V_{π̂[χ]},

where m ∈ A_{P}^{0} \M_{P} (𝔸) . Decomposing g = mnk as above we set

f_{Λ}(g) = e^{⟨Λ+ρ_{P} ,H_{P} (g)⟩}f(g).

We then assume that the Eisenstein series

E(f, Λ, g) = ∑_{γ∈P (F)\GL_{n−1}(F)} f_{Λ}(γg)

is well defined for certain values of Λ . This assumption will be verified later on in those cases, which are important to us. We denote by V_{I_{P,Λ}π̂[χ]} the image of I_{P,Λ}π̂[χ] under Eis.

We fix a non-trivial character τ = ⊗_{v}τ_{v} : F \𝔸 → ℂ^{∗} such that for all finite places the conductor of τ_{v} equals O_{v} . We also denote by τ the induced (generic) character on N_{n}(𝔸) : n ↦ ∑_{i} τ(n_{i,i+1}), n = (n_{ij}) and we denote by W (π, τ) the Whittaker model of π . The isomorphism mapping π into its Whittaker model is given by

(1.2)  \begin{matrix}V_{π} & → & W (π, τ) \\ ϕ & ↦ & ϕ^{τ}(g), \end{matrix}

where

ϕ^{τ}(g) = ∫_{N_{n}(F)\N_{n}(𝔸)} ϕ(ng) τ̄(n) dn.

We want to extend this isomorphism to induced automorphic representations. We denote by w_{P} the longest element in the Weyl group of M_{P} and we abbreviate w_{n,l} = w_{GL_{n}} . Hence, we have

w_{P} = diag(w_{n−2,l}, 1).

The Whittaker model of I_{P,Λ}π̂[χ] is then given as

W (I_{P,Λ}π̂[χ], τ) = W (π̂, τ) ⊗ ℂ_{χ},

where by ℂ_{χ} we understand the 1 -dimensional space, on which GL_{1}(𝔸) acts via χ . We write f(g; h) , f ∈ I_{P,Λ}V_{π̂[χ]} to denote the value of the automorphic function f(g) at h ∈ M_{P} (𝔸) and we use an analogous notation for representations induced from spaces of Whittaker functions. We let Q ≤ GL_{n−1} be any parabolic subgroup containing P , i.e. Q = P or Q = GL_{n−1} . P ∩ M_{Q} ≤ M_{Q} is then a parabolic subgroup with Levi subgroup equal to M_{P} . Hence, we may define

I_{P ∩M_{Q},Λ}^{M_{Q}}π̂[χ] = Ind_{(P ∩M_{Q})(𝔸)}^{M_{Q}(𝔸)} e^{⟨Λ,H_{P} (·)⟩} ⊗ π̂[χ];

analogously we define I_{P,Λ,v}^{Q}π̂[χ] . We note that this definition works for all pairs of parabolic subgroups P, Q in GL_{n−1} satisfying P ≤ Q . If f ∈ I_{P,Λ}^{Q}V_{π̂[χ]} is any section we write f(g; h) to denote the value of the automorphic form f(g) at h ∈ M_{P} (𝔸) ; we use an analogous notation for representations, which are induced from spaces of Whittaker functions. The isomorphism (1.2) then induces a map

P_{Q} : I_{P,Λ}V_{π̂[χ]} →^{∼} Ind_{Q(𝔸)}^{GL_{n−1,𝔸}}(W (I_{P ∩M_{Q},Λ}^{M_{Q}}π̂[χ], τ)),

which sends a section f to

P_{Q}f(g; 1_{M_{Q}}) = ∫_{U_{M_{Q}}(F)\U_{M_{Q}}(𝔸)} f(ug; 1)τ̄(u) du,

where U_{M_{Q}} = N_{n−1} ∩ M_{Q} is the unipotent radical of the Levi component M_{Q} . This defines P_{Q}f in a unique way. If Q = GL_{n−1} we obtain an isomorphism

P_{GL_{n−1}} : I_{P,Λ}π̂[χ] → W (I_{P,Λ}π̂[χ], τ).

We abbreviate I_{P,Λ}^{⋆}π̂[χ] = I_{P,Λ}W (π̂[χ], τ) and P^{∗} = P_{P} . We want to relate P^{∗} to the computation of Fourier coefficients of Eisenstein series. For the moment we let P, Q ≤ GL_{n−1} be any parabolic subgroups satisfying P ≤ Q and π̂[χ] a cuspidal representation of M_{P} (𝔸) . For any place v we define a map

P_{P,v}^{Q} : Ind_{P (F_{v})}^{GL_{n−1,Fv}} e^{⟨Λ,H_{P} (·)⟩} ⊗ W (π̂_{χ,v}, τ_{v}) → Ind_{Q(F_{v})}^{GL_{n−1}(F_{v})}W (I_{P ∩M_{Q},Λ,v}^{M_{Q}} π̂[χ], τ_{v})

by sending a section f_{v} to

P_{P,v}^{Q}f(g) = ∫_{w_{P} w_{Q}(M_{Q}∩U_{P})w_{Q}w_{P} (F_{v})} f_{v}(w_{P} w_{Q}ug; 1)τ̄_{v}(u) du.

We set P_{P}^{Q} = ⊗_{v}P_{P,v}^{Q} and P_{P,v}^{Q} is given by the same integral as above over the adelic points of w_{P} w_{Q}(U_{M_{Q}} ∩ M_{P})w_{Q}w_{P} . We return to our situation and let P be the parabolic subgroup as chosen above and Q = GL_{n−1} . We obtain an isomorphism

P_{P}^{GL_{n−1}} : I_{P,Λ}^{∗}π̂[χ] → W (I_{P,Λ}π̂[χ], τ).

Explicitely, in this case we have

(1.3)  w_{P} w_{Q}(U_{M_{Q}} ∩ M_{P})w_{Q}w_{P} = U_{1,n−2},

where U_{1,n−2} is the unipotent radical of the parabolic subgroup of type (1, n − 2) .

Together with the Eisenstein operator and the assignment ϕ ↦ E^{τ} we obtain a diagram

(1.4)

I_{P,Λ}V_{π̂[χ]}  →^{P^{⋆}} Ind_{P_{𝔸}}^{GL_{n−1,𝔸}}W (π̂[χ], τ)
Eis ↓  ↓ P_{P,v}^{GL_{n−1}}
V_{I_{P,Λ}π̂[χ]} →^{E↦E^{τ}}  W (I_{P,Λ}π̂[χ], τ).

Lemma 2.1 in [Sha 1] then states that (1.3) commutes. As an immediate consequence we see that the local components of Eτ̄(f, Λ, g) of E(f, Λ, g) are given by

(1.5)  Eτ̄ ; v(f, Λ, g) = ∫_{U_{1,n−1}(F_{v})} P^{∗}f_{Λ,v}(w_{P} w_{l,n−1}ng)τ_{v}(n) dn,

where we assume that f = ⊗_{v}f_{v} . We want to apply (1.5) to explicitely compute Eτ̄(f, Λ, g) and to this end we have to fix a section f and hence an Eisenstein series E(f, Λ) . Using the above realizations of I_{P,Λ}π̂[χ] we define a section f_{π̂[χ],Λ}^{⋆} = ⊗_{v}f_{π̂[χ],Λ,v}^{⋆} in I_{P,Λ}^{⋆}π̂[χ] as follows.

• For any finite place v ∉ S′ ∪ {𝔭} the representation I_{P,Λ}π̂[χ] is unramified and we define f_{π̂[χ],Λ,v}^{⋆} to be the spherical function in I_{P,Λ,v}^{⋆}π̂[χ] . Thus, writing g = mnk , p ∈ P (F_{v}) , n ∈ U_{P} (F_{v}) , k ∈ K_{n−1,v} the section is given as

f_{π̂[χ],Λ,v}^{⋆}(g) = e_{v}^{⟨Λ+ρ̌,H_{P,v}(p)⟩} π̂[χ](m) w_{π̂,v}^{0} ⊗ 1,

where w_{π̂,v}^{0} is the unique spherical function in W (π̂_{v}, τ_{v}) satisfying w_{π̂,v}^{0}(e) = 1 .

• At all finite places v ∈ S′ we define f_{π̂[χ],Λ,v}^{⋆} as follows. We set

w = ( \begin{matrix}& 1 \\ & & . & . & . \\ & & & & & 1 \\ 1 \end{matrix} )

and we denote by w_{π̂,v}^{0} the essential vector in W (π̂_{v}, τ_{v}) and by f_{v} the conductor of π̂_{v} . We then define

f_{π̂[χ],Λ,v}^{⋆}(g) = \{ \begin{matrix}π̂[χ](m)w_{π̂,v}^{0} ⊗ 1ωπ̂(k_{n−1,n−1}) & if & g = mnwk ∈ M_{P} (F_{v})N_{P} (F_{v})K_{0}(n − 1, f_{v}) \\ 0 & else. \end{matrix}

It is easily verified that f_{π̂[χ],Λ,v}^{⋆} is well defined, K_{1}(n−1, f_{v}) -invariant on the right and

that f_{π̂[χ],Λ,v}^{⋆} spans the space (Ind_{P_{Fv}}^{GL_{n−1,Fv}} (W (π̂[χ]_{v}, τ_{v})))^{K_{1}(n−1,f_{v})} of K_{1}(n − 1, f_{v}) - invariant vectors.

• For v = 𝔭 we decompose g = mnk ∈ M_{P} (F_{v})U_{P} (F_{v})K_{n−1,v} and we set

(1.6)  f_{π̂[χ],Λ,v}^{⋆}(g) = χ_{v}(k_{n−1,n−1}) e_{v}^{⟨Λ+ρ̌,H_{P,v}(m)⟩}π̂[χ, v](m)w_{π̂,v}^{0} ⊗ 1,

if k = (k_{ij}) ∈ K_{0}(n−1, 𝔭^{e}) and f_{π̂[χ],Λ,v}^{⋆}(g) = 0 else. Here, w_{π̂,𝔭}^{0} is the spherical Whittaker function in W (π̂_{𝔭}, τ_{𝔭}) satisfying w_{π̂,𝔭}^{0}(e) = 1 . Obviously, f_{π̂[χ],Λ,v}^{⋆} is invariant on the right under K_{1}(n − 1, 𝔭^{e}) and it is easy to see that

(Ind_{P_{F𝔭}}^{GL_{n−1,F𝔭}}(W (π̂[χ]_{𝔭}, τ_{𝔭})))^{K_{1}(n−1,f_{𝔭})} = ⟨f_{π̂[χ],Λ,𝔭}^{⋆}⟩_{ℂ}.

• If v is archimedean I_{P,Λ,v}π̂[χ] does not depend on χ_{v} and we let f_{π̂[χ],Λ,v}^{⋆} be an arbitrary but fixed section in I_{P,Λ,v}π̂[χ] , which does not depend on χ .

We set f_{π̂[χ],Λ} = P^{⋆}^{−1}(f_{π̂[χ],Λ}^{⋆}) and E(f_{π̂[χ],Λ}, g) = Eis_{P,π̂[χ],Λ}(f_{π̂[χ],Λ}) and we denote by v_{π̂[χ],Λ} = Eτ̄(f_{π̂[χ],Λ}) the Whittaker function of E(f_{π̂[χ],Λ}) .

With this choice of an Eisenstein series we now turn to the computation of the finite part of its Whittaker function Eτ̄(f_{π̂[χ],Λ}) . We shall use the decomposition Eτ̄ ; fin(f_{π̂[χ],Λ}) = ⊗_{v∉S_{∞}}Eτ̄ ; v(f_{π̂[χ],Λ}) and compute the local Whitaker functions at all finite places.

1.1. The (ramified) places v ∈ S′ − S_{∞} .

Here, the following qualitative result will be enough for our purposes.

Lemma 1.1. For all v ∈ S′ − S_{∞} there is an element g_{v}^{0} ∈ GL_{n−1}(F_{v}) such that Eτ̄ ; v(f_{π̂[χ],Λ})(g_{v}^{0}) does not vanish for almost all (i.e. all but finitely many) unitary characters χ : F^{∗}\𝔸^{∗} → ℂ^{∗} with conductor prime to S′ .

Proof. Since f_{π̂[χ],Λ,v}^{⋆} is not equal to the zero section there is a g_{v}^{0} ∈ GL_{n−1}(F_{v}) such that f_{π̂[χ],Λ,v}^{⋆}(g_{v}^{0}) ≠ 0 . We denote by X_{v}^{nr} = (F_{v}^{∗}/O_{v}^{∗})ˆ the set of all unitary, unramified characters of F_{v} . For all χ_{v} ∈ X_{v}^{nr} there is a bijection

\begin{matrix}I_{P,Λ,v}π[1]_{v} & → & I_{P,Λ,v}π̂_{v}[χ_{v}] \\ f(g) & ↦ & χ_{v}(m_{2})f(g), \end{matrix}

where g = mnk with m = diag(m_{1}, m_{2}) . By its construction the sections f_{π̂[χ],Λ,v}^{⋆} ∈ Ind_{P (F_{v})}^{GL_{n−1}(F_{v})}(W (π̂_{v}, τ_{v})⊗ℂ_{χ_{v}}) correspond to each other under this bijection and Proposition 3.1 in [Sha 2] then implies that the assignment

χ_{v} ↦ v_{π̂[χ],Λ,v}(g_{v}^{0}), χ_{v} ∈ X_{v}^{nr},

with v_{π̂[χ],Λ,v} defined by (1.4) is a holomorpic map. On the other hand, since F_{v}^{∗}/O_{v}^{∗} is discrete, X_{v}^{nr} is compact and any infinite subset X ⊂ X_{v}^{nr} has a culmination point in X_{v}^{nr} . But v_{π̂[χ],Λ,v}(g_{v}^{0}) does not vanish identically and hence, the identity theorem for holomorphic functions implies that v_{π̂[χ],Λ,v}(g_{v}^{0}) can not vanish for infinitely many χ_{v} in X_{v}^{nr} . This proves the Lemma.

For later purpose for all v ∈ S′ we fix a compact open subgroup K_{n−1,v}^{∗} ≤ K_{n−1,v} such that g_{v}^{0} ∈ K_{n−1,v}^{∗} and v_{π̂[χ],Λ,v} is invariant under K_{n−1,v}^{∗} on the right. We set g_{S′}^{0} = ⊗_{v∈S′}g_{v}^{0}

1.2. The (unramified) places v ∉ S′ ∪ {𝔭} .

At places v ∉ S′ ∪ {𝔭} we need to know the values of Eτ̄ ; v(f_{π̂[χ],Λ}) at e . To formulate the result we note that the Lie algebra 𝔞_{P,ℂ} is naturally isomorphic to ℂ^{2} . This induces an identification ℂ^{2} →^{∼} 𝔞_{P,ℂ}^{∗} , which we write as (s_{1}, s_{2}) ↦ Λ_{s_{1},s_{2}} . Explicitely,

Λ_{s_{1},s_{2},𝔸} = e^{⟨Λ_{s_{1},s_{2}},H_{P} (·)⟩} : \binom{m_{1} ∗}{m_{2}} ↦ | det m_{1}|^{\frac{s_{1}}{n−2}} |m_{2}|^{s_{2}}.

For later pupose we set

(1.7)  Λ_{𝔸}(g) = e^{⟨Λ,H_{P} (g)⟩}, Λ_{v}(g) = e_{v}^{⟨Λ,H_{P,v}(g)⟩}

and we also set Λ_{fin} = ⊗_{v≠∞}Λ_{v} . In particular, we have Λ_{𝔸}|_{A_{P}^{0}} = Λ .

Lemma 1.2. For all places v , where π̂ and χ do not ramify, we have

Eτ̄ ; v(f_{π̂[χ],Λ}, e) = L(π̂_{v} × χ_{v}^{−1}, 1 + s_{1} − s_{2})^{−1}.

Proof. Since v is not contained in S′ , there are unramified characters ν_{1}, …, ν_{n−2} : F_{v}^{∗} → ℂ^{∗} such that

π̂_{v} ≅ Ind(ν_{1}, … ν_{n−2}).

We set ν_{n−1} = χ_{v} and \underline{ν} = ν_{1} ⊗ ⋯ ⊗ ν_{n−1} : B_{n−1}(F_{v}) → ℂ^{∗} . We denote by f_{1}^{0} the spherical function in | · |_{v}^{s_{1}} ⊗ Ind_{B_{n−1}(F_{v})}^{GL_{n−2,Fv}}(ν_{1} ⊗ … ⊗ ν_{n−2}) satisfying f_{1}^{0}(e) = 1 ; the

spherical function in Ind_{B_{n−1}(F_{v})}^{GL_{n−1}(F_{v})}(e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ \underline{ν}) then reads

f^{0}(g) = f_{1}^{0}(m_{1})χ_{v}(m_{2})|m_{2}|_{v}^{s_{2}}

for g = mnk ∈ M_{P} (F_{v})N_{P} (F_{v})K_{n−1,v} and m = diag(m_{1}, m_{2}) . Let Q ≤ GL_{n−1} be any parabolic sibgroup containing B_{n−1} . Attached to the pair (B_{n−1}, Q) there is the intertwining map

P_{B_{n−1}}^{Q} : Ind_{B_{n−1,Fv}}^{GL_{n−1,Fv}} (e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ \underline{ν})
→ Ind_{Q(F_{v})}^{GL_{n−1,Fv}} (W (Ind_{B_{n−1}∩M_{Q}(F_{v})}^{M_{Q}(F_{v})}e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ \underline{ν}, τ_{v}))

which sends an element f , which we view as ℂ -valued function, to the section P_{B_{n−1}}^{Q}f , which is uniquely determined by the equation

P_{B_{n−1}}^{Q}f(g; 1_{M_{Q}(F_{v})}) = ∫_{U_{M_{Q}}(F_{v})} f(w_{Q}ng)τ̄_{v}(n) dn.

We note that in case Q = GL_{n−1} the right hand side is just W (I_{P,Λ,v}π̂[χ], τ_{v}) . Using a straightforward calculation we obtain

• P_{B_{n−1},v}^{P}(f^{0})(e; 1) = ∫_{N_{n−2}(F_{v})} f_{1}^{0}(w_{l,n−2}n)τ_{v}(n) dn.

• The diagram

Ind_{B_{n−1}(F_{v})}^{GL_{n−1}(F_{v})}(e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ \underline{ν}) →^{P_{B_{n−1}}^{Q}} Ind_{Q(F_{v})}^{GL_{n−1(Fv)}}(W (e_{v}^{⟨Λ,H_{P,v}(·)⟩} ⊗ \underline{ν}))
P_{B_{n−1}}^{P} ↘  ↓ P_{P}^{GL_{n−1}}
I_{P,Λ,v}^{∗}π̂[χ]

commutes.

Uniqueness of the essential vector up to scalar multiples together with the first property implies that

f_{π̂[χ],Λ,v}^{⋆}(g; 1) = P_{B_{n−1},v}^{P}(f^{0})(g; 1_{M_{P} (F_{v})}) (∫_{N_{n−2}(F_{v})} f_{1}^{0}(w_{l,n−2}n)τ̄(n) dn)^{−1} .

Using the commutativity of (1.3) and applying the above transitivity we obtain

Eτ̄ ; v(f_{π̂[χ],Λ}, e) = P_{P,v}^{GL_{n−1}}(f_{π̂[χ],Λ,v}^{⋆})(e)
= \frac{P_{B_{n},v}^{GL_{n−1}}f_{π̂[χ],Λ,v}^{0}(e)}{P_{B_{n−2},v}^{GL_{n−2}}f_{1}^{0}(e)}.

We may now apply the formula of Casselman-Shalika to compute the factors appearing in the nominator and in the denominator. We denote by t_{v} the matrix

t_{v} = ( \begin{matrix}ν_{1}(ϖ_{v}) \\ & . & . & . \\ & & & & ν_{n−1}(ϖ_{v}) \end{matrix} ) ∈ GL_{n−1}(ℂ).

Moreover, for any pair i < j we define the coroot

H_{ij} = diag(…, ϖ_{v}, …, ϖ_{v}^{−1}, …),

where ϖ_{v} resp. ϖ_{v}^{−1} appear in the i -th resp. j -th position and all other entries are equal to 1 . Applying Theorem 5.4 in [C-S] we then obtain

Eτ̄ ; v(f_{π̂[χ],Λ}, e) = \frac{∏_{i<j}(1 − q_{v}^{−1}|H_{ij}|_{𝔭}^{Λ} t_{v,i}t_{v,j}^{−1})}{∏_{i<j<n−1}(1 − q_{v}^{−1}|H_{ij}|_{𝔭}^{Λ} t_{v,i}t_{v,j}^{−1})}.

Taking into account that ^{L}(GL_{n−2} × GL_{1}) = GL_{n−2}(ℂ)×GL_{1}(ℂ) , this may be rewritten as

Eτ̄ ; v(f_{π̂[χ],Λ}, e) = L(π̂_{v} × χ_{v}, r_{n−2} ⊗ r̃_{1}, 1 + s_{1} − s_{2})^{−1}
= L(π̂_{v} × χ̃_{v}, 1 + s_{1} − s_{2})^{−1},

where r_{i} : GL_{i}(ℂ) → GL_{i}(ℂ) is the standard representation and r̃_{i} denotes the contragredient. Hence, the Lemma is proven.

Our assumptions on the conductor of π̂_{𝔭} imply that L(π̂_{𝔭} ⊗ χ, s) = 1 . Moreover replacing the representation π̂ by a twist η̃ ⊗ π̂ , where η̃ is an idele class character of finite order with sufficiently high conductor fη̃ we may assume that

(1.8.C)  L(π̂_{v} ⊗ χ_{v}, s) = 1

for all places v ∈ S′ . Using Lemma 1.2 we then obtain

(1.9)  v_{π̂[χ],Λ}^{S′∪{𝔭}}(e) = L(π̂ ⊗ χ^{−1}, 1 + s_{1} − s_{2})^{−1},

where v_{π̂[χ],Λ}^{S′} = ⊗_{v∉S′}v_{π̂[χ],Λ,v}

1.3. The place 𝔭 .

In this subsection, to simplify the notation we set r = n − 1 and we use the notation v_{π̂[χ],Λ,v} instead of Eτ̄ ; v(f_{π̂[χ],Λ}) . We want to compute the values of v_{π̂[χ],𝔭} on elements of the torus T_{r}(F_{𝔭}) . In view of the existence of the central character of I_{P,Λ,𝔭}π̂[χ] , it is enough to to this for diagonal matrices of the form

t = diag(1, t_{1}, …, t_{r−1}).

We abbreviate v = (v_{1}, …, v_{r−1}) , where v_{i} = v_{𝔭}(t_{i}) . An easy exercise proves that v_{π̂[χ],𝔭}(t) vanishes unless

(1.10)  −v_{1} ≥ 0, v_{2} − v_{1} ≥ 0 …, v_{r−1} − v_{r−2} ≥ 0,

which we shall assume henceforth. Since the diagram (1.4) commutes we obtain

v_{π̂[χ],Λ,𝔭}(t) = P_{GL_{n−1}}^{P}(f_{π̂[χ],𝔭}^{⋆})(t) = ∫_{U_{1,r−1}(F_{𝔭})} f_{π̂[χ],𝔭}^{⋆}(w_{P} w_{r,l}ut; 1_{r−1}, 1) τ_{𝔭}(u) du.

Here, U_{1,r−1} is the unipotent radical of the parabolic subgroup of type (1, r − 1) and

w_{P} w_{r,l} = (\begin{matrix}& 1_{n−2} \\ 1 \end{matrix}) .

In the following we compute this integral.

Let u = (u_{1}, …, u_{r−1}) as in (2.1) with v_{i} replaced by u_{i} ; by the change of integration variables u_{i} ↦ t_{i}^{−1}u_{i} , i = 1, …, r − 1 we obtain for v_{π̂[χ],Λ,𝔭}(t) the expression

|t|_{𝔭}^{−1} ∫_{F_{𝔭}×···×F_{𝔭}} f_{π̂[χ],𝔭}^{⋆} (( \begin{matrix}0 & t_{1} \\ & ... & ... \\ & & 0 & t_{r−1} \\ 1 & u_{1} & ⋯ & u_{r−1} \end{matrix} ); 1_{r−1}, 1) τ_{𝔭}(t_{1}^{−1}u_{1}) du_{1} … du_{r−1},

where |t|_{𝔭} = ∏_{i=1}^{r−1} |t_{i}|_{𝔭} . We write for short

t̃= diag(t_{1}, …, t_{r−1}, 1)

and

u = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ 1 & u_{1} & ⋯ & u_{r−1} \end{matrix} ) .

Let c = (c_{1}, …, c_{r−1}) ∈ ℤ^{r−1} be a finite sequence of integers. We set

U_{c} = {(u_{1}, …, u_{r−1}) ∈ (F_{𝔭})^{r−1} : v_{𝔭}(u_{i}) = c_{i}, i = 1, …, r − 1}.

and since meas(F_{𝔭}^{r−1} − (F_{𝔭}^{∗})^{r−1}) = 0 the integral decomposes

(1.11)  v_{π̂[χ],Λ,𝔭}(t) = ∑_{c∈ℤ^{r−1}} I(t, c),

where

I(t, c) = |t|_{𝔭}^{−1} ∫_{U_{c}} f_{π̂[χ],𝔭}^{⋆} (t̃u; 1_{r−1}, 1) τ_{𝔭}(t_{1}^{−1}u_{1}) du_{1} … du_{r−1}.

