TEST FUNCTIONS FOR SHIMURA VARIETIES: THE DRINFELD CASE

THOMAS J. HAINES

1. Introduction

Let (G, X, K) be a Shimura datum with reflex field E. Choose a prime number p and a prime ideal 𝔭 of E lying over p. Suppose that K_{p} ⊂ G(ℚ_{p}) is a parahoric subgroup. Also assume that G splits over ℚ_{p}^{un}, and hence that E_{𝔭} is an unramified extension of ℚ_{p}. Let S_{K} denote a model over O_{E,𝔭} of the corresponding Shimura variety. Then it is often the case that S_{K} has bad reduction. The singularities of the special fiber are very complicated in general, and somehow must be understood in order to study the local zeta function of the Shimura variety at 𝔭. Rapoport ([10]) has outlined a strategy to attack this problem. The first part is to find a convenient way to express

tr(F r_{q}; RΨ_{x_{0}}^{I}(\overline{ℚ}_{l})),

where RΨ(\overline{ℚ}_{l}) is the sheaf of nearby cycles on (S_{K})_{\overline{𝔽}_{q}}, q = p^{j} is such that ℚ_{p^{j}} contains E_{𝔭}, x_{0} ∈ S_{K}(𝔽_{q}), F r_{q} is the geometric Frobenius on (S_{K})_{\overline{𝔽}_{q}}, and I is the inertia subgroup of Gal(\overline{ℚ}_{p}/ℚ_{p}). (The study of this trace only directly relates to a semi-simplified version of the local zeta function, but it is nevertheless a first step towards the usual local zeta function.) The sheaf of nearby cycles can often be computed when one understands the global geometry of the special fiber, but in practice the global geometry of the special fiber is too complicated to be dealt with directly. To circumvent this problem Rapoport introduces a “local model” M^{loc} over O_{E,𝔭} and a procedure that attaches to x_{0} ∈ S_{K}(𝔽_{q}) a point x in M^{loc}(𝔽_{q}). We consider now the special case where K_{p} is an Iwahori subgroup. The local model then has a stratification indexed by certain elements of the extended affine Weyl group of G (conjecturally the μ-admissible set; see definition in §2.1). Although the point x is not uniquely determined by x_{0}, it is contained in a well-defined stratum; we can thus also use the symbol x to denote the element of the extended affine Weyl group corresponding to this stratum. The local model at x is locally isomorphic to the special fiber at x_{0}, so by transport of structure we see that we need an expression for the trace of Frobenius on RΨ_{x}^{I}(\overline{ℚ}_{l}). This should have a purely group-theoretic interpretation, if we are eventually going to use the Arthur-Selberg trace formula to express the zeta function in terms of automorphic L-functions.

Such an interpretation has been conjectured by R. Kottwitz. To give his prescription we must fix an unramified extension F containing E_{𝔭} and assume that G is quasisplit over F. Associated to X is a minuscule cocharacter μ (defined up to conjugacy) of the group G_{\overline{E}_{𝔭}}, and by definition of E the conjugacy class of μ is defined over E_{𝔭}. Because G_{F} is quasisplit we can consider μ as a well-defined element of X_{∗}(A)/W_{0}, where A is a maximal F -split torus of G_{F} and W_{0} is the relative Weyl group of G_{F} (see Lemma 1.1.3 of [5]). Let q = p^{j} denote the cardinality of the residue field of F, and let z_{μ} denote the Bernstein function corresponding to μ, which is an element of the center of the Iwahori-Hecke algebra of the p-adic group G(F)

(see definition in §2). Then z_{μ} is a linear combination (over ℤ[q^{1/2}, q^{−1/2}]) of the generators T_{w}, where T_{w} denotes the characteristic function of the Iwahori double coset corresponding to the element w of the extended affine Weyl group of G_{F}. Denote the coefficient of T_{w} by z_{μ}(w). Then we have the following

Conjecture (Kottwitz): If K_{p} is an Iwahori subgroup and x is an 𝔽_{q}-rational point of the local model M^{loc} corresponding to the Shimura datum (G, X, K), then

tr(F r_{q}; RΨ_{x}^{I}(\overline{ℚ}_{l})) = q^{dim(S_{K})/2}z_{μ}(x).

If q^{dim(S_{K})/2}z_{μ} has this property, then it follows that it is the correct “test function” for the Shimura variety, meaning that it gives the function at the prime p which is “plugged into” the twisted orbital integrals that come into the computation of the semi-simple local zeta function when one attempts to use the Arthur-Selberg trace formula.

There is also a version of this conjecture for the general parahoric case: the local model then has a stratification by certain “μ-admissible” elements of the double coset space W_{J}\W̃/W_{J} (here W_{J} is a parabolic subgroup of the extended affine Weyl group W̃). In the statement above the Bernstein function z_{μ} needs to replaced with its image in the parahoric Hecke algebra.

The main result of this paper (Theorem 4.3) is a formula for z_{μ} where μ is any minuscule coweight of any reduced root system. It applies to Hecke algebras with arbitrary parameters (see definition in §2). For simplicity we state the result in the special case corresponding to a split group (the parameter system is then given by L(s) = 1 for all s ∈ S_{a}; see notation in §2).

Theorem 1.1. Let μ be a dominant minuscule coweight of a split connected reductive p-adic group G with root system (X^{∗}, X_{∗}, R, Ř, Π). Let W̃ = X_{∗} ⋊ W denote the extended affine Weyl group of G. Let z_{μ} denote the Bernstein function corresponding to μ. Then

q^{l(a^{μ})/2}z_{μ} = ε_{a^{μ}} ∑_{x : x is μ-adm.} ε_{x}R_{x,a^{t(x)}}(q)T_{x},

where ε_{y} = (−1)^{l(y)}, x has decomposition x = wa^{t(x)} (w ∈ W, t(x) ∈ X_{∗}), and R_{x,a^{t(x)}}(q) is the R-polynomial attached to W̃ in Kazhdan-Lusztig theory (cf. §2).

In this paper we use Theorem 1.1 to prove Kottwitz’ conjecture for a certain class of Shimura varieties with Iwahori type reduction. We examine Shimura varieties attached to the group of unitary similitudes GU(1, d−1) defined by an imaginary quadratic extension E of ℚ in which the prime p splits (the “Drinfeld case”). In this case Rapoport gives formulae for the trace of the Frobenius F r_{q} on the nearby cycles on the corresponding local model ([10] and [9]), the formulae being very explicit under some assumptions on the number q = p^{j} (Proposition 5.1 of §5). Under the same assumptions on q = p^{j} the group G_{ℚ_{p^{j}}} is split and we can use Theorem 1.1 to give an explicit formula for the coefficients q^{l(a^{μ})/2}z_{μ}(x) (comp. Proposition 5.2). Comparing the two explicit formulae, one sees that Kottwitz’ conjecture holds in this case. (Theorem 1.1 is all that is needed for this particular application, because the group in question is split. However, in cases where the group is quasisplit but nonsplit we need to allow for arbitrary parameters and the more general Theorem 4.3 is necessary.)

More precisely, fix an integer d > 2 and let (D, ∗) denote a central simple algebra D of dimension d^{2} over an imaginary quadratic extension E of ℚ, together with a positive involution ∗ which induces the nontrivial automorphism on E. Suppose the prime p splits in E as the product 𝔭_{1}𝔭_{2}, where 𝔭_{1} is the prime ideal of E distinguished in §5. Let G denote the corresponding ℚ-group (defined in §5), and let K = K^{p}K_{p}, where K_{p} is an Iwahori subgroup of G(ℚ_{p}). Let X be chosen as in §5, so that G_{ℝ} is the group GU(1, d−1). With this choice of X the corresponding G(\overline{ℚ}_{p})-conjugacy class \{μ\} of cocharacters of G(\overline{ℚ}_{p}) can be “identified” in a certain precise sense with the cocharacter (1, 0^{d−1}) of Gl_{d}, if we assume that q = p^{j} is such that j · (inv(D_{𝔭_{1}}) ≡ 0 (mod d). Indeed, let F = ℚ_{p^{j}}, where j is chosen as above. Then via the isomorphism of F -groups G_{F} = Gl_{d} × 𝔾_{m} described in §5, we may represent {μ} using the cocharacter μ = (1, 0^{d−1})t where t denotes the cocharacter x ↦ x in X_{∗}(𝔾_{m}) (see §5).

Let S_{K} denote a model over O_{E_{𝔭_{1}}} of the Shimura variety determined by the datum (G, X, K). A first step towards understanding the semi-simple local zeta function of S_{K} is provided by the following theorem.

Theorem 1.2. Let (G, X, K) denote the Shimura datum above, and let M^{loc} denote the corresponding local model. Let j be such that j · (inv(D_{𝔭_{1}})) ≡ 0 (mod d). Then Kottwitz’ conjecture holds for M^{loc} and q = p^{j}.

Corollary 1.3. If D_{𝔭_{1}} is a matrix algebra, then Kottwitz’ conjecture holds for Mloc and any q.