We want to calculate I(D, c) . To any c = (c_{1}, …, c_{r−1}) ∈ ℤ^{r−1} we shall attach a subset J (c) = {i_{1} < … < i_{k}} of {1, …, r − 1} . The evaluation of I(t, c) will then essentially depend on the value of J (c) . We first assume that c_{i} ≤ 0, i = 1, …, r−1 ; in this case we inductively define J (c) as follows: we set i_{0} := r and assuming that i_{h} has already been defined we set

i_{h+1} = min{1 ≤ i < i_{h} : v_{𝔭}(u_{i}) ≤ v_{𝔭}(u_{j}) for all 1 ≤ j < i_{h}}

as long as the set on the right hand side is non-empty. Obviously the last element of this sequence is i_{k} = 1 with k ≤ r−1 . In the general case we first define the sequence c̃ = (c̃_{1}, …, c̃_{r−1}) by setting

c̃_{i} = \{ \begin{matrix}c_{i} & if c_{i} ≤ 0 \\ 0 & if c_{i} > 0, \end{matrix}

and we then define J (c) = J (c̃) .

Using the description (1.6) of the essential vector it is immediate that f_{π̂[χ],𝔭}^{⋆} (t̃u; 1_{r−1}, 1) ≠ 0 only if

(1.12)  v_{𝔭}(c_{r−1}) ≤ −e + min{0, v_{𝔭}(c_{1}), …, v_{𝔭}(c_{r−2})}.

In particular, only those c contribute non-trivially to (1.11) , which satisfy r − 1 ∈ J (c) .

Lemma 1.3. Let u_{1}, …, u_{r−1} ∈ F_{𝔭} and denote by J (c) = {i_{1} > … > i_{k}} the sequence of integers attached to c = (v_{𝔭}(u_{1}), …, v_{𝔭}(u_{r−1})) in the above sense. Then the following decomposition holds

( \begin{matrix}0 & t_{1} \\ & ... & ... \\ & & 0 & t_{r−1} \\ 1 & u_{1} & ⋯ & u_{r−1} \end{matrix} ) = δ^{∗} · ( \begin{matrix}\frac{1}{u_{i_{k}}} & v_{i_{k}} & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ . & . & . \\ & & \frac{u_{i_{k}}}{u_{i_{k−1}}} & v_{i_{k−1}} & ⋯ & ⋯ & ⋯ \\ . & . & . \\ & & & & \frac{u_{i_{2}}}{u_{i_{1}}} & v_{i_{1}} & ⋯ \\ . & . & . \\ & & & & & & u_{i_{1}} \end{matrix} ) · k,

where k ∈ GL_{r}(O_{𝔭}) and δ^{∗} is the diagonal matrix

δ^{∗} = diag((−1)^{k}t_{1}, t_{2}, …, t_{i_{h}−1}, (−1)^{h}t_{i_{h}}, t_{i_{h}+1}, …, t_{r−1}, 1),

with the entry (−1)^{h}t_{i_{h}} , h = 1, …, k appearing in the i_{h} -th position. Furthermore in the above matrix all entries, which are left out are equal to zero, the entries on the diagonal, which are indicated by dots, are equal to 1 , the entries in the i_{h} -th row, which are indicated by dots are arbitrary and

v_{i_{h}} = \{ \begin{matrix}\frac{u_{i_{h}+1}}{u_{i_{h}}} & if & i_{h} < i_{h−1} − 1 \\ 1 & if & i_{h} = i_{h−1} − 1 \end{matrix}

appears in the i_{h} -th column ( i_{0} = r ).

Proof. We first deal with the case v_{𝔭}(u_{i}) ≤ 0 for i = 1, …, r − 1 . Multiplying on the left with the diagonal matrix

diag(t_{1}^{−1}, …, t_{r−1}^{−1}, 1)

we may assume that d_{1} = … = d_{r−1} = 1 . We then proceed by induction on r ∈ ℕ . The case r = 1 being trivial we therefore may assume that the assertion of the Lemma is true for r −1×r −1 -matrices and we have to prove it in the case of r ×r -matrices. Our starting point is the following decomposition

(1.13)  ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ 1 & u_{1} & ⋯ & u_{r−1} \end{matrix} ) = M ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ \frac{1}{u_{i_{1}}} & ⋯ & \frac{u_{i_{1}−1}}{u_{i_{1}}} & 1 & \frac{u_{i_{1}+1}}{u_{i_{1}}} & ⋯ & \frac{u_{r}}{u_{i_{1}}} \\ & & & & 1 \\ . & . & . \\ & & & & & & 1 \end{matrix} ) ,

where

M = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ −\frac{1}{u_{i_{1}}} & ⋯ & ⋯ & −\frac{u_{i_{1}−1}}{u_{i_{1}}} & 1 & −\frac{u_{i_{1}+1}}{u_{i_{1}}} & ⋯ & −\frac{u_{r}}{u_{i_{1}}} \\ & & & & 0 & 1 \\ & & & & ⋮ & ... & ... \\ & & & & & & 0 & 1 \\ & & & & u_{i_{1}} & & & 0 \end{matrix} ) .

Since v_{𝔭}(u_{i_{1}}) ≤ v_{𝔭}(u_{j}) for 1 ≤ j ≤ r , we see that the second matrix on the right hand side of (1.13) is contained in GL_{r}(O_{𝔭}) . We distinguish cases and first assume that i_{h} < i_{h−1} − 1 . The first matrix on the right hand side further decomposes

M = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ −\frac{1}{u_{i_{1}}} & ⋯ & ⋯ & −\frac{u_{i_{1}−1}}{u_{i_{1}}} & −\frac{u_{i_{1}+1}}{u_{i_{1}}} & ⋯ & −\frac{u_{r}}{u_{i_{1}}} & 1 \\ & & & & 1 \\ . & . & . \\ & & & & & & 1 \\ & & & & & & & u_{i_{1}} \end{matrix} ) ( \begin{matrix}1_{i_{1}} \\ & 0 & ⋯ & & ⋯ & 1 \\ & 1 & 0 & & & ⋮ \\ & & 1 \\ & & & ... & ... & ⋮ \\ & & & & 1 & 0 \end{matrix} ) .

The permutation matrix is obviously contained in GL_{r}(O_{𝔭}) and the matrix appearing in the upper left corner of size i_{1} × i_{1} of the first matrix on the right hand side reads

(1.14)  ( 1 ; ... ; 1 ; \frac{−1}{u_{i_{1}}} ) ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ 1 & u_{1} & ⋯ & u_{i_{1}−1} \end{matrix} ) .

On the other hand the sequence of integers J ({u_{1}, …, u_{i_{1}−1}}) attached to {u_{1}, …, u_{i_{1}−1}} is just (i_{2}, …, i_{k}) . Hence, applying the induction hypothesis to the matrix (1.14) we obtain the desired decomposition. In case that i_{1} = i_{0} − 1 we see that M equals

( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ −\frac{1}{u_{r−1}} & ⋯ & ⋯ & −\frac{u_{r−2}}{u_{r−1}} & 1 \\ & & & & u_{r−1} \end{matrix} )

Again, applying the induction hypothesis to the upper left corner of size r − 1 × r − 1 we obtain the desired decomposition in the case of r × r -matrices. The general case now follows immediately: using column operations it is easy to see that there is a matrix n ∈ N_{r}(O_{𝔭}) such that

( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ 1 & u_{1} & ⋯ & u_{r−1} \end{matrix} ) = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & 0 & 1 \\ 1 & ũ_{1} & ⋯ & ũ_{r−1} \end{matrix} ) · n,

and ũ_{i} equals u_{i} if v_{𝔭}(u_{i}) ≤ 0 and equals 1 if v_{𝔭}(u_{i}) > 0 . Hence, we may apply our result to the first matrix on the right hand side and taking into account that by definition J (v_{𝔭}(u_{1}), …, v_{𝔭}(u_{r−1})) = J (v_{𝔭}(ũ_{1}), …, v_{𝔭}(ũ_{r−1})) we obtain the desired decomposition in the general case. Thus the lemma is proven.

We note that, by an easy calculation, condition (2.13) is seen to imply that the matrix k appearing in the Lemma is contained in K_{1}(r, 𝔭^{e}) .

Let J = {i_{1} = r − 1 > … > i_{k−1} > i_{k} = 1} be any subset of {1, …, r − 1} . We denote by J̌ the subset of J consisting of all i ∈ J such that i − 1 ∈ J . In particular J̌ does not contain i_{k} = 1 . Using the Lemma we then obtain the decomposition

t̃u = nδδ^{∗}k.

Here, δ is the diagonal matrix

δ = diag (\frac{1}{u_{i_{k}}} , 1, …, 1, \frac{u_{i_{k}}}{u_{i_{k−1}}} , 1, …, 1, \frac{u_{i_{2}}}{u_{i_{1}}} , 1, …, u_{i_{1}}) ,

where \frac{u_{i_{h+1}}}{u_{i_{h}}} appears in the i_{h} -th position for h ∈ J (c) and

n = ( \begin{matrix}1 & ⋆_{2} & & & ∗ \\ & ... & ... \\ & & 1 & ⋆_{r−1} \\ & & & 1 & 1/u_{i_{1}} \\ & & & & 1 \end{matrix} ) ,

where

⋆_{i} = \{ \begin{matrix}−\frac{t_{i_{h}−1}}{t_{i_{h}}} \frac{u_{i_{h}}}{u_{i_{h+1}}} & if i = i_{h} ∈ J̌(c) \\ (−1)^{i_{h}} \frac{t_{i_{h}}}{t_{i_{h}+1}} \frac{u_{i_{h}+1}}{u_{i_{h}}} & if i ∉ J (c), but i − 1 = i_{h} ∈ J (c) \\ 0 & else. \end{matrix}

We denote by n_{r−1}, δ_{r−1}, δ_{r−1}^{∗} the upper left corner of size r − 1 × r − 1 of n, δ, δ^{∗} . As mentioned above (1.12) implies that k is contained in K_{1}(r, 𝔭^{e}) and we obtain

f_{π̂[χ],𝔭}^{⋆}(t̃u) = f_{π̂[χ],𝔭}^{⋆}(k; n_{r−1}δ_{r−1}δ_{r−1}^{∗}, u_{i_{1}})
= Λ_{𝔭}(δδ^{∗}) τ_{𝔭}(n_{r−1})w_{π̂,𝔭}^{0}(δ_{r−1}δ_{r−1}^{∗}) χ_{𝔭}(u_{i_{1}}).

Hence we have to calculate

I(t, c) = |t|_{𝔭}^{−1} ∫_{T_{r−1}(O_{𝔭})} τ_{𝔭}(n_{r−1}) τ_{𝔭}(t_{1}^{−1}u_{1}) w_{π̂,𝔭}^{0}(δ_{r−1}δ_{r−1}^{∗}) χ_{𝔭}(u_{i_{1}})Λ_{𝔭}(δδ^{∗}) du_{1} … du_{r−1}.

Taking into account that \frac{t_{i_{h}}}{t_{i_{h}+1}} ∈ O_{𝔭} and \frac{u_{i_{h}+1}}{u_{i_{h}}} ∈ O_{𝔭} for i ∈ J (c) − J̌(c) we see that

τ_{𝔭}(n_{r−1}) = ∏_{i_{h}∈J̌(c)} τ_{𝔭} (−\frac{t_{i_{h}−1}}{t_{i_{h}}} \frac{u_{i_{h}}}{u_{i_{h+1}}} ) .

We denote by ϖ^{c} = diag(ϖ^{c_{1}}, …, ϖ^{c_{r−1}}) the diagonal matrix of length r − 1 . Since w_{π̂,𝔭}^{0} is GL_{r−1}(O_{𝔭}) -invariant on the right we have w_{π̂,𝔭}^{0}(δ_{r−1}) = w_{π̂,𝔭}^{0}(Δ(c)) , where

Δ(c) = diag(Δ_{1}, …, Δ_{r−1})

is the diagonal matrix of length r − 1 with entries Δ_{i} = ϖ^{c_{i_{h+1}}−c_{i_{h}}} if i = i_{h} ∈ J (c) and Δ_{i} = 1 if i ∉ J (c) (by the symbol ‘ c_{i_{k+1}} ’ we mean 0 ). Thus, writing u_{i} = ϖ^{c_{i}}ε_{i} , i = 1, …, r − 1 we obtain

I(t, c) = |t|_{𝔭}^{−1}χ_{𝔭}(ϖ^{c_{i_{1}}}) ∫_{T_{r−1}(O_{𝔭})} ∏_{i_{h}∈J̌(c)} τ_{𝔭} (−\frac{t_{i_{h}−1}}{t_{i_{h}}} Δ_{i_{h}} \frac{ε_{i_{h}}}{ε_{i_{h+1}}} ) τ_{𝔭}(t_{1}^{−1}ϖ^{c_{1}}ε_{1})

× w_{π̂,𝔭}^{0}(ϖ^{v}Δ(c)) χ_{𝔭}(ε_{i_{1}})Λ_{𝔭}(diag(ϖ^{v}Δ(c), c_{i_{1}})) dε_{1}, …, dε_{r−1}.

The change of integration variables

ε_{i_{h}} ↦ (−1)^{k−h} ∏_{r=h,... ,k} ε_{i_{r}}, h = k, …, 1

leaves T_{r−1}(O_{𝔭}) stable and yields

I(t, c) = |t|_{𝔭}^{−1} χ_{𝔭}(ϖ^{c_{r−1}}) w_{π̂,𝔭}^{0}(ϖ^{v}Δ(c)) χ_{𝔭}^{k−1}(−1) Λ_{𝔭}(diag(ϖ^{v}Δ(c), c_{i_{1}}))

× ∫_{T_{r−1}(O_{𝔭})} ∏_{i∈J̌(c)} τ_{𝔭} (\frac{t_{i−1}}{t_{i}} Δ_{i}ε_{i}) τ_{𝔭}(t_{1}^{−1}ϖ^{c_{1}}ε_{1}) ∏_{i∈J (c)} χ_{𝔭}(ε_{i}) dε_{1}, …, dε_{r−1}.

We put t_{0} := 1 . Recalling that i_{k} = 1 ∉ J̌(c) we see that the above integral decomposes into the product

∏_{i∈J̌(c)∪\{1\}} ∫_{O_{𝔭}^{∗}} τ_{𝔭} (\frac{t_{i−1}}{t_{i}} Δ_{i}ε_{i}) χ_{𝔭}(ε_{i}) dε_{i} × ∏_{i∈J (c)−(J̌(c)∪\{1\})} ∫_{O_{𝔭}^{∗}} χ_{𝔭}(ε_{i}) dε_{i}.

It follows immediately that I(t, c) vanishes unless J (c) = J̌(c) ∪ {1} . Since J contains r −1 , this equivalent to J = {1, …, r −1} , i.e. i_{h} = r −h, h = 1, …, r −1 and in this case we obtain

I(t, c) = |t|_{𝔭}^{s_{1}−1}|Δ(c)|_{𝔭}^{s_{1}}|c_{i_{1}}|_{𝔭}^{s_{2}} χ_{𝔭}((−1)^{r}ϖ^{c_{r−1}}) w_{π̂,𝔭}^{0}(ϖ^{v}Δ(c))

∏_{i=1,... ,r−1} ∫_{O_{𝔭}^{∗}} τ_{𝔭} (\frac{t_{i−1}ϖ^{c_{i}}}{t_{i}ϖ^{c_{i−1}}} ε_{i}) χ_{𝔭}(ε_{i}) dε_{i}.

On the other hand

∫_{O_{𝔭}^{∗}} τ_{𝔭}(ϖ^{−c}ε)χ_{𝔭}(ε) d^{∗}ε = \{ \begin{matrix}χ_{𝔭}(ϖ^{c}) G(χ_{𝔭}) & if c = e \\ 0 & else \end{matrix}

and we further deduce that I(t, c) does not vanish only if in addition

−v_{𝔭}(t_{1}) + c_{1} = e, v_{𝔭}(t_{i−1}) + c_{i} − v_{𝔭}(t_{i}) − c_{i−1} = e, i = 2, …, r − 1.

Hence, c = (ie + v_{𝔭}(t_{i}))_{i=1,... ,r−1} is the only vector, which contributes to v_{π̂[χ],Λ,𝔭} . In particular Δ(c) = ϖ^{−e+v_{i}−v_{i−1}} , i = 1, …, r − 1 and taking into account that

∏_{i=1}^{r−1} χ_{𝔭}^{−1} (\frac{t_{i−1}}{t_{i}} \frac{ϖ^{c_{i}}}{ϖ^{c_{i−1}}} ) = χ_{𝔭}(t_{r−1})χ_{𝔭}^{−1}(ϖ^{c_{r−1}})

we finally obtain:

Proposition 1.4. Let v_{π̂[χ],Λ,𝔭} be the Whittaker function in W (I_{P,Λ,𝔭}π̂[χ], τ_{𝔭}) attached to the essential vector f_{π̂[χ],Λ,𝔭}^{⋆} defined in (1.5) and put t = diag(1, t_{1}, …, t_{r−1}) . Then, v_{π̂[χ],Λ,𝔭}(t) vanishes unless condition (2.11) is fullfilled and in this case we have

v_{π̂[χ],Λ,𝔭}(t) = χ_{𝔭}((−1)^{r}) G(χ_{𝔭})^{r−1}χ_{𝔭}(t_{r−1}) |t|_{𝔭}^{s_{1}−1}|t_{r−1}|_{𝔭}^{s_{2}−s_{1}}|ϖ|_{𝔭}^{e(r−1)(s_{2}−s_{1})} w_{π̂,𝔭}^{0}(ϖ^{v+e}),

where

e = (e, 2e, …, (r − 1)e).

This formula now solves the problem of computing the values of the essential function on torus elements in that it expresses these values using Whittaker functions on groups in lower rank. In particular, applying it in the case e = 0 it should lead to a new proof of the Casselman-Shalika formula for GL_{n} .

We recall that w_{π̂,𝔭}^{0} is the spherical function satisfying w_{π̂,𝔭}^{0}(e) = 1 . In particular, [C-S], Theorem 5.4 implies that w_{π̂[χ],𝔭}^{0}(t) does not vanish if (1.10) holds. Taking into account that cℓ(F) = 1 we see that χ_{𝔭}(ϖ) = 1 and we finally deduce in the case that s_{1}, s_{2} ∈ ℤ

(1.15)  v_{π̂[χ],𝔭}(ϖ^{v}) = c G(χ_{𝔭})^{n−2}w_{π̂,𝔭}^{0}(ϖ^{v+e}).

Here, c ∈ ℚ^{∗} is a non-vanishing rational number independent of χ .

Collecting our results we finally obtain in the case that s_{1}, s_{2} ∈ ℤ

(1.16) Eτ̄ ; fin(f_{π̂[χ],Λ}, e × ϖ^{v} × g_{S′}^{0}) =

Eτ̄ ; S′−S_{∞}(f_{π̂[χ],Λ}, g_{v}^{0}) c G(χ_{𝔭})^{n−2}w_{π̂,𝔭}^{0}(ϖ^{v+e}) L(π̂[χ] ⊗ χ^{−1}, 1 + s_{1} − s_{2}),

where Eτ̄ ; S′−S_{∞}(f_{π̂[χ],Λ}) = ⊗_{v∈S′−S_{∞}}Eτ̄ ; v(f_{π̂[χ],Λ}) and c ∈ ℚ^{∗} do not vanish.

2 The Rankin-Selberg Convolution

In this section we introduce the global integral which we use to represent the automorphic L -function. Attached to the pair of representations

π × I_{P,Λ} π̂[χ]

there is a Rankin-Selberg convolution. We choose a special vector ϕ ∈ V_{π} , where V_{π} denotes the space of cusp forms on which π can be realized. We shall then evaluate the Zeta integral of the Rankin-Selberg convolution for the pair of automorphic forms (ϕ, E_{f_{πχ,Λ}}) . This integral decomposes into a product

(2.1)  Eτ̄ ; fin(f_{π̂[χ],Λ}, e × ϖ^{v} × g_{S′}^{0}) ∏_{v∉S_{∞}} I_{v}(ϕ_{v}^{τ}),

where I_{v}(ϕ_{v}^{τ}) is a certain local integral involving the Whittaker function of ϕ . Using the caclulations from the last section we obtain the denominator in (2.1). Using the local zeta integral we compute the I_{v}(ϕ_{v}^{τ}) ’s and obtain the nominator in (2.1).

We begin with the definition of ϕ ∈ V_{π} and to this end we choose a Whittaker function w = ⊗_{v}w_{v} ∈ W (π, τ) .

• For any finite place v not equal to 𝔭 and not contained in S′ we let w_{v} be the essential vector in W (π_{v}, τ_{v}) .

• If v is contained in S′ we let w_{v} be the (unique) function whose restriction to GL_{n−1}(F_{𝔭}) via the embedding g ↦ diag(g, 1) is given by

w_{v}|_{K_{n−1,v}^{∗}} = ch_{K_{n−1,v}^{∗}},

where ch_{K_{n−1,v}^{∗}} denotes the characteristic function of K_{n−1,v}^{∗} . The existence of such a Whittaker function is an immediate consequence of Theorem F in [Gel-Kaz].

We set w_{S′} = ∏_{v} w_{v}, v_{π̂[χ],Λ,S′} = ∏_{v} v_{π̂[χ],Λ,v} and g_{S′}^{0} = ∏_{v} g_{v}^{0} , where in each case the product runs over all finite places in S′ .

• At the place v = 𝔭 the representation π_{𝔭} is unramified and hence there are unramified characters α_{1}, …, α_{n} : F_{𝔭} → ℂ^{∗} such that

π_{𝔭} ≅ Ind_{B_{n}(F_{𝔭})}^{GL_{n}(F_{𝔭})}(α_{1}, …, α_{n}).

We abbreviate α(b) = ∏_{i=1}^{n} α_{i}(b_{ii}) for b = (b_{ij}) ∈ B_{n}(F_{𝔭}) . We denote by I(n) ≤ GL_{n}(O_{𝔭}) the Iwahori subgroup consisting of elements k ∈ K_{𝔭}(n) , which are congruent to upper triangular matrices modulo 𝔭 . Let ψ_{𝔭}^{1} ∈ Ind_{B_{n}(F_{𝔭})}^{GL_{n}(F_{𝔭})}(α) be the following I(n) -invariant vector

ψ_{𝔭}^{1}(g) := \{ \begin{matrix}αδ_{B_{n}}^{1/2}(b) & for g = bw_{n,l}β, b ∈ B(n, F_{𝔭}), β ∈ I(n) \\ 0 & else. \end{matrix}

We then define w_{𝔭} ∈ W (π_{𝔭}, τ_{𝔭}) to be the image of ψ_{𝔭}^{1} under the intertwining operator

Ind_{B_{n}(F_{𝔭})}^{GL_{n}(F_{𝔭})}(α_{1}, …, α_{n}) → W (π_{𝔭}, τ_{𝔭}),

which maps an element ψ to the function

w_{ψ}(g) := ∫_{N_{n}(F_{𝔭})} ψ(w_{n,l}ng)τ̄_{𝔭}(n) dn.

(cf. also [J-S 2], (3.2) Proposition).

• For any place v ∈ S_{∞} , we let w_{v} be an arbitrary but fixed Whittaker function in W (π_{v}, τ_{v}) .

We define ϕ ∈ V_{π} to be the cusp form attached to w .

We shall use the following conventions without further mentioning: we identify any matrix h ∈ GL_{n}(F_{𝔭}) with its image in GL_{n}(𝔸) given by embedding it into the 𝔭 - component of the adeles. In particular for any matrix h ∈ GL_{n}(F_{𝔭}) and function f on GL_{n}(𝔸) we denote h · f(g) := f(gh) the right translate of f by h . For m′ > m we identify GL_{m} with its image in GL_{m′} under the embedding

g ↦ (\begin{matrix}g \\ & 1_{m′−m} \end{matrix}) .

We denote by H ∈ GL_{n−1}(F_{𝔭}) the matrix

H = diag(ϖ, ϖ^{2}, …, ϖ^{n−1})

and we put u^{H} = HuH^{−1} . From now on we fix an idele class character

η : F^{∗}\𝔸^{∗} → ℂ^{∗}

of finite order and conductor f_{η} = 𝔭𝔪 , where 𝔪 is prime to 𝔭 . Using the Whittaker function w defined above, we then set

w_{η}(g) := ∑_{u} ∏_{t=1}^{n−2} η_{𝔭}^{t}(u_{t,t+1}) w (g (\begin{matrix}u^{H} \\ & 1 \end{matrix})) ,

where the matrix u = (u_{ij}) runs over all elements of a set of representatives of N_{n−1}(F_{v}) modulo N_{n−1}(F_{v})^{H^{−1}} , which satisfy u_{t,t+1} ∈ (O_{𝔭}/ϖO_{𝔭})^{∗} . We let ϕ_{η} be the cusp form belonging to w_{η} .

We denote by H_{x} := diag(1, …, 1, x) , x ∈ F_{𝔭}^{∗} the diagonal matrix of length n − 1 and we set X_{n−1}^{𝔸} := GL_{n−1}(F)\GL_{n−1}(𝔸) . The integral, which we want to calculate, then reads

J_{π̂[χ],Λ,η}(s) = ∑_{v} χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) ∫_{X_{n−1}^{𝔸}} ϕ_{η} (gv) η ⊗ E(f_{π̂[χ],Λ}, gH_{ϖ^{a−e}}) | det g|^{s−1/2} dg.

In the above expression a is a natural number and v = (v_{1}, …, v_{n−1}) runs over O_{𝔭}/𝔭^{a}×…×O_{𝔭}/𝔭^{a}×(O_{𝔭}/𝔭^{e})^{∗} . Also, we identify v with its image under the embedding

\begin{matrix}F_{𝔭}^{n−1} & → & N_{n}(F_{𝔭}) \\ v = (v_{1}, …, v_{n−1}) & ↦ & ( \begin{matrix}1 & & & & v_{1}/ϖ^{a} \\ & ... & & & ⋮ \\ & & 1 & & v_{n−2}/ϖ^{a} \\ & & & 1 & v_{n−1}/ϖ^{e} \\ & & & & 1 \end{matrix} ) . \end{matrix}

Here, the vector (ϖ^{a}, …, ϖ^{a}, ϖ^{e}) is understood by the range of summation for v . For brevity, we usually write J_{π̂[χ],Λ} instead of J_{π̂[χ],Λ,η}(s) .

Our starting point for the computation of this integral is its decomposition into a product of local integrals, which is valid for Re(s) >> 0 and in our case takes the form

J_{π̂[χ],Λ} = ∏_{v≠𝔭}I(w_{η,v}, η_{v}⊗v_{π̂[χ],Λ,v}, s) ∑_{v_{1},... ,v_{n−1}} χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) I(H_{ϖ^{e−a}}v·w_{η,𝔭}, η_{𝔭}⊗v_{π̂[χ],Λ,𝔭}, s).

In the above equation

I(w_{v}, v_{v}, s) = ∫_{N_{n−1}(F_{v})\GL_{n−1}(F_{v})} w_{v} (\begin{matrix}g \\ & 1 \end{matrix}) v_{v}(g)| det g|_{v}^{s−1/2} dg

is the local zeta integral of the Rankin-Selberg convolution. This reduces the computation of the global integral to the calculation of the local ones appearing in the above decomposition.

• At all places v prime to 𝔭 and S′ , the Whittaker function w_{η,v} = w_{v} is the essential vector and η_{v} ⊗ v_{π̂[χ],Λ,v} is by definition K_{n−1,v} -invariant. As an immediate consequence of (4.1) Théorème in [J-P-S 2] we obtain

I(w_{η,v}, η_{v} ⊗ v_{π̂[χ],Λ,v}, s) = v_{π̂[χ],Λ,v}(e) L(π_{v} × η_{v} ⊗ π̂_{v}, s + s_{1}) L(π_{v} ⊗ χ_{v}η_{v}, s + s_{2}).

• Next, we look to the places v ∈ S′ . From now on we shall assume that for all v ∈ S′ the v -component f_{η,v} of the conductor of η is sufficiently large so that

(2.2.C)  L(π_{v} × η_{v} ⊗ χ_{v}, s) = 1.

On the other hand the definition of v_{π̂[χ],Λ,v} and w_{η,v} = w_{v} immediately yields that

I(w_{v}, η_{v} ⊗ v_{χ,Λ,v}, s) = (vol(K_{v}^{∗}))^{−1} w_{v}v_{π̂[χ],Λ,v}(g_{v}^{0}).

• At all archimedian places v we deduce from [J-P-S 3] that there is a polynomial P = P_{w_{v},v_{π̂[χ],λ,v}}^{η} ∈ ℂ[T] such that

I(w_{v}, η_{v} ⊗ v_{π̂[χ],Λ,v}, s) = P (s)L(π_{v} × η_{v} ⊗ I_{P,Λ,v}π̂[χ], s)
= P (s) ∏_{i=1}^{t−1} L(π_{v} × η_{v} ⊗ π̂_{v}, s)L(π_{v} ⊗ η_{v}χ_{v}, s).

We set P_{w_{∞},v_{π̂[χ],Λ,∞}}^{η} = ∏_{v∈S_{∞}} P_{w_{v},v_{π̂[χ],Λ,v}}^{η} .

• At the place 𝔭 we assume that f_{χ_{𝔭}} = 𝔭^{e} with e ≥ 2 . The character χ_{𝔭}η_{𝔭} is then ramified and since π_{𝔭} is unramified, we see that

L(π_{𝔭} ⊗ χ_{𝔭}η_{𝔭}, s) = 1.

Collecting our results so far we arrive at the equation

J_{π̂[χ],Λ} = P (s)v_{π̂[χ],Λ,S′}(g_{S′}^{0})v_{π̂[χ],Λ}^{S′∪{𝔭}}(e) L_{S′∪{𝔭}}(π × η ⊗ π̂, s + s_{1})L(π ⊗ ηχ, s + s_{2}) J_{π̂[χ],Λ,𝔭},

where

J_{π̂[χ],Λ,𝔭} = ∑_{v} χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) I(H_{ϖ^{e−a}}v · w_{η,𝔭}, v_{π̂[χ],𝔭}, s)

and

v_{π̂[χ],Λ}^{S′∪{𝔭}} = ⊗_{v∉S′∪{𝔭}} v_{π̂[χ],Λ,v}.

On the other hand since 𝔭 ∉ S ∪ S′ we have L(π_{𝔭} × η_{𝔭} ⊗ π̂_{𝔭}, s) = 1 . Again, twisting π̂ with some character we even may assume

(2.3.C)  L(π_{v} × η_{v} ⊗ π̂_{v}, s) = 1

for all v ∈ S′ and replacing Eτ̄ ; v(f_{π̂[χ],Λ}, e) by the expression in Eq. (1.9) we then obtain

(2.4) J_{π̂[χ],Λ} = P (s)v_{π̂[χ],Λ,S}(g_{S′}^{0}) L(π × η ⊗ π̂, s + s_{1})\frac{L(π ⊗ ηχ, s + s_{2})}{L(π̂ ⊗ χ^{−1}, 1 + s_{1} − s_{2})} J_{π̂[χ],Λ,𝔭},

where P = P_{w_{∞},v_{π̂[χ],Λ,∞}} . Hence, to complete the calculation of the integral it remains to compute J_{π̂[χ],Λ,𝔭} on the p -adic group GL_{n−1}(F_{𝔭}) . This will be done in the next subsection.

For later purpose we will assume that (1.8.C) holds for all v ∈ S∪S′ . Thus, concerning η we assume that

(C.1)  f_{η,𝔭} = 𝔭 and L(π_{v} × η_{v}χ_{v}, s) = 1 for v ∈ S′

and concerning π̂ we assume that

(C.2) L(π_{v} ×η_{v} ⊗π̂_{v}, s) = 1 for v ∈ S ∪S′ and L(π̂_{v} ⊗χ_{v}^{−1}, s) = 1 for v ∈ S′

The assumptions can always be achieved by enlargening the conductor of η and or by twisting π̂ with a ramified character. Since π and π̂ are unramified at 𝔭 (C.1) and (C.2) also hold for v = 𝔭 as soon as f_{χ} = 𝔭^{e} with e ≥ 2 . Finally, we note that the conditions on η only depend on π .

2.1 Analysis of the local integral J_{π̂[χ],Λ,𝔭} .

We reduce the calculation of the 𝔭 -component J_{π̂[χ],Λ,𝔭} to the computation of the values of Whittaker functions on elements of the standard torus in GL_{n}(F_{𝔭}) . The idea is to inductively compute J_{π̂[χ],Λ,𝔭} by applying certain operators to it, whose effect we determine explicitely.

Let μ : F_{𝔭}^{∗} → ℂ^{∗} be a character with conductor f_{μ} = 𝔭^{e} with e ≥ 1 and fix an integer m ∈ ℕ, m < n − 1 . For any vector a = (a_{1}, …, a_{m−1}) ∈ ℕ^{m−1} we define the operator

R_{a,μ} : W (π_{𝔭}, τ_{𝔭}) → W (π_{𝔭}, τ_{𝔭})

by setting

R_{a,μ}(w)(g) = ∑_{v} μ(v_{m})w(gv),

where v = (v_{1}, …, v_{m−1}) runs over a system of representatives of O_{𝔭}/ϖ^{a_{1}}O_{𝔭} × … × O_{𝔭}/ϖ^{a_{m−1}}O_{𝔭} × (O_{𝔭}/ϖ^{e}O_{𝔭})^{∗} and we identify v in the argument of w with its image under

(2.5)  \begin{matrix}F_{𝔭}^{m−1} & → & N_{m}(F_{𝔭}) \\ v = (v_{1}, …, v_{m−1}) & ↦ & ( \begin{matrix}1 & & & & v_{1}/ϖ^{a_{1}} \\ & ... & & & ⋮ \\ & & 1 & & v_{m−1}/ϖ^{a_{m−1}} \\ & & & 1 & v_{m}/ϖ^{e} \\ & & & & 1 \end{matrix} ) . \end{matrix}

Again, the vector (ϖ^{a_{1}}, …, ϖ^{e}) is understood by the range of summation for v .

Using the characters η and χ introduced in section 1, we define for m ∈ ℕ

R_{m,η_{𝔭}} = R_{(m,... ,3,2),η_{𝔭}^{m}}.

In the case m = 1 we understand that R_{1,η_{𝔭}} = R_{η_{𝔭}} . For arbitrary w ∈ W (π_{𝔭}, τ_{𝔭}) we then put

w_{η_{𝔭},m} = R_{m,η_{𝔭}} ∘ … ∘ R_{1,η_{𝔭}} w

and

w_{χ_{𝔭}} = R_{(a,... ,a),χ_{𝔭}η_{𝔭}^{n−1}} w_{η_{𝔭},n−2},

where the vector (a, …, a) has length n − 2 . Since

( \begin{matrix}1 & & & u_{1,n−1}/ϖ^{n−2} \\ & ... & & ⋮ \\ & & 1 & u_{n−2,n−1}/ϖ \\ & & & 1 \end{matrix} ) · . . . · \binom{1 u_{1,2}/ϖ}{1} = ( \begin{matrix}1 & u_{1,2} & & u_{1,n−1} \\ & ... & ... & ⋮ \\ & & 1 & u_{n−2,n−1} \\ & & & 1 \end{matrix} )^{H}

we see, that if we specialize w to be the 𝔭 -component of the Whittaker function, which was defined in section 1, we obtain

(2.6)  w_{η_{𝔭},n−2} = w_{η,𝔭}.

Remark 2.1. We note that if w is invariant on the right under I(n) , then the definition of w_{η_{𝔭},m} and w_{χ_{𝔭}} does not depend on the choice of the systems of representatives of O_{𝔭}/ϖ^{a_{i}}O_{𝔭} , which we use in the definition of the operators R_{μ,a} .

Using the remark we obtain

(2.7)  w_{χ_{𝔭}}(g) = ∑_{v} χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) w_{η,𝔭}(gv),

where the vector (v_{1}, …, v_{n−1}) runs over O_{𝔭}/𝔭^{a} × ⋯ × O_{𝔭}/𝔭^{a} × (O_{𝔭}/𝔭^{e})^{∗} . Thus, we find

J_{π̂[χ],Λ,𝔭} = ∫_{N_{n−1}(F_{v})\GL_{n−1}(F_{v})} w_{χ_{𝔭}}(g) v_{π̂[χ],𝔭}(gH_{ϖ^{a−e}}) dg.
= ∫_{N_{n−1}(F_{v})\GL_{n−1}(F_{v})} R_{(a,... ,a),χ_{𝔭}η_{𝔭}^{n−1}} ∘ ⋯ ∘ R_{1,η_{𝔭}}w_{𝔭}(g) v_{π̂[χ],𝔭}(gH_{ϖ^{a−e}}) dg

and it remains to determine the effect of R_{a,μ} on Whittaker functions.

We denote by Ǔ_{a}(m) the subgroup of GL_{m}(F_{𝔭}) consisting of all matrices of the form

(2.8)  k = ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ y_{1} & ⋯ & y_{m−1} & y_{m} \end{matrix} )

satisfying y_{i} ∈ ϖ^{a_{i}}O_{𝔭}, i = 1, …, m − 1 and y_{m} ∈ O_{𝔭}^{∗} . Lemma 2.2. Let w ∈ W (π_{𝔭}, τ_{𝔭}) and assume that there is a character μ′ : F_{𝔭}^{∗} → ℂ^{∗} , such that for any k ∈ Ǔ_{a}(m) as above we have

w(gk) = w(g) μ′(y_{m}).

Then, the support of res_{GL_{m}} R_{a,μ}(w) is contained in

N_{m}(F_{𝔭})GL_{m−1}(F_{𝔭})Ǔ_{a}(m),

and for any g = ng′k , n ∈ N_{m}(F_{𝔭}), g′ ∈ GL_{m−1}(F_{𝔭}) and k ∈ Ǔ_{a}(m) we find

R_{a,μ}(w)(g) = τ_{𝔭}(n) ∏_{i=1}^{m−1} q^{a_{i}}G(μ)μ′μ^{−1}(y_{m}) (w)(g′),

where G(μ) = ∑_{u∈(O_{𝔭}/ϖ^{e}O_{w}p)^{∗}} μ(u)τ_{𝔭}(u/ϖ^{e}) denotes the Gaussian sum. Proof. Let g = (g_{ij}) ∈ GL_{m}(F_{𝔭}) . Using the commutation rule

(\begin{matrix}g \\ & 1 \end{matrix}) v = v′ (\begin{matrix}g \\ & 1 \end{matrix}) ,

where

v′ = (∗, …, ∗, \frac{g_{m,m}v_{m}}{ϖ^{e}} + ∑_{i=1}^{m−1} \frac{g_{m,i}v_{i}}{ϖ^{a_{i}}} )

is the matrix as in (2.5) we find

R_{a,μ} w(g) = ∑_{v_{m}} μ(v_{m})τ_{𝔭} (\frac{g_{m,m}v_{m}}{ϖ^{e}} ) · ∏_{i=1}^{m−1} ∑_{v_{i}} τ_{𝔭} (\frac{g_{m,i}v_{i}}{ϖ^{a_{i}}} ) w(g).

Using

∑_{v_{m}∈(O_{𝔭}/ϖ^{e}O_{w}p)^{∗}} μ(v_{m})τ_{𝔭} (\frac{g_{m,m}v_{m}}{ϖ^{e}} ) = \{ \begin{matrix}μ^{−1}(g_{m,m})G(μ) & g_{m,m} ∈ O_{𝔭}^{∗} \\ 0 & g_{m,m} ∉ O_{𝔭}^{∗} \end{matrix}

as well as the character relation

∑_{v_{i}∈O_{𝔭}/ϖ^{a_{i}}O_{𝔭}} τ_{𝔭} (\frac{g_{m,i}v_{i}}{ϖ^{a_{i}}} ) = \{ \begin{matrix}q^{a_{i}} & g_{m,i} ∈ ϖ^{a_{i}}O_{𝔭} \\ 0 & g_{m,i} ∉ ϖ^{a_{i}}O_{𝔭}, \end{matrix}

we see, that R_{a,μ} w(g) ≠ 0 implies, that g_{m,i} ∈ ϖ^{a_{i}}O_{𝔭} for i = 1, …, m − 1 and g_{m,m} ∈ O_{𝔭}^{∗} . Together with the decomposition

g = ( \begin{matrix}1 & & & \frac{g_{1,m}}{g_{m,m}} \\ & ... & & ⋮ \\ & & 1 & \frac{g_{m−1,m}}{g_{m,m}} \\ & & & 1 \end{matrix} ) (\begin{matrix}g′ \\ & 1 \end{matrix}) ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ g_{m,1} & ⋯ & g_{m,m−1} & g_{m,m} \end{matrix} ) ,

where g′ = (g′_{ij}) ∈ GL_{m−1}(F_{𝔭}) with entries g′_{ij} = g_{ij} − \frac{g_{i,m}}{g_{m,m}} g_{m,j} this proves that g is contained in N_{m}(F_{𝔭})GL_{m−1}(F_{𝔭})Ǔ_{a}(m) . To prove the second part of the claim let g = (g_{ij}) ∈ GL_{m}(F_{𝔭}) and assume that

g = n diag(g′, 1) k,

where n ∈ N_{m}(F_{𝔭}), g′ ∈ GL_{m−1}(F_{𝔭}) and k ∈ Ǔ_{a}(m) as in the lemma. Then g_{m,i} = y_{i}, i = 1, …, m and we obtain

R_{a,μ} w(g) = μ^{−1}(y_{m})G(μ) ∏_{i=1}^{m−1} q^{a_{i}} w(ng′_{k}).

Thus the assumption on the behaviour of w on the right implies the claim, which proves the lemma.

Remark 2.3. In the case m = 1 we use the conventions N_{1}(F_{𝔭}) = {1}, GL_{0} = {1}, Ǔ_{0}(1) = O_{𝔭}^{∗} and res_{GL_{0}} w = w(1_{n}) . In particular, the assertions then read

supp res_{GL_{1}} R_{μ}w ⊂ O_{𝔭}^{∗}

and for ε ∈ O_{𝔭}^{∗}

R_{μ}w(ε) = G(μ)μ′μ^{−1}(ε)w(1_{n}).

Taking into account the recursive definition of the Whittaker functions w_{η_{𝔭},m} and w_{χ_{𝔭}} Lemma 2.2 makes it possible to completely determine w_{η_{𝔭},m} and w_{χ_{𝔭}} . In a first step we compute w_{η_{𝔭},m} ; we prove:

Let w ∈ W (π_{𝔭}, τ_{𝔭}) and assume that w is invariant on the right under I(m) . Then the support of res_{GL_{m}} w_{η_{𝔭},m} is contained in

(2.9)  N_{m}(F_{𝔭})Ǔ_{∗}(m, 𝔭),

where Ǔ_{∗}(m, 𝔭) ≤ I(m) consists of all matrices

k = ( \begin{matrix}k_{1,1} & & 0 \\ ⋮ & ... \\ k_{m,1} & ⋯ & k_{m,m} \end{matrix} )

satisfying k_{ij} ∈ 𝔭^{i−j+1} for i > j and for any such k = (k_{ij}) ∈ Ǔ_{∗}(m, 𝔭) we have

(2.10)  w_{η_{𝔭},m}(k) = η_{𝔭}^{−1}(det k) ∏_{i=1}^{m} q^{i(i+1)/2−1}G(η_{𝔭}^{i}) w(1_{n}).

We will simultaneously prove both parts of the claim using induction on m . The case m = 1 being trivial we immediately assume that the lemma holds for m − 1 . We have w_{η_{𝔭},m} = R_{(m,... ,2),η_{𝔭}^{m}} w_{η_{𝔭},m−1} and Lemma 2.2 yields

supp res_{GL_{m}} w_{η_{𝔭},m} ⊂ N_{m}(F_{𝔭})GL_{n−1}(F_{𝔭})Ū_{(m,... ,2)}(m).

We want to apply Lemma 2.2 to calculate the values of w_{η_{𝔭},m} on this set. Therefore we need to know the behaviour of w_{η_{𝔭},m−1} on the right under Ū_{(m,... ,2)}(m) .

Lemma 2.4. Let w ∈ W (π_{𝔭}, τ_{𝔭}) and assume that w is invariant on the right under I(m) . Then, for any k ∈ Ū_{(m,... ,2)}(m) as in (2.4) we have

w_{η_{𝔭},m−1}(gk) = η_{𝔭}^{m−1}(y_{m}) w_{η_{𝔭},m−1}(g).

We postpone the proof of this lemma and continue the proof of (2.9) and (2.10). Let g = g′_{k} , where g′ ∈ GL_{m−1}(F_{𝔭}) and k ∈ Ū_{(m,... ,2)}(m) . Then the above lemma together with Lemma 2.2 yields

(2.11)  w_{η_{𝔭},m}(g) = ∏_{i=1}^{m−1} q^{i+1}G(η_{𝔭}^{m})η_{𝔭}^{−1}(y_{m}) w_{η_{𝔭},m−1}(g′).

But

Ū_{∗}(m − 1, 𝔭) · Ū_{(m,... ,2)}(m) = Ū_{∗}(m, 𝔭)

and we may therefore apply the induction hypothesis on supp res_{GL_{m−1}}w_{η_{𝔭},m−1} to obtain

supp res_{GL_{m}} w_{η_{𝔭},m} ⊂ N_{m}(F_{𝔭})Ū_{∗}(m, 𝔭).

To prove Eq. (2.10) we note that any k = (k_{ij}) ∈ Ū_{∗}(m, 𝔭) decomposes as

k = (\begin{matrix}k′ \\ & 1 \end{matrix}) ( \begin{matrix}1_{m−1} & 0 \\ k_{m,1} ⋯ & k_{m,m} \end{matrix} ) ,

where k′ = (k′_{ij}) ∈ GL_{m−1}(F_{𝔭}) and k′_{ij} = k_{ij} . Hence, (2.11) becomes

w_{η_{𝔭},m}(k) = q^{m(m+1)/2−1}G(η_{𝔭}^{m})η_{𝔭}^{−1}(y_{m}) w_{η_{𝔭},m−1}(k),

which together with the induction hypothesis on the values of res_{GL_{m−1}}w_{η_{𝔭},m−1} proves Eq. (2.10)(note that η_{𝔭}^{−1}(y_{m})η_{𝔭}^{−1}(det k′) = η_{𝔭}^{−1}(det k) ).

Proof of Lemma 2.4. We say that the matrix A ∈ GL_{m}(F_{𝔭}) is of type (t_{m}) , if the entries satisfy

a_{ij} ∈ ϖ^{−(j−i)}O_{𝔭} if j > i,
a_{ij} ∈ ϖ^{i−j+1}O_{𝔭} if j < i,
a_{i,i} ∈ O_{𝔭}^{∗}.

(2.12) Claim. Let A ∈ GL_{m}(F_{𝔭}) be of type (t_{m}) . Then there is a matrix β_{A} ∈ N̄(m, 𝔭) such that

A = ( \begin{matrix}ε_{1} & ⋯ & u_{ij} \\ & ... & ⋮ \\ & & ε_{m} \end{matrix} ) β_{A}

and ε_{i} ≡ a_{ii} (mod 𝔭), i = 1, …, m and u_{ij} ≡ a_{ij} (mod ϖ^{i−j+1})O_{𝔭}, j > i . Moreover if A runs over a set of matrices of type (t_{m}) such that the entries a_{ij}, j > i run over a system of representatives of ϖ^{−(j−i)}O_{𝔭}/O_{𝔭} and a_{i,i+1} run over ϖ^{−1}O_{𝔭}/O_{𝔭} − O_{𝔭} , then the same is true for the entries u_{ij} .

Using the claim we will prove the lemma. Let k ∈ Ū_{(m,... ,2)}(m) as in (2.8). Then we have

k ( \begin{matrix}1 & ⋯ & u_{ij}/ϖ^{j−i} \\ & ... & ⋮ \\ & & 1 \end{matrix} ) = ( \begin{matrix}1 & ⋯ & u_{ij}/ϖ^{j−i} \\ & ... & ⋮ \\ y′_{1} & ⋯ & y′_{m} \end{matrix} ) ,

where y′_{i} = y_{i} +∑_{h=1}^{i−1} y_{h} \frac{u_{ih}}{ϖ^{h−i}} . The assumption k ∈ Ū_{(m,... ,2)}(m) then implies that the above matrix is of type (t_{m}) . Therefore (2.6) together with the (2.12) yields

w_{η_{𝔭},m−1} (g (\begin{matrix}k \\ & 1_{n−m} \end{matrix})) = ∑_{u_{ij}} ∏_{t=1}^{m−1} η_{𝔭}^{t}(u_{t,t+1}) w (g ( \begin{matrix}ε_{1} & ⋯ & u_{ij}/ϖ^{j−i} \\ & ... & ⋮ \\ & & ε_{m} \\ & & & 1_{n−m} \end{matrix} )) ,

where the summation variables u_{ij} runs over u_{ij} ∈ O_{𝔭}/ϖ^{j−i}O_{𝔭} for j > i + 1 and u_{i,i+1} ∈ (O_{𝔭}/𝔭)^{∗} (note that w is invariant under I(m) ). Taking into account the right invarinace of w under diag(ε_{1}, …, ε_{m}) as well as ε_{t} ≡ 1 (mod 𝔭) for t < m and ε ≡ y′_{m} ≡ y_{m} (mod 𝔭) and changing the summation variables u_{ij} ↦ ε_{j}u_{ij} we find

w_{η_{𝔭},m−1} (g (\begin{matrix}k \\ & 1_{n−m} \end{matrix})) = η_{𝔭}^{m−1}(y_{m}) ∑_{u_{ij}} ∏_{t=1}^{m−1} η_{𝔭}^{t}(u_{t,t+1}) w (g ( \begin{matrix}1 & ⋯ & u_{ij}/ϖ^{j−i} \\ & ... & ⋮ \\ & & 1 \end{matrix} )) .

This finishes the proof of the lemma.

Proof of (2.12) Claim. We use induction on m . The case m = 2 as well as the induction step m − 1 → m are direct consequences of the following decomposition: any matrix A ∈ GL_{m}(F_{𝔭}) decomposes as

A = ( A′ a_{1,m} ; ⋮ ; 0 ⋯ 0 a_{m,m} ) ( \begin{matrix}1 \\ . & . & . \\ \frac{a_{m,1}}{a_{m,m}} & ⋯ & \frac{a_{m,1}}{a_{m,m}} & 1 \end{matrix} ) ,

where the matrix A′ = (a′_{ij}) ∈ GL_{m−1}(F_{𝔭}) has entries a′_{ij} = a_{ij} − a_{i,m} \frac{a_{m,j}}{a_{m,m}} and if A is of type (t_{m}) then A′ is of type (t_{m−1}) .

We call an integer a sufficiently large, if it satisfies the conditions

(2.13.C)  a > n and w_{η_{𝔭},n−2} is invariant under K(n, 𝔭^{a}).

Applying once more Lemma 2.2 and reasoning as in the proof of (2.9) and (2.10) we find:

Proposition 2.5. For a ∈ ℕ sufficiently large, we find

supp res_{GL_{n−1}}w_{χ_{𝔭}} ⊂ N(n − 1, F_{𝔭})Ū_{∗0}(n − 1, 𝔭^{a}),

where Ū_{∗0}(n − 1, 𝔭^{a}) ≤ I(n − 1) is the subgroup of matrices

k = \binom{k′ 0}{y_{1} ⋯ y_{n−1}} ,

satisfying k′ ∈ Ū_{∗}(n − 2, 𝔭) , y_{1}, …, y_{n−2} ∈ 𝔭^{a} and y_{n−1} ∈ O_{𝔭}^{∗} . Moreover, for k ∈ Ū_{∗0}(n − 1, 𝔭^{a}) as above we have

w_{χ_{𝔭}}(k) = χ_{𝔭}^{−1}(y_{n−1})η_{𝔭}^{−1}(det k) ∏_{i=1}^{n−2} q^{i(i+1)/2−1}G(η_{𝔭}^{i}) q^{(n−2)a}G(χ_{𝔭}η_{𝔭}^{n−1}) · w(1_{n}).

Using the Proposition we may now evaluate J_{π̂[χ],Λ,𝔭} for integers a , which are sufficiently large: since

H_{ϖ^{e−a}} Ū_{∗0}(n − 1, 𝔭^{a})H_{ϖ^{a−e}} ⊂ K_{0}(n − 1, 𝔭^{e})

we deduce that

v_{π̂[χ],𝔭}(gkH_{ϖ^{a−e}}) = v_{π̂[χ],𝔭}(gH_{ϖ^{a−e}})χ_{𝔭}(k_{n−1,n−1})η_{𝔭}(det k)

for k = (k_{ij}) ∈ Ū_{∗0}(n − 1, 𝔭^{a}) . Hence, using Proposition 2.5 we obtain (2.14)

J_{π̂[χ],Λ,𝔭} = ∫_{Ū_{∗0}(n−1,𝔭^{a})} dg ∏_{i=1}^{n−2} q^{i(i+1)/2−1}G(η_{𝔭}^{i})q^{(n−2)a} G(χ_{𝔭}η_{𝔭}^{n−1}) w_{𝔭}(e)v_{π̂[χ],𝔭}(H_{ϖ^{a−e}}).

In order to complete the computation of J_{π̂[χ],Λ,𝔭} we are left with determining the restriction of the Whittaker functions to the standard torus in GL_{n} .

2.2. Calculation of the values of w_{𝔭} .

Lemma 2.6. Let α^{w} denote the conjugate of the character α under the longest element w = w_{n,l} of the Weyl group of GL_{n} . For t = diag(t_{1}, …, t_{n}) ∈ T_{n}(F_{𝔭}) we have

w_{𝔭}(t) = \{ \begin{matrix}α^{w}δ_{B_{n}}^{1/2}(t) & if & v_{𝔭}(t_{1}) ≥ … ≥ v_{𝔭}(t_{n}) \\ 0 & & else. \end{matrix}

In particular, w_{𝔭}(e) = 1 .

Proof. We have to compute the integral

(2.15)  w_{𝔭}(t) = ∫_{N_{n}(F_{𝔭})} ψ_{𝔭}^{1}(w_{n,l}n t) τ̄_{𝔭}(n) dn

and we start by determining the set of matrices n = (n_{ij}) ∈ N_{n}(F_{𝔭}) such that w_{n,l}nt ∈ supp(ψ_{𝔭}^{1}) . Taking into account the desription of the support of ψ_{𝔭}^{1} (see section 1) this is equivalent to

w_{n,l}nt = b^{−1}w_{n,l}β

for some b = (b_{ij}) ∈ B(n, F_{𝔭}) and β = (β_{ij}) ∈ I(n) . This may be rewritten as

( \begin{matrix}t_{n}b_{1,1} & ⋯ & t_{n−j+1}b_{ij} \\ & ... & ⋮ \\ & & t_{1}b_{nn} \end{matrix} ) ( \begin{matrix}& & 1 \\ & 1 & ⋮ \\ 1 & ⋯ & \frac{t_{j}}{t_{n−i+1}} n_{(n−i+1),j} \end{matrix} ) = ( \begin{matrix}β_{n,1} & ⋯ & β_{n,n} \\ ⋮ & & ⋮ \\ β_{1,1} & ⋯ & β_{1,n} \end{matrix} )

and an easy induction on i = n, …, 1 proves that this is equivalent to b_{ij} ∈ t_{n−j+1}^{−1}ϖO_{𝔭} for j > i , v_{𝔭}(b_{ii}) = v_{𝔭}(t_{n−i+1}^{−1}) for i = 1, …, n and n_{ij} ∈ \frac{t_{i}}{t_{j}} O_{𝔭} . Consequently, the support of the integral (2.15) is contained in

{(n_{ij}) ∈ N(n, F_{𝔭}) : n_{ij} ∈ t_{i}/t_{j}O_{𝔭}} = tN_{n}(O_{𝔭})t^{−1}

and since α is unramified the integral reads

w_{𝔭}(t) = ∫_{tN_{n}(O_{𝔭})t^{−1}} αδ_{B_{n}}^{1/2}(w_{n,l}tw_{n,l}) ∑_{i=1}^{n−1} τ̄_{𝔭}(n_{i,i+1}) ∏_{j>i} dn_{ij}.

Using character relations for τ_{𝔭} we see that the integral vanishes unless v_{𝔭}(t_{i}) ≥ v_{𝔭}(t_{j}) for i < j . On the other hand if this is true then τ_{𝔭}(n_{i,i+1}) = 1 and we obtain

w_{𝔭}(t) = vol(tN_{n}(O_{𝔭})t^{−1}) αδ_{B_{n}}^{1/2}(w_{n,l}tw_{n,l}).

The volume reads vol(tN_{n}(O_{𝔭})t^{−1}) = δ_{B_{n}}(t) . Together with the identity

δ_{B_{n}}(w_{n,l}tw_{n,l}) = δ_{B_{n}}^{−1}(t)

this completes the proof of the proposition.

2.3. The final formula.

Collecting all our results, we finally obtain

Theorem 2.7. Assume that f_{χ} = 𝔭^{e} with e ≥ 2 and χ_{∞} = id . Then we have

\frac{L(π ⊗ χη, s + s_{2})}{L(π̂ ⊗ χ^{−1}, 1 + s_{2} − s_{1})} = ♣ P (s)^{−1} L(π × η ⊗ π̂, s + s_{1})^{−1} J_{π̂[χ],Λ,η}(s),

where

♣ = c G(χ_{𝔭}η_{𝔭}^{n−1})^{−1}G(χ_{𝔭})^{2−n}|f_{χ}|_{𝔭}^{−1}v_{π̂[χ],S}(g_{S}^{0})^{−1}w_{π̂,𝔭}^{0}(ϖ^{v′})^{−1}.

with c ∈ ℚ^{∗} . If we want to indicate the dependence on π̂[χ] we write ♣_{π̂,χ}

Remark 2.8. The formula in Theorem 2.7 computes the values L(π ⊗ χη, s) of the automorphic L -function on GL_{n} using certain period integrals attached to automorphic forms on GL_{n} as well as values of L -functions on groups GL_{m} of lower rank. Hence, the complete calculation will be by induction on the rank of the group.

In the next section we shall evaluate the period integral J_{π̂[χ],Λ}(s) for s = 1/2 in a different way.

3 The period Integral

In this section we compute the Fourier transform of the twisted values L(π ⊗χη, 1/2) . As a result we find a distribution dP_{Λ} , whose p -adic Mellin transform computes the values of the L -function.

The computation will be based on a certain period integral P_{Λ,η}(ε; 𝔭^{e}) , which we introduce first. We denote by X_{e} the set of all characters μ : F^{∗}\𝔸^{∗} → ℂ^{∗} satisfying the properties: μ_{v} = id for all v ∈ S_{∞} and the conductor f_{μ} divides 𝔭^{e} . This set is in bijection

X_{e} ↔ {μ_{𝔭} : (O_{𝔭}/ϖ^{e}O_{𝔭})^{∗} → ℂ^{∗} : μ_{𝔭}(O^{∗}) = 1}

with the set of O_{F}^{∗} -invariant characters on (O_{𝔭}/ϖ^{e}O_{𝔭})^{∗} , the bijection being given by sending a character μ to its 𝔭 -component μ_{𝔭} and restricting to O_{𝔭}^{∗} . In particular we may choose a local prime ϖ = ϖ_{𝔭} such that η_{𝔭}(ϖ) = 1 . We use the following notation: if G is a finite group and V a vector space, we denote by G^{V} the set of all maps f : G → V . Also we write A(n − 1) = A(GL_{n−1}(ℚ)/GL_{n−1}(𝔸)) . The duality of X_{e} and O_{e} := O_{𝔭}^{∗}/((1 + ϖ^{e}O_{𝔭})O_{F}^{∗}) then defines in the usual way a Fourier transform

F : X_{e}^{A(n−1)} → O_{e}^{A(n−1)},

sending f : G → A(n − 1) to

F_{x}(f) = ∑_{μ∈X_{e}} μ^{−1}(x)f_{μ}.

We denote by D the diagonal matrix of length n − 1

D = ( \begin{matrix}ϖ \\ & . & . & . \\ & & & & ϖ \\ & & & & & 1 \end{matrix} ) ∈ T_{n−1}(F_{𝔭}),

and we fix a complex number κ . We set π̂[μ] = π̂ ⊗ μ . In section 1 we defined for primitive characters μ in X_{e} automorphic forms E(f_{π̂⊗μ,Λ}, g) and we now extend this definition to all characters μ ∈ X_{e} with conductor f_{μ} = 𝔭^{e_{μ}} setting

(3.1)  E_{π̂[μ],Λ,𝔭^{e}}^{κ}(g) = E_{π̂[μ],Λ,𝔭^{e}}(g) = κ^{−(e−e_{μ})}E(f_{π̂[μ],Λ}, gD^{−(e−e_{μ})}).

The family {E_{π̂[μ],Λ,𝔭^{e}}} then defines an element of X_{e}^{A(n−1)} . Accordingly, the transform F_{x}{E_{π̂[μ],Λ,𝔭^{e}}} is an element of O_{e}^{A(n−1)} , which we also view in the obvious way as element of (O_{𝔭}/ϖ^{e}O_{𝔭})^{∗})^{A(n−1)} . The following properties are immediate. For δ ∈ O_{𝔭}^{∗} we have

(3.2)  F_{x}{E_{π̂[μ],Λ,𝔭^{e}}}(gH_{δ}) = F_{δ^{−1}x}{E_{π̂[μ],Λ,𝔭^{e}}}(g).

For every χ ∈ X_{e} the inverse transform reads

(3.3)  E_{π̂[χ],Λ,𝔭^{e}} = F_{χ}{F_{x}{E_{π̂[μ],Λ,𝔭^{e}}}}
= \frac{1}{|(O_{𝔭}/ϖ^{e}O_{𝔭})^{∗}|} ∑_{x∈(O_{𝔭}/ϖ^{e}O_{𝔭})^{∗}} χ_{𝔭}(x) F_{x}\{E_{π̂[μ],Λ,𝔭^{e}}\}.

Concerning the first property we observe that

D^{e−e_{μ}}K_{0}(n − 1, 𝔭^{e})D^{−(e−e_{μ})} ⊂ K_{0}(n − 1, 𝔭^{e_{μ}}),

and since E_{π̂[μ],Λ} is by definition the essential vector in I_{P,Λ,𝔭}π̂[μ] we obtain for k = (k_{ij}) ∈ K_{0}(n − 1, 𝔭^{e})

E_{π̂[μ],Λ,𝔭^{e}}(gk) = E_{π̂[μ],Λ,𝔭^{e}}(g) μ_{𝔭}(k_{n−1,n−1}).

The definition of the Fourier transform then immediately implies the first claim.

For any vector v = (v_{1}, …, v_{n−1}) such that v_{1}, …, v_{n−2} ∈ O_{𝔭} and v_{n−1} ∈ O_{𝔭}^{∗} and any ε ∈ (O_{𝔭}/𝔭^{e}O_{𝔭})^{∗} we introduce the period integral

P_{Λ,η}(v, ε; 𝔭^{e}) := ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) v) η ⊗ F_{ε+ϖ^{e}O_{𝔭}}\{E_{π̂[μ],Λ,𝔭^{e}}\}(g) dg.

Here, in accordance with (2.5) we identify v with the unipotent matrix

v = ( \begin{matrix}1 & & & & v_{1}/ϖ^{a} \\ & ... & & & ⋮ \\ & & ... & & v_{n−2}/ϖ^{a} \\ & & & 1 & v_{n−1}/ϖ^{e} \\ & & & & 1 \end{matrix} )

To simplify the notation we mostly drop the index η and we write E_{ε,Λ,𝔭^{e}} = F_{ε+ϖ^{e}O_{𝔭}}{E_{π̂[μ],Λ,𝔭^{e}}} .

Lemma 3.1. i) Assume that e ≥ a . Then for any matrix

u = ( \begin{matrix}1 & & & o_{1} \\ & ... & & ⋮ \\ & & 1 & o_{n−2} \\ & & & 1 \end{matrix} ) ∈ N_{n−1}(O_{𝔭})

the following relation holds:

P_{Λ,η}(v, ε; 𝔭^{e}) = P_{Λ,η}(uv, ε; 𝔭^{e}).

ii) For any δ ∈ O_{𝔭}^{∗} we have

η_{𝔭}^{n−1}(δ) P_{Λ,η}(H_{δ} v, ε; 𝔭^{e}) = P_{Λ,η}(v, δε; 𝔭^{e}).

Here, we view v = (v_{1}, …, v_{n−1}) as column vector.

Proof. i) We set u′ := H_{ϖ^{−(e−a)}}uH_{ϖ^{e−a}} . Since E_{ε,𝔭^{e}} is invariant under N_{n−1}(O_{𝔭}) we obtain by the change of variable g ↦ gu

P_{Λ}(v; 𝔭^{e}) = ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) u′v) η ⊗ E_{1,𝔭^{e}}(g) dg.

On the other hand the expression in (2.6) for w_{η_{𝔭},n−2} together with the independence of this expression of the choice of representatives in O_{𝔭}/ϖ^{e}O_{𝔭} (cf. Remark 2.1) proves, that w_{η_{𝔭},n−2} is invariant under all matrices diag(ũ, 1) , where

ũ = ( \begin{matrix}1 & & & o_{1} \\ & ... & & ⋮ \\ & & 1 & o_{n−2} \\ & & & 1 \end{matrix} ) ∈ N(n − 1, O_{𝔭}).

Therefore, the same holds for ϕ_{η} and together with the identity

(\begin{matrix}u′ \\ & 1 \end{matrix}) v = ( \begin{matrix}1 & & & & (v_{1} + o_{1}v_{n−1})/ϖ^{a} \\ & ... & & & ⋮ \\ & & 1 & & (v_{n−2} + o_{n−2}v_{n−1})/ϖ^{a} \\ & & & 1 & v_{n−1}/ϖ^{e} \\ & & & & 1 \end{matrix} ) ( \begin{matrix}1 & & & o_{1}ϖ^{e−a} & 0 \\ & ... & & ⋮ & ⋮ \\ & & ... & o_{n−2}ϖ^{e−a} & ⋮ \\ & & & 1 & 0 \\ & & & & 1 \end{matrix} )

we obtain

P_{Λ}(v; 𝔭^{e}) = P_{Λ}((v_{1} + o_{1}v_{n−1}, …, v_{n−2} + o_{n−2}v_{n−1}, v_{n−1}); 𝔭^{e}),

which is the claim. ii) Recalling the definition of the period integral we find

P_{Λ,η}(H_{δ}v, ε; 𝔭^{e})

= ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) ( \begin{matrix}1 & & & & v_{1}/ϖ^{a} \\ & ... & & & ⋮ \\ & & ... & & v_{n−2}/ϖ^{e} \\ & & & 1 & δv_{n−1}/ϖ^{e} \\ & & & & 1 \end{matrix} )) η ⊗ E_{ε,𝔭^{e}}(g) dg.

On the other hand Eq. (2.10) implies

(3.4)  ϕ_{η} (g (\begin{matrix}H_{δ} \\ & 1 \end{matrix})) = ϕ_{η}(g)η_{𝔭}^{n−2}(δ).

Together with the change of integration variable g ↦ gH_{δ^{−1}} we obtain

P_{Λ,η}(H_{δ}v, ε; 𝔭^{e}) = η_{𝔭}^{2−n}(δ) ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) v) η ⊗ E_{ε,𝔭^{e}}(gH_{δ^{−1}}) dg.

The assertion then follows using Eq. (3.2). Thus, the Lemma is proven.

Corollary 3.2 For e ≥ a the product η_{𝔭}^{n−1}(v_{n−1}) P_{Λ,η}(v, ε; 𝔭^{e}) does not depend on v . From now on we shall assume that e ≥ a . For any ε ∈ (O_{𝔭}/𝔭^{e})^{∗} we therefore may set

P_{Λ}(ε; 𝔭^{e}) := η_{𝔭}^{n−1}(v_{n−1}) P_{Λ}(v, ε; 𝔭^{e})

and we define

(3.5)  dP_{Λ}(ε + 𝔭^{e}O_{𝔭}) := (κq^{n−1}γ)^{e}P_{Λ}(ε; 𝔭^{e}),

where

(3.6)  γ = γ_{π} = αδ_{B_{n}}^{1/2}(diag(1, ϖ^{−1}, …, ϖ^{−1})).

It is then immediate by the following distribution relation for the periods P_{Λ}(ε; 𝔭^{e}) that dP_{Λ,η} defines a distribution on O_{𝔭}^{∗} . Proposition 3.3. The periods P_{Λ,η}(ε; 𝔭^{e}) satisfy the distribution relation

P_{Λ,η}(ε; 𝔭^{e}) = κq^{n−1}γ ∑_{{ε′∈(O𝔭/𝔭^{e+1}O𝔭)^{∗} \atop ε′≡ε (mod 𝔭^{e})}} P_{Λ,η}(ε′; 𝔭^{e+1}).

Proof. The distribution relation for the period integral follows from the distribution relations, which are satisfied by the Eisenstein series E_{ε,𝔭^{e}} = F_{ε+ϖ^{e}O_{𝔭}}{E_{π̂[μ],Λ,𝔭^{e}}} and the cusp form ϕ_{η} :

• The sections E_{ε,𝔭^{e}} satisfy the distribution relation

(3.7)  E_{ε,𝔭^{e}}(gD^{−1}) = κ ∑_{ε′} E_{ε′,𝔭^{e+1}}(g),

where the sum runs over all elements ε′ ∈ (O_{𝔭}/𝔭^{e+1}O_{𝔭})^{∗} which are congruent to ε modulo ϖ^{e}O_{𝔭} . Since η_{𝔭}(ϖ) = 1 the same relation holds for η ⊗ E_{ε,𝔭^{e}}

• We have

(3.8)  ϕ(g) = γ ∑_{w_{1},... ,w_{n−1}} ϕ(gD^{∗}w),

where D^{∗} = diag(ϖ, …, ϖ, 1) is the diagonal matrix of length n , w is the matrix

(3.9)  w = ( \begin{matrix}1 & & & w_{1}/ϖ \\ & ... & & ⋮ \\ & & 1 & w_{n−1}/ϖ \\ & & & 1 \end{matrix} ) .

and w_{1}, …, w_{n−1} run over O_{𝔭}/𝔭O_{𝔭} .

To prove the first distribution relation one simply evaluates the left hand side by plugging in the definition of E_{ε′,e+1}(g) and using character relations. The second one is an easy consequence of the relation

ψ_{𝔭}^{1}(g) = αδ_{B_{n}}^{1/2}(diag(1, ϖ^{−1}, …, ϖ^{−1})) ∑_{w_{1},... ,w_{n−1}} ψ_{𝔭}^{1}(gD′w)

for the section ψ_{𝔭}^{1} , which we chose in section 1.

We may now prove the proposition. Let v ∈ N(n, F_{𝔭}) be as in the definition of the periods. Using the expression for w_{η,n−2} and the distribution relation for ϕ we find

ϕ_{η}(gv) = ∑_{u_{ij},v_{k}} ∏_{t=1}^{n−2} η_{𝔭}^{t}(u_{t,t+1})χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1})ϕ(gv′)

= γ ∑_{u_{ij},v_{k}} ∏_{t=1}^{n−2} η_{𝔭}^{t}(u_{t,t+1})χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) ∑_{w_{1},... ,w_{n−1}} ϕ(gv′D^{∗}w),

v′ = ( \begin{matrix}1 & u_{1,2}/ϖ & ⋯ & u_{1,n−1}/ϖ^{n−2} & v_{1}/ϖ^{a} \\ & ... & & ⋮ & ⋮ \\ & & 1 & u_{n−2,n−1}/ϖ & v_{n−2}/ϖ^{a} \\ & & & 1 & v_{n−1}/ϖ^{e} \\ & & & & 1 \end{matrix} )

and w and w_{1}, …, w_{n−1} are as in (3.9). A small computation proves

v′D^{∗}w = D^{∗} ( \begin{matrix}1 & ⋯ & ⋯ & u_{ij}/ϖ^{j−i} & x_{1} \\ & ... & & ⋮ & ⋮ \\ & & 1 & ⋮ & x_{n−2} \\ & & & 1 & x_{n−1} \\ & & & & 1 \end{matrix} )

where x_{i} = \frac{v_{i}}{ϖ^{a+1}} + ∑_{h=i+1}^{n−1} \frac{w_{h}}{ϖ} \frac{u_{ih}}{ϖ^{h−i}} + \frac{w_{i}}{ϖ} and x_{n−1} = \frac{v_{n−1}+ϖ^{e}w_{n−1}}{ϖ^{e+1}} . Thus we get

ϕ_{η}(gv) = γ ∑_{w_{1},... ,w_{n−1}} ϕ_{η}(gD^{∗}w′),

where

w′ = ( \begin{matrix}1 & & & x_{1} \\ & ... & & ⋮ \\ & & 1 & x_{n−1} \\ & & & 1 \end{matrix} ) .

We turn to the calculation of P_{Λ}(ε; 𝔭^{e}) and we choose v as follows: v_{1} = … = v_{n−2} = 0 and v_{n−1} = 1 . Then, replacing ϕ_{η}(gH_{ϖ^{e−a}}) by the above expression and changing the variable g ↦ gD^{−1} we find

P_{Λ}(ε; 𝔭^{e}) = γ ∑_{w_{1},... ,w_{n−1}} ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e+1−a}} \\ & 1 \end{matrix}) w′) η ⊗ E_{ε,𝔭^{e}}(gD^{−1}) dg.

Next we use the distribution relation (3.7) as well as (3.2) to obtain

P_{Λ}(ε; 𝔭^{e}) = γκ ∑_{w_{1},... ,w_{n−1}} ∑_{ε′≡ε (mod 𝔭^{e})} ∫_{X_{n−1}^{𝔸}} ϕ_{η} (gH_{ϖ^{e+1−a}}w′) η ⊗ E_{ε′,𝔭^{e+1}}(g) dg.

Using the periods this may be rewritten

P_{Λ}(ε; 𝔭^{e}) = γκ ∑_{w_{1},... ,w_{n−1}} ∑_{ε′≡ε (mod 𝔭^{e})} P_{Λ}((ϖ^{a}x_{1}, …, ϖ^{a}x_{n−2}, ϖ^{e+1}x_{n−1}), ε′; 𝔭^{e+1}).

In the above equation, ϖ^{a}x_{1}, …, ϖ^{a}x_{n−2} ∈ O_{𝔭} and ϖ^{e+1}x_{n−1} = ε + ϖ^{e}w_{n−1} ∈ 1 ∗ 𝔭^{e}O_{𝔭} . Therefore using Lemma 3.1 and taking into account that η_{𝔭}|_{1+𝔭O_{𝔭}} = 1 we arrive at

P_{Λ}(ε; 𝔭^{e}) = γκq^{n−1} ∑_{w_{n−1}∈O_{𝔭}/ϖO_{𝔭}} P_{Λ}((0, …, 0, 1), ε′(ε + ϖ^{e}w_{n−1})^{−1}; 𝔭^{e+1}).

Changing the summation variable ε in the obvious way the assertion of the proposition follows.

Like the periods P_{Λ,η}(v; 𝔭^{e}) the distribution dP_{Λ,η} depends on the choice of the components at infinity w_{∞} = ⊗_{v∈S_{∞}}w_{v} and v_{π̂[χ],Λ,∞} = ⊗_{v∈S_{∞}}v_{π̂[χ],Λ,v} and if we want to indicate this dependence we write dP_{Λ,η} = dP_{w_{∞},v_{π̂[χ],Λ,∞}}^{Λ,η} .

Proposition 3.4. For all characters χ with conductor f_{χ} = 𝔭^{e} , ( e > a ), infinity component χ_{v} = id for all v ∈ S_{∞} and satisfying χ(x) = 1 for all x ∈ V_{F} we have

∫_{O_{𝔭}^{∗}} χ_{𝔭} dP_{Λ,η} = q^{a(n−2)}(γκq^{n−1})^{e} J_{π̂[χ],Λ,η}(1/2).

Proof. Using (3.3) J_{π̂[χ],Λ}(1/2) may be written

J_{π̂[χ],Λ}(1/2) = ∑_{v_{1},... ,v_{n−1}} ∑_{ε∈(O_{𝔭}/𝔭^{e})^{∗}} χ_{𝔭}η_{𝔭}^{n−1}(v_{n−1}) χ_{𝔭}(ε)

× ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) v) η ⊗ E_{ε,𝔭^{e}}(g) dg.

Changing the summation variable g ↦ gH_{ε} and using Lemma 3.1, the right hand side becomes

J_{π̂[χ],Λ}(1/2) = ∑_{v_{1},... ,v_{n−1}} ∑_{ε∈(O_{𝔭}/𝔭^{e})^{∗}} χ_{𝔭}(εv_{n−1})

× ∫_{X_{n−1}^{𝔸}} ϕ_{η} ((\begin{matrix}g \\ & 1 \end{matrix}) (\begin{matrix}H_{ϖ^{e−a}} \\ & 1 \end{matrix}) ( \begin{matrix}1 & & & & 0 \\ & ... & & & ⋮ \\ & & 1 & & 0 \\ & & & 1 & 1/ϖ^{e} \\ & & & & 1 \end{matrix} )) η ⊗ E_{εv_{n−1},𝔭^{e}}(g) dg.

The change of the summation variable ε ↦ εv_{n−1}^{−1} then yields

J_{π̂[χ],Λ}(1/2) = ∑_{v_{1},... ,v_{n−1}} ∑_{ε∈(O_{𝔭}/𝔭^{e}O_{𝔭})^{∗}} χ_{𝔭}(ε) P_{Λ}((0, …, 0, 1), ε; 𝔭^{e})
= ∑_{v_{1},... ,v_{n−1}} χ_{𝔭}(ε) P_{Λ}(ε; 𝔭^{e})
= (κγq^{n−1})^{−e}q^{a(n−2)} ∑_{ε∈(O_{𝔭}/𝔭^{e}O_{𝔭})^{∗}} χ_{𝔭}(ε)dP_{Λ}(ε + 𝔭^{e}O_{𝔭}).

Hence, the assertion and the proposition is proven.

Together with Theorem 2.7 we immediately obtain the representation of the values of the L -function as a p -adic Mellin transform.

Theorem 3.5. Assume that the conductor f_{η} is sufficiently large as in (C.1) . Let χ : F^{∗}\𝔸^{∗} → ℂ^{∗} be any character with conductor f_{χ} = 𝔭^{e} for e ≥ a and infinity component χ_{∞} = id . Then the following holds

(γκq^{n−1})^{e}♣ ∫_{O_{𝔭}^{∗}} χ_{𝔭} dP_{Λ,η} = P_{w_{∞},v_{π̂[χ],Λ,∞}}(1/2) \frac{L(π ⊗ χη, 1/2 + s_{2})}{L(π̂ ⊗ χ^{−1}, 1 + s_{1} − s_{2})}.

We recall that in the formulation of the Theorem Λ = (s_{1}, s_{2}) .

Remark 3.6. The distribution dP_{Λ,η} is O_{F}^{∗} -invariant: for all o ∈ O_{F}^{∗} we have

dP_{Λ,η}(oε + 𝔭^{e}) = dP_{Λ,η}(ε + 𝔭^{e}).

In particular, dP_{Λ,η} is even.

4 Algebraic automorphic representations.

To simplify the exposition from now on we shall assume F = ℚ , i.e. 𝔸 denotes the ring of rational adeles. We introduce the type of cuspidal representions of GL_{n}(𝔸) for which we shall examine the values of the belonging L -function. These are algebraic automorphic representations, which in addition satisfy a regularity condition. Using results of Clozel it turns out that these representations have realizations as submodules of the cohomology of the symmetric space with coefficients in a finite dimensional representation of GL_{n}(ℂ) .

Let π be a cuspidal automorphic representation of GL_{n}(𝔸) , whose infinity component is of the following type. If n = 2m is even we assume that there is a w ∈ ℤ and integers l_{1} > l_{2} > … > l_{m} > 0 satisfying l_{i} ≡ l_{j} (mod 2) such that

(4.1)  π_{∞} ≅ Ind_{P (ℝ)}^{GL_{n}(ℝ)}(| · |_{ℝ}^{w/2} ⊗ D_{l_{1}}, …, | · |_{ℝ}^{w/2} ⊗ D_{l_{m}}),

where P_{n} ≤ GL_{n} is the parabolic subgroup of type (2, 2, …, 2) . If n = 2m + 1 is odd there is a w ∈ ℤ and integers l_{1} > l_{2} > ⋯ > l_{m} > 0 satisfying l_{i} ≡ l_{j} (mod 2) such that

(4.2)  π_{∞} ≅ Ind_{P (ℝ)}^{GL_{n}(ℝ)}(| · |_{ℝ}^{w/2} ⊗ D_{l_{1}}, …, | · |_{ℝ}^{w/2} ⊗ D_{l_{m}}, | · |_{ℝ}^{w})

where P_{n} ≤ GL_{n} is the parabolic subgroup of type (2, 2, …, 2, 1) .

We diagonally embed ℂ ↪ ℂ^{n} and we identify ℂ with its image. We denote by ∼ the equivalence relation on ℤ^{n} , which is induced by the standard action of the Weyl group W_{GL_{n}} = S_{n} of GL_{n} . We set

(4.3)  l = (l_{1}, l_{2}, …, −l_{2}, −l_{1})

if n is even and

(4.4)  l = (l_{1}, l_{2}, …, 0, …, −l_{2}, −l_{1})

if n is odd. The type at infinity p = (p_{i}) of the representation π (cf. [C], p. 107) then reads

p = (w/2 + 1/2l − (n − 1)/2)_{∼}.

In particular, we deduce that π is algebraic in the sense of [C], Définition 1.8, p. 89 if and only if

l + w ≡ 1 (mod 2)

if n is even and l ≡ 0 (mod 2) and w ≡ 0 (mod 2).

if n is odd. Moreover, since l_{i} ≠ l_{j} for i ≠ j we see that in this case π is automatically regular in the sense of [C], Définition 3.12, p. 111. We set

m_{π} = w/2 + 1/2l − ρ̌,

where ρ̌ denotes the half sum of positive simple roots of GL_{n} with respect to T_{n} . If there is no danger of confusion we drop the index and simply write m = (m_{i}) . Explicitely, we have

m = (\frac{w}{2} + \frac{l_{1}}{2} − \frac{n − 1}{2}, \frac{w}{2} + \frac{l_{2}}{2} − \frac{n − 3}{2}, …, \frac{w}{2} − \frac{l_{1}}{2} + \frac{n − 1}{2})

if n is even and

m = (\frac{w}{2} + \frac{l_{1}}{2} − \frac{n − 1}{2}, \frac{w}{2} + \frac{l_{2}}{2} − \frac{n − 3}{2}, …, \frac{w}{2}, …, \frac{w}{2} − \frac{l_{1}}{2} + \frac{n − 1}{2})

if n is odd. In particular, if π is algebraic we deduce that m_{π} is an integral and dominant weight (not necessarily regular). Moreover, m = (m_{i}) ∈ ℤ^{n} is pure of weight w ∈ ℤ , which means that there is a w ∈ ℤ such that

m_{i} + m_{n+1−i} = w, i = 1, …, n/2

if n is even and

m_{i} + m_{n+1−i} = w, i = 1, …, (n − 1)/2 and m_{(n+1)/2} = w/2

if n is odd.

On the other hand, if m ∈ ℤ^{n} is dominant and satisfies the purity condition then there is a representation π_{∞} of GL_{n}(ℝ) such that m = m_{π} : for any vector l = (l_{i}) of the form (4.3) resp. (4.4) and w ∈ ℤ we write

(4.5)  π_{∞}(w, l)

to denote the representation (4.1) resp. (4.2). Let w be the weight of m and define l = (l_{i}) by

(4.6)  l = 2m + 2ρ̌ − w;

we obtain m = m_{π} if π_{∞} = π_{∞}(w, l) (Explicitely l_{i} = 2m_{i} − 2(i − 1) + n − 1 − w , i = 1, …, m = n ). Of course, π is necessarily algebraic and regular.

From now on we shall assume that π is a cuspidal automorphic representation of GL_{n}(𝔸) with infinity component as follows.

• If n = 2m is even there are integers l_{1} > l_{2} > ⋯ > l_{m} > 0 satisfying l_{i} ≡ l_{j} (mod 2) such that

(Coh.a)  π_{∞} ≅ | · |^{w/2} ⊗ Ind_{P (ℝ)}^{GL_{n}(ℝ)}(D_{l_{1}}, D_{l_{2}}, …, D_{l_{m}})

where P_{n} ≤ GL_{n} is the parabolic subgroup of type (2, 2, …, 2) and w = 0 if the l_{i} are odd and w = −1 if the l_{i} are even.

• If n = 2m + 1 is odd there are even integers l_{1} > l_{2} > ⋯ > l_{m} > 0 such that

(Coh.b)  π_{∞} ≅ Ind_{P (ℝ)}^{GL_{n}(ℝ)}(D_{l_{1}}, D_{l_{2}}, …, D_{l_{m}}, id)

where P_{n} ≤ GL_{n} is the parabolic subgroup of type (2, 2, …, 2, 1) (cf. [C], p. 114).

The representation π is algebraic and regular and hence, m = m_{π} ∈ ℤ^{n} is dominant; we denote by ρ the finite dimensional representation of GL_{n}(ℂ) acting on the vector space M_{m} with highest weight m with respect to the standard torus. We define

m̃ = −w_{n,l}m = −w + m

and m̃ = (m̃_{i}) is the highest weight of the contragredient representation (ρ̃, M̃_{m}) . Lemme 3.14 in [C] then implies that

(4.7)  H^{•}(𝔤_{n}, O_{n}(ℝ)Z_{n}^{0}(ℝ), π_{∞} ⊗ ϵ_{∞} ⊗ ρ̃) ≠ 0

for b_{n} ≤ • ≤ t_{n} . Here,

b_{n} = \{ \begin{matrix}\frac{n^{2}}{4} & if n is even \\ \frac{n^{2}−1}{4} & if n is odd, \end{matrix}

t_{n} = \{ \begin{matrix}\frac{(n+1)^{2}−1}{4} − 1 & if n is even \\ \frac{(n+1)^{2}}{4} − 1 & if n is odd \end{matrix}

and

ϵ_{∞} = \{ \begin{matrix}id & n even \\ id & n ≡ 1 (mod 4) \\ sgn & n ≡ 3 (mod 4). \end{matrix}

Since O_{n}(ℝ) normalizes SO_{n}(ℝ) the group of connected components π_{0} = O_{n}(ℝ)/SO_{n}(ℝ) acts on H^{•}(𝔤_{n}, SO_{n}(ℝ)Z_{n}^{0}(ℝ), π_{∞} ⊗ ρ̃) via right translation. Obviously, π_{0} ≅ ℤ/2ℤ and we denote by id resp. sgn the trivial resp. the non-trivial representation of π_{0} . Eq. (4.7) then implies that π_{0} acts on the relative Lie algebra cohomology via ϵ_{∞} , i.e. we obtain

H^{b_{n}}(𝔤_{n}, SO_{n}(ℝ)Z_{n}^{0}(ℝ), π_{∞} ⊗ ρ̃) ≅ \{ \begin{matrix}sgn ⊕ id & if & n ≡ 0 (mod 2) \\ id & if & n ≡ 1 (mod 4) \\ sgn & if & n ≡ 3 (mod 4). \end{matrix}

The claim in the case n odd follows immediately from [C], Lemme 3.14, while in the case n even one has to take into account that the restriction of π to the group of matrices with positive determinant decomposes

π|_{GL_{n}^{+}(ℝ)} = π^{+} ⊕ π^{−}

and π^{+} ≅ π^{−} as GL_{n}^{+}(ℝ) -modules. In particular, the cohomology groups are two resp. one dimensional. The non-vanishing of the cohomology at infinity yields an embedding of the finite part of π into the cohomology as follows. We choose a basis {ω′_{1}, …, ω′_{dim X_{n}^{1}}} of the dual of the tangent space (𝔤_{n}/𝔨_{n}𝔷_{n})^{∗} of X_{n}^{1} = GL_{n}(ℝ)/K_{n,∞} as well as a basis {a} of M_{m} . We let

ξ_{π,∞} = ∑_{j=(j_{1},... ,j_{bn})} ∑_{a} ω′_{j} w_{∞,j,a} ⊗ a

be a generator of

H^{b_{n}}(𝔤_{n}, K_{n,∞}Z_{n}^{0}(ℝ), W (π_{∞}, τ_{∞}) ⊗ ρ)

if n is odd and of

H^{b_{n}}(𝔤_{n}, K_{n,∞}Z_{n}^{0}(ℝ), W (π_{∞}, τ_{∞}) ⊗ ρ)^{+}

if n is even, where the superscript + indicates the +1 -eigenspace for the action of π_{0} . For any compact open subgroup K ≤ GL_{n}(ℤ̂) we introduce the manifolds

S_{n}(K) = GL_{n}(ℚ)\GL_{n}(𝔸)/KK_{n,∞}.

The finite dimensional representation (ρ̃, M̃_{m}) defines a locally constant sheaf M̃_{m} = M̃_{m,K} on S_{n}(K) and there are cohomology groups

H_{?}^{•}(S_{n}(K), M̃_{m,K}), ? = sing, c, cusp.

Passing to the limit we define the cohomology groups

H_{?}^{•}(S̃_{n}, M̃_{m}) = inj lim_{K} H_{?}^{•}(S_{n}(K), M̃_{m,K}), ? = sing, c, cusp.

These groups are modules under the canonical action of GL_{n}(𝔸_{f}) × π_{0} .

For any Whittaker function w_{f} ∈ W (π_{f}, τ_{f}) the product w_{f}ξ_{π,∞} now defines an element of H^{b_{n}}(𝔤_{n}, K_{n,∞}Z_{n}^{0}(ℝ), W (π, τ) ⊗ ρ) and together with the embedding π ↪ L_{0}^{2}(GL_{n}(ℚ)\GL_{n}(𝔸)) this yields an injection

\begin{aligned}ξ_{π,∞} : & W (π_{f}, τ_{f}) & ↪ & H_{cusp}^{b_{n}}(S̃_{n}, Ṽ) \\ & w_{f} & ↦ & ω. \end{aligned}

In coordinates, the image of w_{f} reads

ω = ∑_{j=(j_{1},... ,j_{bn})} ∑_{a} ω′_{j} ϕ_{j,a} ⊗ a,

where ϕ_{j,a} is the cusp form corresponding to w_{f}w_{∞,j,a} . We remark that the cuspidal cohomology defines a subspace of the cohomology with compact supports

H_{cusp}^{b_{n}}(S̃_{n}, M_{m}) ≤ H_{c}^{b_{n}}(S̃_{n}, M̃_{m})

(cf. [C], p 123).

The critical points. – We finish this section with the determination of the critical points of the representation π ⊗ η . Here, η : ℚ^{∗}\𝔸^{∗} → ℂ^{∗} is an idele class character of finite order and we set ♭ = 0 if η_{∞} = id and ♭ = 1 if η_{∞} = sgn .

Lemma 4.1. If n is even, the critical points l for π ⊗ η are

l = −\frac{l_{m} + w}{2} + 1, −\frac{l_{m} + w}{2} + 2, …, \frac{l_{m} − w}{2} − 1, \frac{l_{m} − w}{2} ∈ \frac{1}{2} + ℤ,

independent of η . Here, w was defined to be 0 if the l_{i} are odd and −1 if the l_{i} are even. If n is odd, the critical points for η ⊗ π are all integers

l = −\frac{l_{m}}{2} + 1, −\frac{l_{m}}{2} + 2, …, \frac{l_{m}}{2} − 1, \frac{l_{m}}{2} ∈ ℤ,

which satisfy the congruence

l ≡ \{ \begin{matrix}♭ & (mod 2) & if l ≥ 1 \\ ♭ + 1 & (mod 2) & if l ≤ 0. \end{matrix}

Proof. We first assume n = 2m to be even. Since D_{l} ⊗ sgn ≅ D_{l} we see that π_{∞} ⊗ sgn ≅ π_{∞} and we may assume η_{∞} = id . We denote by W_{ℝ} the Weil group of ℝ . Using the notation of [Kn], ch. 3.6 the representation π_{∞}^{w} of W_{ℝ} attached to π_{∞} via local Langlands correpondence reads

π_{∞}^{w} = (l_{1}, w/2) ⊕ (l_{2}, w/2) ⊕ ⋯ ⊕ (l_{m}, w/2).

Taking into account that L(π_{∞}, s) = L(π_{∞}^{w}, s) we deduce from the formulas in [Kn], ch. 3.6 that

L(π_{∞}, s) = ∏_{i=1}^{m} 2(2π)^{−(s+w/2+l_{i}/2)}Γ (s + \frac{l_{i} + w}{2}) .

Since the Γ -Function has its poles at s = 0, −1, −2, … the claim follows. If n = 2m+1 is odd we find for the representation of the Weil group corresponding to π_{∞}⊗η_{∞}

(π_{∞} ⊗ η_{∞})^{w} = (l_{1}, 0) ⊕ (l_{2}, 0) ⊕ ⋯ ⊕ (l_{m}, 0) ⊕ (♭, 0).

and hence, the local factor of the L -function reads

L(π_{∞} ⊗ η_{∞}, s) = ∏_{i=1}^{m} 2(2π)^{−(s+l_{i})}Γ (s + \frac{l_{i}}{2} ) π^{(s+1)/2}Γ (\frac{s + ♭}{2}) .

Since the l_{i} are even if n is odd, the statement about the poles of the Γ -function again yields the claim. This completes the proof of the lemma.

Remark 4.2. 1) In terms of the weight m_{π} = (m_{i}) attached to π the set of critical points reads as follows. If n is even the critical points are

−m_{[n/2]} + 1/2, …, 1/2, …, m_{[n/2]} + 1/2 − w ∈ 1/2 + ℤ,

if n is odd these are all integers in the range

−m_{[n/2]}, …, 0, 1, …, m_{[n/2]} + 1,

which satisfy the same congruence as in Lemma 4.1.

2) The functional equation implies

L(π ⊗ χη, l) = L(| · |^{w} ⊗ π̃ ⊗ χ^{−1}η^{−1}, 1 − l − w)

and | · |^{w} ⊗ π̃ again is of the form described in (Coh.a) resp. (Coh.b) . Hence it is sufficient to examine the values at critical points l ≤ (1 − w)/2 . In all cases this implies l ≤ 1/2 .

We say that (l, η) is a critical pair for π if l is one of the critical points for π ⊗ η as described in Lemma 4.1.

In our further treatment of the L -function attached to twists of the representation π we shall have to distinguish the following cases.

Definition 4.3. Let π = π(w, l) be a representaion of GL_{n}(𝔸) as in (Coh) and η an idele class character of finite order. We say that

π is of type I if n ≡ 0 (mod 2) and l ≡ 0 (mod 2)
type II  n ≡ 0 (mod 2)  l ≡ 1 (mod 2)
type III  n ≡ 1 (mod 2)  l ≡ 0 (mod 2).

From now on we shall assume that n = 2m is even and π is as in (Coh) . For later purpose we assume that in addition to (Coh) the vector m_{π} = (m_{i}) attached to π satisfies the following condition; we set Δ_{i} = m_{i} − m_{i+1} , i = 1, …, n/2 − 1 .

(Reg) Δ_{i} > 0, i < n/2 − 1  if π is of type I Δ_{i} > 2, i < n/2 − 2, Δ_{n/2−1} ≥ n  π  II.

Thus, in case I we assume that m is regular, while in case II we make a slightly stronger assumption.

5 Eisenstein series and the cohomology at infinity

We shall describe a topological realization of the induced automorphic representation I_{P,Λ}π̂[χ] as certain subspace of the cohomology of the symmetric space attached to GL_{n−1} consisting of classes, which do not vanish at infinity. This will be based on the theory of Eisenstein cohomology and hence the construction of such a space starts from a Hecke invariant subspace in the cohomology of the boundary of the symmetric space. As a result we find classes ω_{Eis,π̂[χ],Λ} in the cohomology of the symmetric space of GL_{n−1} , which turn out to be rational.

5.1. A Hecke-invariant subspace in the cohomology of the boundary.

We want to describe the cohomology of the boundary as module under the Hecke algebra in terms of the (cuspidal) cohomology of groups of lower rank.

We let S̄_{n−1}(K) be the Borel-Serre compactification of S_{n−1}(K) , which is a manifold with boundary and corners. Using the above notations we may describe the boundary ∂S̄_{n−1}(K) as follows. We let P be the set of all parabolic subgroups P ≤ GL_{n−1} , which are defined over ℚ . The group GL_{n−1} acts on P via conjugation and ∂S̄_{n−1}(K) then is the disjoint union of faces

∂S̄_{n−1}(K) = ⋃_{P} e(P, K),

where P runs over a set of representatives of P modulo GL_{n−1}(ℚ) . We denote by ρ_{λ} : GL_{n−1} → GL(M_{λ}) the irreducible representation of GL_{n−1} of highest weight λ acting on a ℂ -vector space M_{λ} . Using [C] p. 122 we deduce that there is a finite extension E_{λ} of ℚ and a E_{λ} -vector space M_{E_{λ}} such that M_{λ} = M_{E_{λ}} ⊗ℂ and M_{E_{λ}} is stable under ρ_{λ} . For any field E/E_{λ} we set M_{E} = M_{E_{λ}} ⊗ E and we denote by M_{E} resp. M the local system on S_{n−1}(K) corresponding to M_{E} resp. Mℚ̄ . M_{λ} extends to a sheaf on the compactification S̄_{n−1}(K) and hence also on the boundary ∂S̄_{n−1}(K) , which we also denote by M_{λ} .

The face e(P, K), P ∈ P is homotopy equivalent to the space

(5.1)  P (ℚ)A_{P}^{0} \GL_{n−1}(𝔸)/K_{n−1,∞}K

(cf. [Sch], p.39, 44 non-adelically). We want to describe this quotient more explicitely: we let Ξ be the finite set of double cosets P (𝔸_{f})\GL_{n−1}(𝔸_{f})/K and for any compact subgroup K^{M_{P}} ≤ M_{P} (𝔸_{f}) we set

0S_{n−1}^{M_{P}}(K^{M_{P}}) = M_{P} (ℚ)A_{P}^{0} \M_{P} (𝔸)/K^{M_{P}} K_{n−1,∞}^{M_{P}}.

Furthermore, for any ξ ∈ Ξ we denote by K^{M_{P}} (ξ) the image of

ξKξ^{−1} ∩ P (𝔸_{f})

in M_{P} (𝔸_{f}) . The face e(P, K) then is isomorphic to a disjoint union of manifolds

e(P, K) = ⋃_{ξ∈Ξ} e_{ξ}(P, K)

such that each of the manifolds e_{ξ}(P, K) has a locally trivial, canonical fibration

e_{ξ}(P, K) → ^{0}S_{n−1}^{M_{P}}(K^{M_{P}} (ξ))

with fibre isomorphic to N_{P} (ℚ) ∩ K^{M_{P}} (ξ)\N_{P} (ℝ) . We denote by H^{•}(𝔫_{P} , M_{λ}) the locally constant sheaf on ^{0}S_{n−1}^{M_{P}}(K^{M_{P}}) attached to the algebraic M_{P} -module H^{•}(𝔫_{P} , M_{λ}) . Right translation defines a M_{P} (𝔸_{f}) -module structure on

H^{i}(^{0}S̃_{n−1}^{M_{P}}, H^{j}(𝔫_{P} , M_{λ})),

and passing to the limit over all K ≤ GL_{n−1}(𝔸_{f}) , the above fibration induces an isomorphism of GL_{n−1}(𝔸_{f}) -modules

H^{•}(ẽ(P), M_{λ}) ≅ Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H^{•}(^{0}S̃_{n−1}^{M_{P}}, H^{•}(𝔫_{P} , M_{λ}))

(cf. [Ha 1], p. 88). We denote by W_{n−1} = W_{GL_{n−1}} resp. W_{M_{P}} the Weyl group of GL_{n−1} resp. M_{P} and we define the relative Weyl group

W^{P} = {w ∈ W_{n−1} : w^{−1}(α) ∈ Φ_{n−1}^{+} for all α ∈ Δ(M_{P} , T_{n−1})}.

The set W^{P} is a canonical choice of a system of representatives for the quotient

W^{P} ↔ W_{M_{P}} \W_{n−1}.

The structure of H^{•}(𝔫_{P} , M_{λ}) as M_{P} -module is then given by the formula of Kostant:

H^{i}(𝔫_{P} , M_{λ}) = ⨁_{{w∈W^{P} \atop ℓ(w)=i}} H^{i}(𝔫_{P} , M_{λ})(w · λ),

where w · λ = w(λ + ρ̌) − ρ̌ denotes the twisted action of the Weyl group on weights and H^{i}(𝔫_{P} , M_{λ})(w · λ) is isomorphic to the unique M_{P} -module with highest weight w · λ . For brevity we write H_{w·λ} = H^{ℓ(w)}(𝔫_{P} , M_{λ})(w · λ) and obtain

H^{r}(ẽ(P), M_{λ}) = ⨁_{{w∈W^{P} \atop ℓ(w)+p=r}} Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H^{p}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ}).

The same decomposition holds after tensoring with ℂ and we define

H_{cusp}^{r}(ẽ(P), M_{λ,ℂ}) = ⨁_{{w∈W^{P} \atop ℓ(w)+p=r}} Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H_{cusp}^{p}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ}).

The cuspidal cohomology decomposes under the action of the Hecke algebra into a direct sum

H_{cusp}^{•}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ}) = ⨁_{π′} H_{cusp}^{p}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ})(π′)_{ℂ},

where π′ ⊂ L_{0}^{2}(M_{P} (ℚ)A_{P}^{0} \M_{P} (𝔸)) runs over all unitary, cuspidal representations and H_{cusp}^{p}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ})(π′)_{ℂ} is the π′_{f} -isotypical component. We finally obtain

(5.2)  H_{cusp}^{r}(ẽ(P), M_{λ,ℂ}) = ⨁_{{w∈W^{P} \atop ℓ(w)+p=r}} ⨁_{π′} Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H^{p}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ})(π′)_{ℂ}.

We set

H_{cusp}^{r}(ẽ(P), M_{λ,ℂ})[w, π′]_{ℂ} = Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H_{cusp}^{r−ℓ(w)}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ})(π′)_{ℂ}.

We want to extend classes in the cohomology of the face e(P, K) to a neighbourhood of this face in S_{n−1}(K) . Such a neighbourhood is given by the space

e(P, K) × A_{P}^{0} = P (ℚ)\GL_{n−1}(𝔸)/KSO_{n−1}(ℝ)

and we describe a lifting of cohomology classes

Φ : H^{•}(e(P, K), M_{λ}) → H^{•}(e(P, K) × A_{P}^{0} , M_{λ}).

For Λ ∈ 𝔞_{P,ℂ}^{∗} we set

ω_{Λ}(g) = e^{⟨Λ+ρ̌,H_{P} (g)⟩}ω(g).

We know that ω ∈ H_{cusp}^{•}(^{0}S̃_{n−1}^{M_{P}}, M_{λ,ℂ}) is contained in some H^{•}(^{0}S̃_{n−1}^{M_{P}}, H_{w·λ,ℂ})(π′)_{ℂ} and Korollar 3.5 in [Sch] then implies that if we choose for Λ the linear form

Λ_{w} = −w(λ + ρ̌)|_{A_{P}^{0}}

we obtain a closed form

ω_{Λ_{w}} ∈ H^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ}).

The assignment ω ↦ ω_{Λ_{w}} defines the map Φ . We set

H_{cusp}^{r}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[w, π′]_{ℂ} = Φ(H_{cusp}^{r}(ẽ(P), M_{λ,ℂ})[w, π′]_{ℂ}).

This is a Hecke-invariant subspace of the cohomology of ẽ(P) × A_{P}^{0} .

Indeed, following [Sch], ch. 3.6 we see that

H_{cusp}^{r}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[w, π′]_{ℂ} ≅ H^{r−ℓ(w)}(𝔤_{n−1}, K_{n−1,∞}, I_{𝔸}^{a}(ℂ_{(w·Λ+ρ̌)_{𝔸}} ⊗ V (π′)) ⊗ M_{λ}).

Here, to be short, we have used the following notations: we abbreviate I_{𝔸_{f}} = Ind_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})} and we also set I_{ℝ} = Ind_{P (ℝ)}^{GL_{n−1}(ℝ)} and I_{𝔸} = Ind_{P (𝔸)}^{GL_{n−1}(𝔸)} . We use an analogous notation for the non-unitary induction. Moreover Λ_{𝔸} is define in (1.7) and by ℂ_{Λ_{𝔸}} we understand the 1 -dimensional space on which A_{P} (𝔸) acts via Λ_{𝔸} . Analogously we define ℂ_{Λ_{f}} and ℂ_{Λ} . In particular, since the cuspidal spectrum of M_{P} (𝔸) satisfies multiplicity-1 we deduce

(5.3)  H_{cusp}^{r}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[w, π′]_{ℂ} ≅ mult_{π′,∞} I_{P,Λ_{w},f}π′_{f}

where

mult_{π′,∞} = dim H^{r}(𝔤_{n−1}, K_{n−1,∞}Z_{n−1}^{0}(ℝ), I_{ℝ}(ℂ_{w·Λ} ⊗ π′_{∞}) ⊗ M_{λ})

(note that ρ̌|_{A_{P}^{0}} = ρ_{P} ).

Thus, to any choice of data P ∈ P , π′ ∈ L_{0}^{2}(M_{P} (ℚ)A_{P}^{0} \M_{P} (𝔸)) and w ∈ W^{P} we can assign a subspace in the cohomology of ẽ(P) × A_{P}^{0} . We want to apply this to construt non-trivial subspaces in H_{cusp}^{r}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ}) for a special weights λ attached to the critical points of cuspidal representations on GL_{n}(𝔸) : we let π be a cuspidal representation of GL_{n}(𝔸) for n even, satisfying (Coh) and (Reg) and η and idele class character of finite order and l ≤ 1/2 any critical point for η ⊗ π . Thus, in case n is even η does not matter. We denote by m = (m_{i}) and w the quantities attached to π as in section 4; depending on m and l we then choose a weight λ = (λ_{i}) ∈ ℤ^{n−1} such that the following conditions hold:

(W.1) λ is integral and dominant.

(W.2) The entries of λ satisfy

λ_{n−i} = −λ_{i}  for i = 1, …, n/2 − 1 if π is of type I
λ_{n−i} = −λ_{i} − 2  i = 1, …, n/2 − 1  π  II.

(W.3) m_{1} ≥ λ_{1} ≥ m_{2} ≥ λ_{2} ≥ ⋯ ≥ m_{n−1} ≥ λ_{n−1} ≥ m_{n} ,

(W.4)

λ_{n/2} = −l + 1/2

We want to verify that there is always a weight λ = (λ_{i}) satisfying (W.1) - (W.4) .

• π is of type I. We define λ = (λ_{i}) by

λ_{i} = \{ \begin{matrix}−l + 1/2 & if & i = n/2 \\ m_{i+1} + 1 & if & i = 1, …, n/2 − 1 \\ m_{i} & if & i = n/2 + 1, …, n − 1. \end{matrix}

Trivially (W.4) holds. Since m is of weight −1 λ satisfies (W.2) . Obviously, (W.1) holds for all λ_{i} , i ≥ n/2 and in view of (W.2) it holds for all i . It remains to verify (W.3) . Obviously, λ_{i} ≥ m_{i+1} for i = 1, …, n/2 − 1 and since m is regular we also have m_{i} ≥ λ_{i} for i = 1, …, n/2 − 1 . Furthermore, Remark 4.2 implies m_{n/2} ≥ λ_{n/2} ≥ 0 . (W.2) together with the fact that m has weight −1 and m_{i} − m_{i+1} ≥ 1 then shows that m_{i} ≥ λ_{i} ≥ m_{i+1} for all i = 1, …, n − 1 , i.e. (W.3) holds.

• π is of type II. We define λ by

λ_{i} = \{ \begin{matrix}−l + 1/2 & if & i = n/2 \\ m_{n/2} + n − 2 & if & i = n/2 − 1 \\ m_{i+1} + 1 & if & i = 1, …, n/2 − 2 \\ −λ_{n−i} − 2 & if & i = n/2 + 1, …, n − 1. \end{matrix}

Taking into account that m has weight 0 , m_{i} − m_{i+1} ≥ 4 and m_{n/2−1} − m_{n/2} ≥ n it is easily verified that (W.1) − (W.4) hold.

In this section only the first two properties of λ (and partly the fourth one) will play a role. Indeed, relying on (W.1) and (W.2) and following the construction described above we shall prove that for a suitable choice of ŵ ∈ W_{P} the spaces

H^{•}(ẽ(P) × A_{P}^{0} , M_{λ})[π̂[χ], ŵ]

do not vanish. Here, π̂[χ] is the representation defined in section 1 and π̂ is assumed to be of a special type at infinity. In the next subsection we shall exhibit certain classes ω_{λ,π′,ŵ} ∈ H^{b_{n−2}}(ẽ(P), M_{λ})[π′, ŵ] . The Eisenstein classes attached to these classes turn out to be vital for the calculation of the critical values of the automorphic L -function in the next section. This then will rest on (W.3) and (W.4) .

In detail, we let

• P ≤ GL_{n−1} be the parabolic subgroup of section 1, i.e. P is the maximal parabolic subgroup of type (n−2, 1) . The relative Weyl group consists of the elements W^{P} = {W_{M} ·w_{i}, i = 1, …, n − 1} , where w_{i} is the cycle

w_{i} = ( \begin{matrix}1_{i−1} \\ & 0 & 1 \\ & ⋮ & ... & ... \\ & 0 & ⋯ & 0 & 1 \\ & 1 & 0 & ⋯ & 0 \end{matrix} ) .

We let

• ŵ = w\frac{n}{2} .

Using (W.2) we obtain for the operation of ŵ on the weight λ

(5.4.I)  ŵ · λ = (λ_{1}, …, λ_{\frac{n}{2} −1}, −λ_{\frac{n}{2} −1} − 1, …, −λ_{1} − 1, λ\frac{n}{2} + \frac{n − 2}{2})

if π is of type I and

(5.4.II)  ŵ · λ = (λ_{1}, …, λ_{\frac{n}{2} −1}, −λ_{\frac{n}{2} −1} − 3, …, −λ_{1} − 3, λ\frac{n}{2} + \frac{n − 2}{2})

if π is of type II.

We denote by λ|_{1} = (λ_{1}, …, λ_{n−2}) and λ|_{2} = λ_{n−1} the restrictions of λ . The above expressions show that ŵ · λ|_{1} satisfies the purity condition with weight

w = \{ \begin{matrix}−1 & if & π & is of type & I \\ −3 & & π & & II. \end{matrix}

Hence, according to section 4 there is a cuspidal representation π̂ with non-trivial cohomology with coefficient system of weight w · λ|_{1} . In detail, as in section 4 we set

l′ = 2(ŵ · λ|_{1} + ρ̌) − w ∈ ℤ^{n−2}.

We remark that l′ = (l′_{i}) where

(5.5)  l′_{i} ∈ 2ℤ.

We choose a cuspidal automorphic representation π̂ of Z_{n−2}^{0}(ℝ)\GL_{n−2}(𝔸) such that π̂ satisfies (C.1) and

(5.6)  π̂_{∞} ≅ ϵ̂_{∞} ⊗ π_{∞}(w, l′)

where ϵ̂_{∞} = sgn^{k} with

k ≡ l − \frac{n − 1}{2} (mod 2).

Using (4.6) this is equivalent to

(5.7)  m_{ϵ̂_{∞}⊗π̂} = ŵ · λ|_{1},

where w is the weight of ŵ·λ|_{1} . In particular π̂ has non-vanishing cohomology. Since the group GL_{n−2} admits an involution of Cartan type, the (extension to non-trivial coefficients of the) results in [B-L-S] imply the existence of a cuspidal representation satisfying (5.6) (cf. [B-L-S], p. 34).

As in section 1 we define

• π̂[χ] = π̂ ⊗ χ.

As a consequence we mention for later use that the weights, which we contructed to prove the existence of weights satisfying (W.1) − (W.4) also satisfy

(W.5)  ŵ · λ|_{1} is regular.

Again, π_{0} = O_{n}(RR)/SO_{n}(ℝ) acts on the cohomology of the boundary via right translationand hence, H_{cusp}^{•}(ẽ(P)×A_{P}^{0} , M_{λ,ℂ}) has the structure of an GL_{n}(𝔸_{f})×π_{0} - module. We write sgn to denote the non-trivial character of π_{0} ≅ ℤ/2ℤ .

Lemma 5.1. We have

H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ} ≅ sgn^{k} ⊗ I_{P,Λŵ,f} π̂[χ],

where k ≡ −l + (n − 1)/2 (mod 2) .

Proof. In view of Eq. (5.3) to prove the Lemma we have to determine mult_{π̂[χ],∞} . We first determine the (^{0}𝔪, O_{n−1}(ℝ)^{M_{P}}) -cohomology of π̂[χ]_{∞} .

The discussion in section 4 implies that m_{π(w,l′)} = ŵ · λ|_{1} and we deduce from [C], Lemme 3.14 that π(w, l′) has non-vanishing cohomology with coefficients in M_{ŵ·λ|_{1}} ; in particular since π̂_{∞} = ϵ̂_{∞} ⊗ π_{∞}(w, l′) we obtain

H^{i}(𝔰𝔩_{n−2}, O_{n−2}(ℝ), π̂_{∞} ⊗ ϵ′_{1,∞} ⊗ M_{ŵ·λ|_{1}}) = \{ \begin{matrix}ℂ & if & i = b_{n−2} \\ 0 & if & i < b_{n−2} \end{matrix}

where ϵ′_{1,∞} = sgn^{k} with

k ≡ l − \frac{n − 1}{2} (mod 2)

and b_{n} is the bottom degree, in which cohomology of a cuspidal representation on GL_{n} can occur (cf. section 4). Similarly,

ŵ · λ|_{2} = −l + (n − 1)/2

and hence,

H^{i}(0, ±1, χ_{∞} ⊗ ϵ′_{2,∞} ⊗ x^{ŵ·λ|_{2}}) = \{ \begin{matrix}ℂ & if & i = 0 \\ 0 & if & i > 0, \end{matrix}

where ϵ′_{2,∞} = sgn^{k} with k satisfying

(5.8)  k ≡ −l + (n − 1)/2 (mod 2).

We set ϵ′_{∞} = ϵ′_{1,∞} ⊗ ϵ′_{2,∞} , which we view as character of M_{P} (ℝ) . If n is even we choose ϵ′_{1,∞} = ϵ′_{2,∞} . We then obtain ϵ′_{∞} = sgn^{k} , where k is as above.

The Lie algebra ^{0}𝔪 resp. the group O_{n−1}(ℝ)^{M} decompose ^{0}𝔪 = 𝔰𝔩_{n−2} × 0 and O_{n−1}(ℝ)^{M} = O_{n−2}(ℝ) × {±1} and applying the Kunneth rule yields an isomorphism

H^{•}(^{0}𝔪, O_{n−1}(ℝ)^{M}, π̂[χ]_{∞} ⊗ ϵ′_{∞} ⊗ M_{ŵ·λ}) ≅ H^{•}(𝔰𝔩_{n−2}, O_{n−2}(ℝ), π̂_{∞} ⊗ ϵ′_{∞} ⊗ M_{ŵ·λ|_{1}}),

which is canonical up to the choice of a generator of H^{0}(0, ±1, χ_{∞} ⊗ ϵ′_{2,∞} ⊗ ŵ · λ|_{2}) . In particular, we have

(5.9)  H^{i}(^{0}𝔪, O_{n−1}(ℝ)^{M_{P}} , π̂[χ]_{∞} ⊗ ϵ′_{∞} ⊗ M_{ŵ·λ}) ≅ \{ \begin{matrix}ℂ & if & i = b_{n−2} \\ 0 & if & i < b_{n−2} \end{matrix}

and using Wigner’s lemma we deduce

(5.10)  χ_{π̂[χ]_{∞}} = χ_{−ŵ·λ+ρ̌}|_{𝔟_{ℂ}},

where χ_{π̂[χ]_{∞}} denotes the infinitesimal character of π̂[χ]_{∞} . The second condition in [B-W], Theorem 3.3 (i) therefore is satisfied and since the inducing parameter is Λŵ = −ŵ(λ + ρ̌)|_{A_{P}^{0}} the first condition in [B-W], Theorem 3.3 (i) is trivially satisfied. We deduce form [loc. cit.]

H^{b_{n−2}+ℓ(ŵ)}(𝔤_{n−1}, O_{n−1}Z_{n−1}^{0}(ℝ), I_{ℝ}(ℂ_{Λŵ} ⊗ ϵ′_{∞} ⊗ π̂[χ]_{∞}) ⊗ M_{λ})

= ⨁_{i,j} H^{i}(^{0}𝔪, O_{n−1}^{M}(ℝ), ϵ′_{∞} ⊗ π̂[χ]_{∞} ⊗ M_{ŵ·λ}) ⊗ ⋀^{j} 𝔞_{P,ℂ}^{∗},

where i, j run over all non-negative integers satisfying i + j = b_{n−2} . Together with (5.9) we deduce mult_{π̂[χ],∞} = 1 . On the other hand, the assertion on the action of π_{0} is an immediate consequence of (5.8) and the proof of the Lemma is complete.

We denote by Eπ̂ the field of definition of π̂_{f} . Using [C], Théorème 3.13 the assumptions on π̂_{∞} imply that Eπ̂ is a finite extension of ℚ . Obviously, π̂[χ]_{f} is defined over Eπ̂(χ) .

Lemma. 5.2. Let λ be an integral and dominant weight. For any w ∈ W^{P} we have

H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{ℂ} = H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{E_{π′}} ⊗ ℂ,

H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{E_{π̂[χ]}}
= H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{ℂ} ∩ H^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,E_{π̂[χ]}}).

Proof. The weight λ is assumed to be integral, which implies that the character a ↦ |a|_{𝔸,f}^{Λ_{w}}, a ∈ S_{P} (𝔸) is ℚ -valued. Thus, since induction is rational, we see that Ind_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}(| · |_{𝔸,f}^{Λ_{w}}| ⊗ ϵ′_{f} ⊗ π̂[χ]_{f}) is defined over Eπ̂(χ) . The cohomology carries a structure of Aut(ℂ/ℚ) × GL_{n−1}(𝔸_{f}) -module and hence, the assertion of the Lemma follows from the following statement

(bMD) H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{ℂ} is a direct summand, which does not weakly intertwine with its complement.

This is the Manin-Drinfeld principle for the boundary (cf. [Ha 2]). We look to the cohomology of ẽ× A_{P}^{0} as module under the Hecke algebra. In the same way as in the case of the cohomology of ẽ(P) one establishes an isomorphism

H^{•}(ẽ(P) × A_{P}^{0} , M_{λ}) ≅ Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}H^{•}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , H^{•}(𝔫_{P} , M_{λ})).

The space H_{cusp}^{b_{n−2}}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , H_{w·λ})((Λ_{w} + ρ̌)_{𝔸} ⊗ π̂[χ]) appears as a direct summand in H^{b_{n−2}}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , H^{ℓ(w)}(𝔫_{P} , M_{λ})) and therefore

H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P)×A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{ℂ} = I_{𝔸_{f}}^{a} H_{cusp}^{b_{n−2}}(^{0}S̃_{n−1}^{M_{P}}×A_{P}^{0} , H_{w·λ})((Λ_{w} + ρ̌)_{𝔸}⊗π̂[χ])

is a direct summand in H^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ}) . Thus, we are left with proving that there is no weak intertwining with its complement.

We assume that there is an irreducible subquotient (ξ_{f}, V_{ξ_{f}}) in H^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ}) , which is isomorphic to a constituent of the Jordan-Hoelder series of I^{a} (Λ_{w} + ρ̌)_{f} ⊗ π̂[χ]_{f} . We denote by ξ the automorphic representation, whose component at infinity is isomorphic to ξ_{f} and V_{ξ} is the subquotient of the space of automorphic forms corresponding to ξ . Eq. (5.2) implies that there is a w′ ∈ W_{P} and a subquotient (ξ_{P,f}, V_{ξ_{P,f}}) in

H^{b_{n−2}+ℓ(w)−ℓ(w′)}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , H_{w′·λ,ℂ})

such that V_{ξ_{f}} is a subquotient of the induced representation Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})}V_{ξ_{P,f}} . The

cohomology of ^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} can be entirely described by automorphic forms

H^{•}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , H_{w′·λ,ℂ}) ≅ H^{•}(𝔪, K_{n−1,∞}^{M_{P}}, A(M_{P} (ℚ)\M_{P} (𝔸)) ⊗ H_{w′·λ})

and [L], Proposition 2 shows that there is a parabolic subgroup Q ≤ M_{P} and a cuspidal representation (ξ_{Q}, V_{ξ_{Q}}) , V_{ξ_{Q}} ⊂ L_{0}^{2}(M_{Q}(ℚ)\M_{Q}(𝔸)) of the Levi component M_{Q} of Q such that V_{ξ_{P,f}} is a subquotient of Ind^{a}_{Q(𝔸_{f})}^{M_{P} (𝔸_{f})} V_{ξ_{Q,f}} . Using induction in stages we

see that the representations Ind^{a}_{P (𝔸_{f})}^{GL_{n−1}(𝔸_{f})} (Λ_{w} + ρ̌)_{f} ⊗ π̂[χ]_{f} and Ind^{a}_{Q(𝔸_{f})}^{GL_{n−1}(𝔸_{f})} V_{ξ_{Q,f}} are weakly intertwining. Since these representations are admissible we deduce from [J-S] (4.4) Theorem that the pairs (P, | · |_{𝔸}^{Λ_{w}+ρ̌} ⊗ π̂[χ]) and (Q, ξ_{Q}) are associate. Since Q ≤ P and

W^{(M_{P})} = N_{GL_{n−1}}(M_{P})/M_{P} = 1

this implies P = Q and ξ_{Q} ≅ (Λ_{w} + ρ̌)_{𝔸} ⊗ π̂[χ] , i.e. V_{ξ_{P}} = (Λ_{w} + ρ̌)_{𝔸} ⊗ V_{π̂[χ]} . In other words we have

V_{ξ_{f}} ⊂ H^{b_{n−2}+ℓ(w)}(^{0}S̃_{n−1}^{M_{P}} × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w′].

Applying the Kunneth rule we see that

H^{•}(𝔪, K_{n−1,∞}^{M_{P}}, | · |^{Λŵ+ρ̌} ⊗ π̂[χ] ⊗ H_{w′·λ})

≅ H^{•}(^{0}𝔪, K_{n−1,∞}^{M_{P}}, π̂[χ] ⊗ H_{w′·λ}) ⊗ H^{•}(𝔪, K_{n−1,∞}^{M_{P}}, (Λŵ + ρ̌)_{𝔸} ⊗ H_{w′·λ}).

Since V_{ξ_{f}} does not vanish Wigner’s Lemma on the infinitesimal character immediately yields that

w′(λ + ρ̌)_{A_{P}^{0}} + Λŵ = 0

and similarly the non-vanishing of the subspaces in cohomology V_{ξ_{f}} and V_{π̂[χ]} implies

−χ_{w(λ+ρ̌)|_{𝔟_{ℂ}}} = χ_{π̂[χ]} = −χ_{w′(λ+ρ̌)|_{𝔟_{ℂ}}} = 0

(compare Eq. (5.10)). The regularity of λ + ρ̌ then leads to w = w′ , i.e. V_{ξ_{f}} is a subquotient of

H_{cusp}^{b_{n−2}+ℓ(w)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{ℂ}

This proves (bMD) and also the Lemma.

5.2. Definition of the class ω_{λ,π̂[χ],ŵ} .

We denote by W (τ) the space of Whittaker functions on GL_{n−1}(𝔸) with respect to τ and W (π, τ) ⊂ W (τ) is the π -isotypical component. We also denote by

\begin{aligned}β : & Aut(ℂ/ℚ) & → & Gal(ℚ̄/ℚ) & → & ℤ̂^{∗} \\ & σ & ↦ & σ|ℚ̄ & ↦ & t_{σ} = ∏_{v≠∞} t_{σ,v} \end{aligned}

the cyclotomic character and we set

t_{σ,n−1} = t_{σ} = diag(t_{σ}^{−(n−2)}, t_{σ}^{−(n−3)}, …, 1).

This defines a map

β_{n−1} : Aut(ℂ/ℚ) → T_{n−1}(ℤ̂).

The automorphism group Aut(ℂ/ℚ) acts on W (τ_{f})

wσ̃(g) = w(t_{σ}g)^{σ}

and since π̂[χ]_{f} is defined over E_{π̂[χ]} we obtain

W (π̂[χ]_{f}, τ_{f}) = W (π̂[χ]_{f}, τ_{f})_{E_{π̂[χ]}} ⊗ ℂ,

where the subscript E_{π̂[χ]} indicates the space of Aut(ℂ/E_{π̂[χ]}) -invariants. As a consequence we see that there is a Aut(ℂ/ℚ) -operation on induced representations: every σ ∈ Aut(ℂ/ℚ) defines a σ -linear map

\begin{aligned}σ̃ : & I_{𝔸_{f}}| · |_{𝔸,f}^{Λ} ⊗ W (π̂[χ]_{f}, τ_{f}) & → & I_{𝔸_{f}}| · |_{𝔸,f}^{Λ}W (π̂[χ]_{f}^{σ}, τ_{f}) \\ & f & ↦ & f σ̃(g) = (f(g))σ̃ = f(g; t_{σ}h_{1}, h_{2})^{σ}. \end{aligned}

In particular, I_{𝔸_{f}}(| · |_{𝔸,f}^{Λ} ⊗ W (π̂[χ]_{f}, τ_{f})_{E_{π̂[χ]}}) is a GL_{n−1}(𝔸_{f}) -invariant subspace. We recall that I_{P,Λŵ}^{⋆} π̂[χ] = I_{𝔸}| · |_{𝔸}^{Λŵ} ⊗ W (π̂[χ], τ) . We want to choose a generator ξ_{π̂,∞}^{⋆} of the 1 -dimensional vector space H^{b_{n−2}+ℓ(ŵ)}(𝔤_{n−1}, K_{n−1,∞}, I_{P,Λŵ,∞}^{⋆}π̂[1] ⊗ M_{λ}) . We consider the diagram

I_{P,Λŵ,f}^{⋆} π̂[1] = I_{𝔸_{f}} | · |_{𝔸,f}^{Λŵ} ⊗ W (π̂[1]_{f}, τ_{f}) ↓ ξ_{π̂,∞}^{⋆}

H^{b_{n−2}+ℓ(ŵ)}(𝔤, K_{n−1}^{∞}, I_{P,Λŵ} π̂[1] ⊗ M_{λ}) →^{P_{coh}^{⋆}^{−1}} H^{b_{n−2}+ℓ(ŵ)}(𝔤, K_{n−1}^{∞}, I_{𝔸}| · |^{Λŵ} ⊗ V_{π̂[1]} ⊗ M_{λ})
∥
H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[ŵ, π̂[1]].

Here, we have set 𝔤 = 𝔤_{n−1} ; the map P_{coh}^{⋆} on cohomology is induced by the map P^{⋆} defined in section 1, i.e. P_{coh}^{⋆} : ∑_{i} f_{i} ω_{i} ↦ ∑_{i} P^{⋆}(f_{i}) ω_{i} and the first isomorphism is determined by the choice of a generator ξ_{π̂,∞}^{⋆} . In particular, any choice of a system of generators ξ_{π̂,∞}^{⋆} defines a square of maps with the horizontal arrows being given by the composition of the maps in the above diagram

(⋆)  \begin{matrix}H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[ŵ, π̂[1]] & → & I_{P,Λŵ,f}^{⋆} π̂[1] \\ σ ↓ & & ↓ σ̃ \\ H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[ŵ, π̂[1]^{σ}] & →^{P_{coh}^{⋆}^{−1}∘ξ_{π̂^{σ},∞}^{⋆}} & I_{P,Λŵ,f}^{⋆} π̂[1]^{σ}. \end{matrix}

Here, we use the action of Aut(ℂ/ℚ) on the infinity component of the representation π̂[χ] (cf. [C], Définition 3.6); indeed, we have π[χ]_{∞}^{σ} = π[χ]_{∞} .

Lemma 5.3 The submodule ℂ[GL_{n−1}(ℚ_{v})] · f_{π̂[χ],Λ,v}^{∗} of I_{P,Λ,v}^{∗}π̂[χ] is defined over Eπ̂[χ]

ℂ[GL_{n−1}(ℚ_{v})] · f_{π̂[χ],Λ,v} = Eπ̂(χ)[GL_{n−1}(ℚ_{v})] · f_{π̂[χ],Λ,v} ⊗ ℂ.

Moreover, the Eπ̂(χ) -structure on ℂ[GL_{n−1}(ℚ_{v})] · f_{π̂[χ],Λ,v} is unique up to homotheties.

Proof. Analyzing the proof of Lemme I.1 in [W] one sees that to prove the Lemma it is sufficient that the following properties hold

(I_{P,Λ,v}^{∗}π̂[χ])^{K_{1}(n−1,f_{v})} = ⟨f_{π̂[χ],Λ,v}^{∗}⟩.

and f_{π̂[χ],Λ,v}^{∗} generates ℂ[GL_{n−1}(ℚ_{v})] · f_{π̂[χ],Λ,v}^{∗} . The second property is trivially satisfied, while the first one is immediate by the definition of the f_{π̂[χ],Λ,v}^{∗} . Thus, the Lemma is proven.

As a consequence of Lemma 5.3 we see that it is possible to choose the generators ξ_{π̂,∞}^{∗} in such a way that on the GL_{n−1}(𝔸_{f}) -submodules generated by f_{π̂[χ],Λ,f}^{∗} the diagram (⋆) is defined over Eπ̂(χ) and commutes.

Since χ_{∞} = id we deduce embeddings

I_{P,Λŵ,f}^{⋆} π̂[χ] → H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[π̂[χ], ŵ]

for all π̂[χ] = π̂ ⊗ χ . Since the spaces W (π̂[χ]_{f}, τ_{f}) = W (π̂_{f}, τ_{f}) ⊗ ℂ_{χ} and W (π̂[1]_{f}, τ_{f}) = W (π̂_{f}, τ_{f}) ⊗ ℂ_{1} can be canonically identified we obtain a natural identification

I_{P,Λŵ,f}^{⋆} π̂[χ] = I_{P,Λŵ,f}^{⋆} π̂[1],

which is equivariant for the action of Aut(ℂ/ℚ) . The diagram

I_{P,Λŵ,f} π̂[1] ξ_{π̂,∞}^{⋆}
∥  ↘
I_{P,Λŵ,f} π̂[χ] →^{ξ_{π̂,∞}^{⋆}} H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[π̂[χ], ŵ]

obviously commutes and we deduce that the map, which is induced by ξ_{π̂,∞}^{⋆} on I_{P,Λŵ,f}^{⋆} π̂[χ] too is Aut(ℂ/ℚ) -equivariant.

In section 1 we defined vectors f_{π̂[χ]}^{⋆} ∈ I_{P,Λŵ,f}^{⋆} π̂[χ] for all χ ∈ X_{p} . The mulitiplicity-1 property, which is satisfied for all finite places v and the equality w_{π̂,v}^{0}(t_{σ,v}) = w_{π̂,v}^{0}(1)^{σ} = 1 , which holds because t_{σ,v} ∈ K_{1}(n − 1, f_{v}) then show

(f_{π̂[χ],v}^{⋆})^{σ} = f_{π̂[χ]^{σ},v}^{⋆}

for all σ ∈ Aut(ℂ/E_{λ}) and all finite places v .

We now define

ω_{λ,π̂[χ],ŵ} = P_{coh}^{⋆}^{−1} ∘ ξ_{π̂,∞}^{⋆}(f_{π̂[χ]}^{⋆}) ∈ H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], w]_{Eπ̂(χ)}.

Obviously we have ω_{λ,π̂[χ],ŵ}^{σ} = ω_{λ,π̂[χ]^{σ},ŵ} .

Since the elements of the relative Weyl group have length

ℓ(w_{i}) = n − 1 − i

we see that ω_{λ,π̂[χ],ŵ} is a cohomology class in degree

b_{n−2} + ℓ(ŵ) = \frac{n^{2} − 2n}{4}.

We choose basis B_{n−1} = {ω_{1}, …, ω_{dim X_{n−1}^{1}}} of the dual (𝔤_{n−1}/𝔨_{n−1})^{∗} of the tangent space of the symmetric space X_{n−1}^{1} = GL_{n−1}^{+}(ℝ)/K_{n−1,∞} as well as a basis {b} of M_{λ} and we write

ξ_{π̂,∞}^{⋆} = ∑_{i=(i_{1},... ,i_{b_{n−2}+ℓ(ŵ)})} ∑_{b} f_{i,b,∞}^{⋆} ω_{i} ⊗ b.

Here, ω_{i} = ω_{i_{1}} ∧ … ∧ ω_{b_{n−2}+ℓ(ŵ)} . Following the above construction we then we find

ω_{λ,π̂[χ],ŵ} = ∑_{i,b} f_{π̂[χ],i,b} ω_{i} ⊗ b,

where we have set f_{π̂[χ],i,b}^{⋆} = f_{π̂[χ],f}^{⋆}f_{i,b,∞}^{⋆} and f_{π̂[χ],i,b} = P^{⋆}^{−1}(f_{π̂[χ],i,b}^{⋆}) .

5.3. Eisenstein cohomology.

Using the theory of Eisenstein series we may extend classes from the boundary to the whole space S_{n−1}(K) . For the moment we let P ≤ GL_{n−1} be any parabolic subgroup and π′ be a cuspidal representation in L_{0}^{2}(M_{P} (ℚ)\M_{P} (𝔸)) . The cylinder e(P, K) × A_{P}^{0} is then isomorphic to e(P, K) × A_{P}^{0} ≅ P (ℚ)\GL_{n−1}(𝔸)/KK_{n−1,∞} and we therefore may define the map

Eis : H^{•}(ẽ(P) × A_{P}^{0} , M_{λ}) → H^{•}(S̃_{n−1}, M_{λ})

by assigning to ω ∈ H^{•}(ẽ(P) × A_{P}^{0} , M_{λ}) the class

Eis(ω) = ∑_{γ∈P (ℚ)\GL_{n−1}(ℚ)}γ ∘ ω.

This operator is defined by analytic continuation except for possible poles which we want to discuss a little closer. Since we may realize the induced representation I_{𝔸}(| · |_{𝔸}^{Λ} ⊗ π′) , Λ ∈ 𝔞_{P,ℂ}^{∗} as subspace of C^{∞}(P (ℚ)\GL_{n−1}(𝔸)/K) we obtain a map

Eis_{P,π′,Λ} : H^{•}(𝔤_{n−1}, K_{n−1,∞}Z_{n−1}^{0}(ℝ), I_{𝔸}(| · |_{𝔸}^{Λ} ⊗ π′))
→ H^{•}(𝔤_{n−1}, K_{n−1,∞}, A(GL_{n−1}(ℚ)\GL_{n−1}(𝔸)).

It is a consequence of the general theory that Eis_{P,π′,Λ} only has a finite number of simple poles in the range

{Λ ∈ 𝔞_{P,ℂ}^{∗} : Im(Λ) = 0, ⟨ρ̌_{P} , αˇ ⟩ ≥ ⟨Λ, αˇ ⟩ > 0, α ∈ Δ(P, A_{P})},

where αˇ denotes the dual root. We go back to the situation of the previous subsection, i.e. from now on P ≤ GL_{n−1} denotes again the parabolic subgroup of type (n−2, 1) . A small calculation proves that

ŵ(λ + ρ̌) = (λ_{1} + \frac{n − 2}{2}, …, λ_{\frac{n}{2} −1} + 1, −λ_{\frac{n}{2} −1} − 1, …, −λ_{1} − \frac{n − 2}{2}, λ\frac{n}{2} )

if π is of type I and

ŵ(λ + ρ̌) = (λ_{1} + \frac{n − 2}{2}, …, λ_{\frac{n}{2} −1} + 1, −λ_{\frac{n}{2} −1} − 3, …, −λ_{1} − \frac{n + 2}{2}, λ\frac{n}{2} )

if π is of type II. In particular using (W.2) and (W.4) we deduce

(5.11)  Λŵ = \{ \begin{matrix}(0, l − \frac{1}{2}) & if & π & is of type & I \\ (n − 2, l − \frac{1}{2}) & & π & & II. \end{matrix}

The element Λŵ obviously is real. On the other hand since l ≤ 1/2 we have ⟨Λŵ, αˇ ⟩ > 0 for α ∈ Δ(P, A_{P}) . Our choice of the parabolic subgroup P together with Proposition 5.3 in [F-Sch] then implies that Eis_{P,π̂[χ],Λ} is regular at Λŵ . As a result we obtain the well defined map

(5.12) Eis_{P,π̂[χ],Λŵ} : H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P)×A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ} → H_{Eis}^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}).

Using this map we finally define a special class in the cohomology of S̃_{n−1} : we denote by ω_{Eis,π̂[χ],Λŵ} the image of ω_{λ,π̂[χ],ŵ} under the above map. Using Lie algebra cocycles we have

ω_{Eis,π̂[χ],Λŵ} = ∑_{i} E_{π̂[χ],i,b} ω_{i} ⊗ b,

where we have set E_{π̂[χ],i,b} = Eis(f_{π̂[χ],i,b}) .

For the later computation of the values of the L -function using cohomology we shall introduce the Whittaker function corresponding to the cohomology class ω_{Eis,π̂[χ],Λŵ} . We denote by v_{π̂[χ],i,b} the Whittaker function of E_{π̂[χ],i,b} and may then visualize our situation in the picture

f_{π̂[χ],i,b} ∈ I_{P,Λŵ} π̂[χ] →^{P_{π̂[χ]}^{⋆}} I_{P,Λŵ}^{⋆} π̂[χ]  ϶ f_{π̂[χ],i,b}^{⋆}
Eis ↓  ↓ P_{GL_{n−1}}^{P}
Eis(f_{π̂[χ],i,b}) ∈ V_{I_{P,Λŵ} π̂[χ]} →^{E↦E^{τ}} W (I_{P,Λŵ} π̂[χ], τ) ϶ v_{π̂[χ],i,b}.

This is the same diagram as (1.3) and we want to transfer it to cohomology. To this end we choose a generator ξ_{π̂,∞} of H^{b_{n−2}+ℓ(ŵ)}(𝔤_{n−1}, K_{n−1,∞}, I_{P,Λŵ,∞}^{⋆}π̂[χ] ⊗ M_{λ}) . We obtain a chain of maps

W (I_{P,Λŵ,f} π̂[χ])
↓ ξ_{π̂,∞}
H^{b_{n−2}+ℓ(ŵ)}(𝔤, K_{n−1}^{∞}, W (I_{P,Λŵ} π̂[χ], τ) ⊗ M_{λ}) →^{{E↦E^{τ} }^{−1}} H^{b_{n−2}+ℓ(ŵ)}(𝔤, K_{n−1}^{∞}, V_{I_{P,Λŵ} π̂[χ]} ⊗ M_{λ})
∩
H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}).

Again we choose the ξ_{π̂,∞} ’s such that the diagram

W (I_{P,Λŵ,f} π̂[χ], τ_{f})_{E_{π̂[χ]}} → H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})_{E_{π̂[χ]}}
σ ↓  ↓ σ
W (I_{P,Λŵ,f} π̂[χ]^{σ}, τ_{f})_{E_{π̂[χ]}} → H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})_{E_{π̂[χ]}}

is defined over E_{λ} . We write

ξ_{π̂,∞} = ∑_{i,b} v_{i,b,∞} ω_{i} ⊗ b

We then obtain a corresponding diagram in cohomology

f_{π̂[χ],f}^{⋆} ∈  I_{P,Λŵ,f}^{⋆} π̂[χ]  →^{P_{coh}^{⋆}^{−1}∘ξ_{π̂,∞}^{⋆}} H^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ})[ŵ, π̂[χ]]
P_{GL_{n−1},f}^{P} ↓  ↓ Eis
v_{π̂[χ],f} ∈ W (I_{P,Λŵ,f} π̂[χ], τ_{f})_{E_{π̂[χ]}} →^{{E↦E^{τ} }^{−1}∘ξ_{π̂,∞}}  H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})_{E_{π̂[χ]}},

which commutes at up to some scalar. More precisely, we define v_{π̂[χ],f} to be the finite part in the decomposition v_{π̂[χ],i,b} = v_{π̂[χ],f}v_{i,b,∞} ; the section f_{π̂[χ],f}^{⋆} is then being mapped to v_{π̂[χ],f} under the map given by the composition of the three arrows in the above diagram.

5.4. Non-triviality of ω_{Eis,π̂[χ],Λŵ} .

We proceed by computing the restriction of ω_{Eis,π̂[χ],Λŵ} to the boundary of S̃_{n−1} . As an application we shall derive the non-triviality of the Eisenstein cohomology class as well as obtain the rationality of ω_{Eis,π̂[χ],Λŵ} . Since the boundary is a disjoint union

∂S_{n−1}(K) = ⋃_{P ∈P,P ≠GL_{n−1}} e(P, K)

for any P ≠ GL_{n−1} there is a restriction map

r_{P} : H^{•}(S̃_{n−1}, M_{λ}) → H^{•}(ẽ(P), M_{λ}).

Let ω ∈ H^{•}(S̃_{n−1}, M_{λ}) . The space e(P, K) × Ā_{P}^{0} is a neighbourhood of E(P, K) in S_{n−1}(K) and the restriction ω|_{a·e(P,K)} does not depend on a ∈ A_{P}^{0} , hence we may set ω|_{e(P,K)} = ω|_{a·e(P,K)} for some a ∈ A_{P}^{0} . For any parabolic subgroup P′ in GL_{n−1} we define the constant term

ω|_{P′} = ∫_{N_{P′}(ℚ)\N_{P′}(𝔸)} ω(ng) dn.

In coordinates we have

ω|_{P′} = ∑_{i} f_{i}_{P′} ω_{i_{1}} ∧ … ∧ ω_{b_{n−2}+ℓ(ŵ)},

where ω = ∑_{i} f_{i} ω_{i} . The restriction is then given by

r_{P′}([ω]) = [ω_{P′}]|_{e(P,K)}

(cf. [Sch], 1.10 Satz).

We recall the general formula for the constant term. Let P ≤ GL_{n−1} a parabolic subgroup, π′ a cuspidal representation of the Levi component M_{P} and Λ ∈ 𝔞_{P,ℂ} . Let furthermore f_{Λ} be a section in I_{P,Λ}π′ and let P′ ≤ GL_{n−1} a parabolic subgroup. We denote by {P }_{ass} resp. {P }_{conj} the equivalence class of all parabolic subgroups P′ ≤ GL_{n−1} , which are associate resp. conjugate to P and W (A_{P} , A_{P′}) is the set of all inner automorphisms s : A_{P} → A_{P′} which induce an isomorphism M_{P} → M_{P′} . We then obtain

Eis_{P,π′,Λ}(f_{Λ})_{P′} = ∑_{s∈W (A_{P} ,A_{P′})} c(s, Λŵ)f_{Λ},

where c(s, Λ) is the standard intertwining operator. Representing cohomology classes by (𝔤_{n−1}, K_{n−1∞}) -cocycles we may write

Eis_{P,π′,Λ}(ω)|_{P} = ∑_{s∈W (A_{P} ,A_{P′})} c(s, Λ)ω_{Λ},

where ω_{Λ} ∈ H^{•}(𝔤_{n−1}, K_{n−1,∞}Z_{n−1}^{0}(ℝ), I_{𝔸}(Λ_{𝔸} ⊗ π′)) is as above and

c(s, Λ)ω_{Λ} = ∑ c(s, λ)f_{i} ω_{i}.

We want to apply this to compute the restriction of ω_{Eis,π̂[χ],Λŵ} to the boundary. We have {P_{(n−2,1)}}_{ass} = {P_{(n−2,1)}}_{conj} ∪ {P_{(1,n−2)}}_{conj} and W (P_{(n−2,1),}P_{(n−2,1)}) = {1} and W (P_{(n,1),}P_{(1,n)}) = {w_{l,n−1}} . Using the general theory of the constant term we see that ω_{Eis,π̂[χ],Λŵ}|_{P′} vanishes unless P′ = P_{(1,n−2)} or P′ = P_{(n−2,1)} and in these cases we obtain

ω_{Eis,π̂[χ],Λŵ}|_{P_{(n−2,1)}} = ω_{π̂[χ],Λŵ}

and

ω_{Eis,π̂[χ],Λŵ}|_{P_{(1,n−2)}} = c(w_{n−1,l}, Λ) ω_{π̂[χ],Λŵ}|_{P′}.

To put it into one line we have

⨁_{{P′∈P \atop P′≠GL_{n−1}}} r_{P′}ω_{Eis,π̂[χ],Λŵ} = ω_{π̂[χ],Λŵ}|_{e(P_{(n−2,1)},K)} + c(w_{n−1,l}, Λ) ω_{π̂[χ],Λŵ}|_{e(P_{(1,n−2)},K)}.

In particular, r_{∂S̃_{n−1}}ω_{Eis,π̂[χ],Λŵ} and hence ω_{Eis,π̂[χ],Λŵ} do not vanish.

As another application we want to prove that the map

Eis_{P,π̂[χ],Λŵ} : H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ} → H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})

already is defined over E_{π̂[χ]} . Let σ ∈ Aut(ℂ, E_{λ}) , i.e. M_{λ}^{σ} = M_{λ} . Since

(I^{a}(Λ_{ŵ,𝔸} ⊗ π̂[χ]))^{σ} ≅ I^{a}(Λ_{ŵ,𝔸} ⊗ π̂[χ]^{σ}),

the multiplicity-1 result on the isotypical component (cf. Lemma 5.1) implies

(H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ})^{σ} = H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ]^{σ}, ŵ]_{ℂ}.

We consider the diagram

H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ} →^{Eis_{P,π̂[χ],Λŵ}} H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})
σ ↓  σ ↓
H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ^{σ}]_{ℂ} →^{Eis_{P,π̂[χ]^{σ},Λŵ}} H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}).

The family of operators Eis_{P,π̂[χ],Λŵ} is defined over E_{λ} if and only if the above diagram commutes for all σ ∈ Aut(ℂ, E_{λ}) . To prove this, following [Ha 3], ch. 4.3 we look at the differnce Δ_{σ} = σ ∘ Eis_{P,π̂[χ],Λŵ} − Eis_{P,π̂[χ]^{σ},Λŵ} ∘ σ . If Δ_{σ} does not vanish it defines a GL_{n−1}(𝔸_{f}) -equivariant map

Δ_{σ} : H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{ℂ} → H_{∂S_{n−1}}^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}),

where H_{∂S_{n−1}}^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}) = ⋂_{{P′∈P \atop P′≠GL_{n−1}}} ker r_{P′} . The space of classes, which vanish

on the boundary decomposes

H_{∂S_{n−1}}^{•}(S̃_{n−1}, M_{λ,ℂ}) = H_{res}^{•}(S̃_{n−1}, M_{λ,ℂ}) ⊕ H_{cusp}^{•}(S̃_{n−1}, M_{λ,ℂ}).

On the other hand we assume that λ is regular and hence H_{res}^{•}(S̃_{n−1}, M_{λ,ℂ}) = 0 (cf. [F], Theorem 19.II). Thus, Δ_{σ} intertwines I_{𝔸_{f}}(Λ_{ŵ,𝔸} ⊗ π̂[χ]_{f}) with the space of cusp forms, which is impossible. We record our result:

Theorem 5.4. The operators

Eis_{P,π̂[χ],Λŵ} : H_{cusp}^{b_{n−2}+ℓ(ŵ)}(ẽ(P) × A_{P}^{0} , M_{λ,ℂ})[π̂[χ], ŵ]_{E_{π̂[χ]}} → H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,E_{π̂[χ]}})

are defined over E_{π̂[χ]} and for all σ ∈ Aut(ℂ/E_{λ}) we have (Eis_{P,π̂[χ],Λŵ})^{σ} = Eis_{P,π̂[χ]^{σ},Λŵ} .

Corollary 5.5. For all σ ∈ Aut(ℂ/E_{λ}) we have (ω_{Eis,π̂[χ],Λŵ})^{σ} = ω_{Eis,π̂[χ]^{σ},Λŵ} . In particular,

ω_{Eis,π̂[χ],Λŵ} ∈ H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,E_{π̂[χ]}}).

Proof. Immediate from (ω_{ŵ,π̂[χ]})^{σ} = ω_{ŵ,π̂[χ]^{σ}} .

6 The critical values of automorphic L -functions

In this last section we finally turn to the computation of the values L(π ⊗ χη, l) for critical pairs (η, l) . To this end we assume that n is even and π is a cuspidal representation of GL_{n}(𝔸) satisfying (Coh) and (Reg) . Thus, π has non-vanishing cohomology with regular coefficients. Our method is as follows. We select a weight λ such that the p -adic Mellin transform of dP_{η,l}^{Λŵ} computes the values of the L - function at critical pairs (η, l) . For these weights λ we then compute the values of dP_{η,l}^{π} .

6.1. The topological calculation of the distribution dP_{Λŵ,η}^{π} .

Let m and w be the quantities attached to π as in section 4. We let (η, l) be any critical pair for π , where we may assume that l ≤ 1/2 . Depending on m and l we fix a weight λ = (λ_{1}, …, λ_{n−1}) which satisfies the conditions (W.1) - (W.5) .

In this subsection we shall state the consequences of properties (W.3) and (W.4) of such a choice of λ .

• (W.4) yields the computation of the critical values of the L -function. We denote by v_{π̂[χ],i,b} = ⊗_{v}v_{π̂[χ],i,b,v} the Whittaker function of Eis(f_{π̂[χ],i,b}) . We define the distribution

dP_{Λŵ,η,π,π̂[χ]}^{coh} = ∑_{i,j} ∑_{a,b} sgn[i, j]tr(a ⊗ b) dP_{w_{∞,j,a},v_{∞,i,b}}^{Λŵ,η}

as well as the polynomial

P_{l,♭} = ∑_{i,j} ∑_{a,b} sgn[i, j] tr(a ⊗ b) P_{w_{j,a,∞},v_{i,b,∞}}^{η}.

Here, sgn[i, j] denotes the signum of the permutation [i, j] . Note that v_{i,b,∞} depends on λ , hence on l = λ_{n/2} . We will mostly drop the indices π, π̂[χ] to facilitate the notation. Eq. (5.11) together with Theorem 3.5 then yields

(6.1)  (γκq^{n−1})^{e}♣ ∫_{ℤ_{p}^{∗}} χ_{p}η_{p}^{n−1} dP_{Λŵ,η}^{coh} = P_{l,♭}(1/2)\frac{L(π ⊗ χη, l)}{L(π̂ ⊗ χ^{−1}, −l + 3/2 + Δ)},

where Δ = 0 in case I, Δ = n − 2 in case II. Thus, the p -adic Mellin transform of dP_{Λŵ,η}^{coh} computes the critical value L(π ⊗ χη, l) .

• (W.3) yields a cohomological interpretation of dP_{Λŵ,η}^{coh} . Using [F-H], 6.12 we immediately obtain (cf. [M])

Lemma 6.1. Let ρ|_{GL_{n−1}} be the restriction of ρ to GL_{n−1} . Then the restriction

ρ|_{GL_{n−1}} = ρ_{λ} ⊕ V

contains the representation ρ_{λ} with highest weight λ as a direct summand.

As a consequence we see that there is a non-trivial pairing tr : M̃_{m}|_{GL_{n−1}} × M_{λ} → ℂ . This pairing enables us to construct the distribution dP_{Λŵ,η}^{coh} in a topological way as follows. We introduce for any compact open subgroup K ≤ GL_{n−1}(ℤ̂) the manifold

F_{n}(K) = GL_{n}(ℚ)\GL_{n}(𝔸)/KK_{n,∞} ≅ ℝ_{+}^{∗} × S_{n}(K).

The embedding GL_{n−1} ↪ GL_{n} , g ↦ diag(g, 1) induces mappings

i : F_{n−1}(K(n − 1, N)) → S_{n}(K(n, N))

where K(n, N) ≤ GL_{n}(𝔸_{f}) denotes the principal congruence subgroup of level N as well as a map

j^{∗} : (𝔤_{n}/𝔨_{n})^{∗} → (𝔤_{n−1}/𝔰𝔬_{n−1})^{∗}

In particular, since i is proper we obtain a map in cohomology with compact supports

i^{∗} : H_{c}^{•}(S̃_{n}, M̃_{m}) → H_{c}^{•}(F̃_{n−1}, i^{∗}M̃_{m}).

To simplify the exposition we assume that the basis vectors ω_{i} and ω′_{j} are chosen so that

j^{∗}(ω′_{j}) = \{ \begin{matrix}ω_{j} & if & j ≤ dim X_{n−1} + 1 \\ 0 & else. \end{matrix}

We denote by p : F_{n−1}(K) → S_{n−1}(K) the canonical projection and by

p^{∗} : H^{•}(S̃_{n−1}, M_{λ}) → H^{•}(F̃_{n−1}, p^{∗}M_{λ})

the induced map on cohomology.

The assignment of finite dimensional representations M ↦ η ⊗ M induces an assignment of sheafs on S_{n−1}(K)

ψ : M ↦ M^{ψ}.

and hence a map

H^{•}(S̃_{n−1}, M) → H^{•}(S̃_{n−1}, M^{ψ})

which we denote as ω ↦ ψ ⊗ ω . We note that if ℚ(η) ≤ E we have M_{E}^{η} = M_{E} . In particular we obtain a class

η ⊗ ω_{Eis,π̂[χ],Λŵ} ∈ H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,E_{π̂[χ]}(η)}).

The pairing tr induces a pairing of the associated sheafs

tr : i^{∗}M̃_{m,ℂ} ⊗ p^{∗}M_{λ,ℂ} → \underline{ℂ}.

and the formulas for b_{n} as well as bŵ ; n−2 = b_{n−2} + ℓ(ŵ) show that

b_{n} + b_{n−2} + ℓ(ŵ) = \frac{n^{2} − n}{2} = dim(X_{n−1} × ℝ).

Together with the cup product we obtain the same diagram as in [M]

(6.2)

H_{c}^{b_{n}}(F̃_{n−1}, i^{∗}M̃_{m,ℂ}) × H^{bŵ ; n−2}(F̃_{n−1}, p^{∗}M_{λ,ℂ}) →^{tr∘∪} H_{c}^{dim F̃_{n−1}}(F̃_{n−1}, \underline{ℂ}) = ℂ
↑ i^{∗}  ↑ p^{∗}
H_{c}^{b_{n}}(S̃_{n}, M̃_{m,ℂ})  H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ}).

From this we deduce the existence of a pairing

\begin{aligned}⟨ , ⟩ : & H_{c}^{b_{n}}(S̃_{n−1}, M_{m,ℂ}) × H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−2}, M_{λ,ℂ}) & → & ℂ \\ & (ω, ω′) & ↦ & tr i^{∗}ω ∪ p^{∗}ω′. \end{aligned}

As in [M] we denote by

r_{u}^{∗} : H_{c}^{b_{n}}(S̃_{n}, M_{m}) → H_{c}^{b_{n}}(S̃_{n}, M_{m}),

the right translation with u ∈ GL_{n}(𝔸_{f}) . We set

u_{p^{e}} = ( \begin{matrix}1 & & & 0 \\ & ... & & ⋮ \\ & & 1 & 1/p^{e} \\ & & & 1 \end{matrix} ) .

The construction of the classes ω_{Eis,π̂[χ],Λŵ} applies to any representation π[μ] = π̂ ⊗μ , where μ ∈ X_{e} and μ has conductor f_{μ} = p^{e} . As in section 3 we extend this definition to all characters μ ∈ X_{e} and set

ω_{Eis,π(μ),Λŵ,p^{e}} = κ^{−(e−e_{μ})} r_{D^{−(e−eμ)}}^{∗}(ω_{Eis,π(μ),Λŵ}).

In the same way as in section 3 we define a Fourier transform

F : X_{e}^{H^{•}(S̃_{n−1},M_{λ})} → O_{e}^{H^{•}(S̃_{n−1},M_{λ})}

and we set

ω_{Eis,ε+p^{e}ℤ_{p},Λŵ} = F_{ε+p^{e}ℤ_{p}}(ω_{Eis,π̂⊗μ,Λŵ,p^{e}}).

All this is analogous to the constructions in [M] and with the same proof as in [M] we obtain the following description of the values of dP_{Λ_{hatw},η}^{coh} .

Proposition 6.2. For any ε ∈ ℤ_{p}^{∗} and e ≥ 2 we have

dP_{Λŵ,η}^{coh}(ε + p^{e}ℤ_{p}) = (γκq^{n−1})^{−e} ⟨η ⊗ ω_{Eis,ε+p^{e}ℤ_{p},Λŵ}, r_{u_{p^{e}}}^{∗}ω⟩.

Remark 6.3. Since the p -adic Mellin transform of dP_{Λŵ,η}^{coh} computes the values of our L -function (cf. Eq. (6.1)) this yields a topological calculation of special values of L(π, s) .

The group of connected components.– We consider the action of the group of connected components ℤ/2ℤ = O_{n−1}(ℝ)/SO_{n−1}(ℝ) = O_{n}(ℝ)/SO_{n}(ℝ) on dP_{Λŵ,η}^{coh} . We denote by −1 the non-trivial element in ℤ/2ℤ . Using differential forms we find dP_{Λŵ,η}^{coh}(ε + p^{e}ℤ_{p}) = J(ω, ω_{Eis,ε+p^{e}ℤ_{p},Λŵ}) , where

J(ω, ω_{Eis,ε+p^{e}ℤ_{p},Λŵ}) = ∫_{F_{n−1}(K)} ω ((\begin{matrix}g \\ & 1 \end{matrix}) u_{p^{e}}) ∧ ω_{Eis,ε+p^{e}ℤ_{p},Λŵ}(g).

Changing integration variables we see that

J(ω, ω_{Eis,ε+p^{e}ℤ_{p}}, Λŵ) = J(−1 · ω, −1 · ω_{Eis,ε+p^{e}ℤ_{p}}, Λŵ).

On the other hand recalling Eq. (4.7) and Lemma 5.1 we see that

−1 · r_{u_{p^{e}}}^{∗}ω ∧ −1 · ω_{Eis,ε+p^{e}ℤ_{p},Λŵ}, = (−1)^{k} r_{u_{p^{e}}}^{∗}ω ∧ ω_{Eis,ε+p^{e}ℤ_{p},Λŵ},

where

k ≡ ♭ + l + (n − 1)/2 (mod 2).

Hence, dP_{Λŵ,η}^{coh} and therefore P_{l,♭} vanish unless

♭ ≡ l + (n − 2)/2 (mod 2),

which is satisfied.

6.2. Definition of the complex period Ω(π) .

We denote by E_{m} the finite extension of ℚ over which the finite dimensional representation ρ : GL_{n} → M_{m} is defined and we denote by M_{m,E_{m}} ≤ M_{m} a GL_{n} -stable E_{m} -subspace. We let M_{m} be the sheaf on S_{n}(K) corresponding to M_{m,E_{m}} . Using Theorem 3.13 and 3.19 in [C] we see that π_{f} is defined over a finite extension E_{π} of ℚ (containing E_{m} ). Moreover π_{f} can be realized on a E_{π} -subspace in the Betti cohomology:

H^{•}(S̃_{n}, M_{m})(π_{f}) = H^{•}(S̃_{n}, M_{m})(π_{f})_{E_{π}},

where

H^{•}(S̃_{n}, M_{m})(π_{f})_{E_{π}} = H^{•}(S̃_{n}, M_{m})(π_{f}) ∩ H^{•}(S̃_{n}, M_{m,E_{π}}).

In the same way as in section 5 we define a twisted action of Aut(ℂ/ℚ) on the space of Whittaker functions on GL_{n}(𝔸_{f}) and obtain

W (π_{f}, τ_{f}) = W (π_{f}, τ_{f})_{E_{π}} ⊗ ℂ,

where W (π_{f}, τ_{f})_{π_{E}} is the space of Aut(ℂ/E_{π}) -invariants.

To define the complex periods Ω(π) in a sensful way we first have to normalize the choice of the generators ξ_{π,∞} . To this end we have to make the

Assumption. The special value P_{l,♭}(1/2) does not vanish.

Once this assumption is made we normalize our choose a scalar multiple of the generator ξ_{π,∞} such that P_{l,♭}(1/2) = 1 . We consider the diagram:

W (π_{f}, τ_{f}) →^{ξ_{π,∞}} H^{b_{n−1}}(S̃_{n}, M_{m,ℂ})(π)_{E_{π}}
σ̃ ↓  ↓ σ
W (π_{f}^{σ}, τ_{f}) →^{ξ_{π^{σ},∞}} H^{b_{n−1}}(S̃_{n}, M_{m,ℂ})(π^{σ})_{E_{π}}.

We choose Ω̃(π) ∈ ℂ^{∗} such that the arrows Ω̃(π)ξ_{π,∞} are defined over E_{π} and the above diagram commutes for all σ ∈ Aut(ℂ/E_{m}) . The existence of such a choice follows form [W], Lemme I.1 together with Schur’s lemma.

The weight λ and hence I_{P,Λŵ,∞}π̂[χ] and v_{π̂[χ],i,b} still depend on l . Therefore the normalization of the ξ_{π,∞} ’s and also Ω̃(π) depend on l . We write Ω̃_{l}(π) to indicate this.

An Application of the functional equation.– We want to reformulate Eq. (6.1) using the functional equation for the L -function attached to the representation π̂ ⊗ χ^{−1} . In its general form the functional equation reads

L(π̂ ⊗ χ^{−1}, s) = ε(π̂ ⊗ χ^{−1}, s)L(π̃′_{1} ⊗ χ, 1 − s).

The ε -factor decomposes

ε(π̂ ⊗ χ^{−1}, s) = (D_{F}^{n−2}N_{F} (f_{π̂⊗χ})^{1/2−s}W (π̂ ⊗ χ^{−1}).

where D_{F} is the discriminant and N_{F} the absolute Norm of the number field F . Since in our case F = ℚ and the conductors of π̂ and χ are coprime we have D_{F} = 1 and N_{F} (f_{π̂⊗χ}) = fπ̂f_{χ} . Moreover, since in addition χ is unramified at infinity we deduce from [Ba-Ra] Proposition 4.1

W (π̂ ⊗ χ^{−1}) = ωπ̂(f_{χ})χ(fπ̂)W (π̂)W (χ)^{n−2}.

Taking into account that W (χ) = f_{χ}^{−1/2}G(χ_{p}) we find

L(π̂ ⊗ χ^{−1}, l) = f 2l−3/2 ; π̂f_{χ}^{2l−n/2−1/2}ωπ̂(f_{χ})χ(fπ̂)W (π̂)G(χ_{p})^{n−2}L(π̃′_{1} ⊗ χ, 1 − l).

We note that the root number W (π̂) is contained in Eπ̂ and satisfies W (π̂)^{σ} = W (π̂^{σ}) . This is an easy consequence of [C], Lemme 4.6 and the local functional equation. Thus, except for the Gaussian sum, all terms appearing in the functional equation are contained in E_{π̂[χ]} and behave equivariantly under the action of Aut(ℂ/ℚ) .

Before replacing L(π̂ ⊗ χ^{−1}, −l + Δ) in Eq. () by the above expression we calculate the behaviour of the factor ♣ = ♣_{π̂,χ} under automorphisms. We recall the definition

♣ = c G(χ_{p}η_{p}^{n−1})^{−1}G(χ_{p})^{2−n}|f_{χ}|_{p}^{−1}v_{π̂[χ],S}(g_{S}^{0})^{−1}w_{π̂,p}^{0}(ϖ^{v′})^{−1}.

The Gaussian sum G(χ) is contained in ℚ(χ) . Moreover, w_{π̂,p}^{0} was defined the spherical Whittaker function satisfying w_{π̂,p}(e) = 1 . This uniqueness property immediately implies wσ̃ ; π̂,p = w_{π̂^{σ},p} for σ ∈ Aut(ℂ/ℚ) . Analogously, the considerations in section

4.3 show that v_{π̂[χ],v} ∈ W (π̂[χ]_{v}, τ_{v})_{E_{π̂[χ]}} and vσ̃ ; π̂[χ],v = v_{π̂[χ]^{σ},v} for all σ ∈ Aut(ℂ/ℚ)

and v ∈ S . We choose a finite abelian extension K = ℚ(ζ) large enough so that

Aut(ℂ/K) ⟶^{β_{n−1}} ∏_{v∈S}(K_{n−1,v}^{∗})^{g_{v}^{0}} ∩ T_{n−1}(ℤ̂) ≤ T_{n−1}(ℤ̂).

and calculate for σ ∈ Aut(ℂ/K) and v ∈ S

v_{π̂[χ],v}(g_{v}^{0})^{σ} = vσ̃ ; π̂[χ],v(t_{σ,n−1}^{−1}g_{v}^{0}) = v_{π̂[χ]^{σ},v}(g_{v}^{0}),

because v_{π̂[χ],v} is invariant on the right under K_{n−1,v}^{∗} . In particular, we obtain v_{π̂[χ],S}(g_{S}^{0}) ∈ KE_{π̂[χ]} . Since w_{π̂,p} is K_{n−2,v} -invariant we deduce in the same way that w_{π̂,p}(t)^{σ} = w_{π̂^{σ},p}(t) for all σ ∈ Aut(ℂ/ℚ) and t ∈ T_{n−2}(F_{p}) ; in particular w_{π̂,p}(ϖ^{v′}) ∈ Eπ̂ .

Enlargening K we may assume that ℚ(η) ≤ K and altogether we have seen that c^{−1}♣_{π̂,χ} ∈ KE_{π̂[χ]} and

(c^{−1}G(χ_{p})^{n−2}G(χ_{p}η_{p}^{n−1})♣_{π̂,χ})^{σ} = c^{−1}G(χ_{p}^{σ})^{n−2}G(χ_{p}^{σ}η_{p}^{n−1})♣_{π̂^{σ},χ^{σ}}

for all σ ∈ Aut(ℂ/K) . To complete the discussion of the factors appearing in the computation of the values of the L -function, we note that we may choose κ ∈ ℚ and that below we shall see that γ_{π} is contained in E_{π} , satisfying γ_{π}^{σ} = γ_{π^{σ}} for σ ∈ E_{m} . Thus combining (6.1) with the functional equation we find

(6.3) ∫_{ℤ_{p}^{∗}} χ_{p} dP_{Λŵ,η}^{coh}

= P_{l,♭}(1/2) ˆ♣ G(χ_{p})L(π × η ⊗ π̂[χ], 1/2 + Δ) \frac{L(π ⊗ χη, l)}{L(| · |^{−1/2} ⊗ π̃′_{1} ⊗ χ, l − Δ)},

where ˆ♣ = ˆ♣_{π,π̂[χ]} ∈ E_{π}E_{π̂[χ]} is equivariant with respect to the action of σ ∈ Aut(ℂ/E_{m}E_{λ}) .

We want to use Eq. (6.3) to deduce the properties of the critical values of the L - function L(π ⊗ ηχ, l) . To this end we have to make sure that all the special values of L -functions appearing in (6.3) are regular. First we note that all the L -functions are holomorphic in ℂ ; in case of the convolution this is proven in [M-W], Appendice. Also we may assume that L(π ⊗ ηχ, l) does not vanish. Eq (6.3) implies that the product of all three values is regular, hence we know that all three special values are regular as soon as we know that

L(π × η ⊗ π̂, 1/2 + Δ) ≠ 0.

• π is of type I. The value in question equals L(π ×|·|^{1/2} ⊗π̂, 0) , where |·|^{1/2} ⊗π is unitary. (C.2) implies that the local factors outside the places where π or π̂ ramify are equal to one. Hence, the functional equation together with non-vanishing of the convolution at s = 1 (cf. [Sha 3], 2.2, e) and h)) shows that L_{fin}(π × η ⊗ π̂, 1/2) does not vanish. Since the Γ -factors at infinity can not contribute a zero we see that L(π × ⊗π̂, 1/2) does not vanish.

• π is of type II. Since we assume n ≥ 3 the point 1/2 + Δ lies in the range of convergence of the Euler product, whence the claim.

The ratio of L -function on GL_{n} and GL_{n−2} .– We continue to cite from [M].

Lemma 6.4. There is a finite abelian extension K = ℚ(ζ) and a complex number Ω′ ∈ ℂ^{∗} such that Ω′w_{f} ∈ W (π_{f}, τ_{f})_{E_{π}(ζ)} and (Ω′w_{f})σ̃ = Ω′w_{f} for all σ ∈ Aut(ℂ/E_{m}(ζ)) . Here, w_{f} denotes the finite part of the Whittaker function defined in section 1. This in particular implies γ_{π} ∈ E_{π} and γ_{π}^{σ} = γ_{π^{σ}} for σ ∈ Aut(ℂ/E_{m}) .

Proof. This is the same proof as in [M].

We set Ω′_{l}(π) = Ω̃_{l}(π)Ω′ and as an immediate consequence we obtain

\frac{1}{Ω′_{l}(π)}ω ∈ H_{c}^{b_{n}}(S̃_{n}, M_{m,E_{π}(ζ)}).

On the other hand the averaging process over all characters μ ∈ X_{e} immediately yields

ω_{Eis,ε+p^{e}ℤ_{p},Λŵ} ∈ H^{b_{n−2}+ℓ(ŵ)}(S̃_{n−1}, M_{λ,ℂ})_{Eπ̂}.

We obtain

Proposition 6.5. There is a finite abelian extension K = ℚ(ζ) such that for any U ≤ ℤ_{p}^{∗} the following holds:

dP_{Λŵ,η}^{coh}(U) ∈ Ω′_{l}(π) · E_{π}Eπ̂(ζ)

and

{dP_{Λŵ,η,π,π̂[χ]}^{coh}(U)/Ω′_{l}(π)}^{σ} = dP_{Λŵ,η,π^{σ},π̂[χ]^{σ}}^{coh}(U)/Ω′_{l}(π^{σ})

for all σ ∈ Aut(ℂ/E_{m}E_{λ}(ζ)) .

Proof. This follows immediately from Proposition 5.3 and the fact that the diagram (6.2) defining ⟨ , ⟩ and hence the pairing ⟨ , ⟩ itself are defined over ℚ .

We define

Ω_{l}(π, π̂) = Ω′_{l}(π)L(π × η ⊗ π̂, 1/2 + Δ)

and we abbreviate

𝕃(π ⊗ χη, l) = \frac{G(χ_{p})^{[n/2]}}{Ω_{l}(π, π̂)} \frac{L(π ⊗ χη, l)}{L(| · |^{−1/2} ⊗ π̃′_{1} ⊗ χ, l − Δ)}.

Together with Eq. (6.3) we arrive at the following result on the special values of L -functions on GL_{n} .

Theorem 6.6. Let n be even and let π be a cuspidal representation of GL_{n}(𝔸) satisfying (Coh) and (Reg) . Let (l, η) , l ≤ 1/2 be a critical pair for π . Let furthermore λ ∈ ℤ^{n−1} be any weight satisfying (W.1) - (W.5) . Then there is a finite abelian extension K of E_{m}E_{λ} such that for all characters χ with infinity component χ_{∞} = id and conductor f_{χ} = p^{e} , e ≥ 2 the following holds:

𝕃(π ⊗ χη, l) ∈ KE_{π}Eπ̂(χ)

and for all σ ∈ Aut(ℂ/K) we have

𝕃(π ⊗ χη, l)^{σ} = 𝕃(π^{σ} ⊗ χ^{σ}η, l).

6.3. The induction step.

We want to apply Theorem 6.6 to inductively establish our final result on the values at critical points of the L -function.

Theorem 6.7 Let π be a cuspidal representation of GL_{n}(𝔸) , n even, which is unramified at 𝔭 and satisfies (Coh) and (Reg) . Let η : ℚ^{∗}\𝔸^{∗} → ℂ^{∗} be an idele class character of finite order and conductor satisfying (C.1) . Let l ≤ 1/2 be any critical point for η ⊗ π . Then there is a complex period Ω_{l}(π) ∈ ℂ^{∗} and a finite extension K/ℚ such that for all χ ∈ X_{𝔭}

\frac{L(π ⊗ ηχ, l)}{Ω_{l}(π)} ∈ KE_{m}

and for all σ ∈ Aut(ℂ/KE_{m}) we have

(G(χ)^{[n/2]} \frac{L(π ⊗ χη, l)}{Ω_{l}(π)})^{σ} = G(χ^{σ})^{[n/2]} \frac{L(π^{σ} ⊗ χ^{σ}η, l)}{Ω_{l}(π^{σ})}.

In detail, to prove the Theorem, we proceed as follows. We write m = m_{π} for the weight attached to π . We choose a weight λ = (λ_{1}, …, λ_{n−1}) satisfying (W.1) - (W.5) . Hence, λ depends on l and π . We then choose a cuspidal representation π̂ of GL_{n−2}(𝔸) satisfying (5.6) with respect to λ . If now | · |^{−1/2} ⊗ π̂[χ] again satisfies (W.1) - (W.5) resp. (Reg) and l turns out to be a critical point for | · |^{−1/2} ⊗ π̂[χ] we may apply Theorem 6.6 to inductively prove the assertion of the Theorem on the critical value L(π ⊗ ηχ, l) . Since n is even the induction is over all even integers n, n−2, …, 2 and we are reduced to the computation of L(π̂[χ]⊗χ, l) , where π̂[χ] is a cuspidal representation of GL_{2}(𝔸) with non-vanishing cohomology and l is critical for π̂[χ] . In this case the statement about the values at critical points is classical.

Thus it remains to verify that | · |^{−1/2} ⊗ π̂[χ] satisfies (Coh) and (Reg) and that l is a critical point for | · |^{−1/2} ⊗ π̂[χ] . We shall seperately treat the cases introduced in Definition 4.3, into which the representation π falls.

• π is of type I. By definition | · |^{−1/2} ⊗ π̂_{∞} = π(−1/2, l′) with l′ ≡ 0 (mod 2) , hence, | · |^{−1/2} ⊗ π̂ is of type I for GL_{n−2} . (W.5) immediately implies that (Reg) holds for π̂ . Moreover,

m_{|·|^{−1/2}⊗π̂} = w · λ|_{1} = (λ_{1}, …, λ_{\frac{n−2}{2}} , −λ_{\frac{n−2}{2}} − 1, …, −λ_{1} − 1).

Remark 4.2 then shows that all points l ∈ 1/2+ℤ in the range −λ_{n/2−1}+1/2 ≤ l ≤ 1/2 are critical for | · |^{−1/2} ⊗ π̂ . Since λ_{n/2−1} ≥ m_{n/2} and −m_{n/2} + 1/2 ≤ l ≤ 1/2 we deduce that l is critical for | · |^{−1/2} ⊗ π̂ .

• π is of type II. Again, |·|^{−1/2}⊗π̂ is of type I and (W.5) implies (Reg) . Moreover, l ≥ m_{n/2} + 1/2 implies that l − (n − 2) ≥ −m_{n/2} + 1/2 − (n − 2) = λ_{n/2−1} + 1/2 (compare the special choice of λ ) and hence l−(n−2) is a critical point for |·|^{−1/2}⊗π̂ by Remark 4.2.

type II  type I  n
↘  ↓
type I  n − 2

↓

⋮  ⋮

↓

type I  4

↓

type I  2

(n even)

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