One can also use Theorem 1.1 to predict the trace of Frobenius on nearby cycles for the local models attached to other Shimura varieties. When a calculation of this trace is possible, Kottwitz’ conjecture can be verified using a stratum-by-stratum comparison of explicit formulae, as in Theorem 1.2. For example, consider the Shimura variety with Iwahori type reduction attached to the group GU(2, 3) determined by an imaginary quadratic extension E of ℚ in which the prime p splits. This comes from a central simple algebra D over E as above. We again consider only j such that the condition in Theorem 1.2 holds. This means that we are essentially dealing with the local model M^{loc} attached to GL_{5} and μ = (1, 1, 0, 0, 0) (see the proof of Theorem 1.2 in §5). Let τ ∈ W̃ denote element indexing the “most singular stratum” of the local model; it turns that a^{t(τ)} = a^{(0,0,0,1,1)} = s_{3}s_{4}s_{2}s_{3}s_{1}s_{2}τ (here s_{i} is the transposition (i i + 1), for 1 ≤ i ≤ 4). Then Theorem 1.1 and Kottwitz’ conjecture predict that tr(F r_{q}; RΨ_{τ}^{I}(\overline{ℚ}_{l})) is

ε_{τ} ε_{a^{μ}}R_{τ,s_{3}s_{4}s_{2}s_{3}s_{1}s_{2}τ} (q) = (1 − q)^{4}(1 + q^{2}).

A calculation by U. Görtz of the trace of Frobenius on nearby cycles for this “most singular point” produced exactly this expression. Also, Görtz calculated the trace of Frobenius on the nearby cycles for all 33 strata for the local model attached to Gl_{4} and μ = (1, 1, 0, 0). Comparing the results with the formula for z_{μ} in Theorem 1.1, he verifed that Kottwitz’ conjecture holds for the local model of the Shimura variety with Iwahori type reduction attached to the group GU(2, 2) determined by an imaginary quadratic extension of ℚ in which p splits.

We now outline the contents of the paper. In §2, we give further notation and prove some elementary lemmas. In §3 we present an efficient method of computing Bernstein functions for minuscule coweights as linear combinations of the basis elements T̃_{w}, w ∈ W̃ (or equivalently, the elements T_{w}). In §4 we prove the main theorem (Theorem 4.3). Moreover we prove that when μ is minuscule, the support of z_{μ} is precisely the μ-admissible set (Proposition 4.6). In §5 we discuss the Shimura varieties in the Drinfeld case, and deduce the truth of Kottwitz’ conjecture for the special case in Theorem 1.2 above, using Rapoport’s formulae (Proposition 5.1) and the explicit formula for z_{μ} in this case (Proposition 5.2).

2. Notation

For the most part we will use the notation in [8], except that the affine Weyl groups and Hecke algebras we consider will be “dual” to Lusztig’s.

2.1. The Affine Weyl Group of a Root System. Let (X^{∗}, X_{∗}, R, Ř, Π) be a (based) root system, where Π denotes the simple positive roots. Let R^{+} (resp. R^{−}) denote the set of positive (resp. negative) roots; we often use α > 0 to denote α ∈ R^{+}. We assume throughout this paper that the root system is reduced.

Corresponding to α ∈ Π we have the simple reflection s_{α}, acting on X_{∗} (resp. X^{∗}) by s_{α}(x) = x − ⟨α, x⟩α̌ (resp. s_{α}(y) = y − ⟨y, α̌⟩α). The Weyl group W_{0} is the subgroup of GL(X_{∗}) (or GL(X^{∗})) generated by S = {s_{α} | α ∈ Π}. It is known that (W_{0}, S) is a finite Coxeter group.

Let ⪯ denote the partial order on X_{∗} (resp. X^{∗}) defined by λ ⪯ μ ⇔ μ − λ is a linear combination with ≥ 0 integer coefficients of elements of {α̌ | α ∈ Π} (resp.{α | α ∈ Π}) . Let Π_{m} denote the set of β ∈ R such that β is a minimal element of R ⊂ X^{∗} with respect to ⪯. If the root system is irreducible, Π_{m} = {−α̃}, where α̃ is the unique highest root.

Let W̃ be the semidirect product W_{0}⋉X_{∗} = {wa^{x} | w ∈ W_{0}, x ∈ X_{∗}} (a is a fixed symbol). The multiplication is given by w′a^{x′}wa^{x} = w′wa^{w^{−1}(x′)+x}. Define a function l : W̃ → ℤ by the formula

l(wa^{x}) = ∑_{α∈R^{+} : w(α)∈R^{−}} |⟨α, x⟩ + 1| + ∑_{α∈R^{+} : w(α)∈R^{+}} |⟨α, x⟩|.

Define an action of W̃ on X^{∗}×ℤ by wa^{x}(y, k) = (w(y), k−⟨y, x⟩). Let R̃ = R̃^{+}∪R̃^{−} ⊂ X^{∗}×ℤ be defined by

R̃^{+} = {(α, k) | α ∈ R, k > 0} ∪ {(α, 0) | α > 0},
R̃^{−} = {(α, k) | α ∈ R, k < 0} ∪ {(α, 0) | α < 0}.

Let

Π̃ = {(α, 0) | α ∈ Π} ∪ {(α, 1) | α ∈ Π_{m}} ⊂ R̃^{+},
S_{a} = {s_{α} | α ∈ Π} ∪ {s_{α}aα̌ | α ∈ Π_{m}} ⊂ W̃.

There is a bijection Π̃ ↔ S_{a} (A ↔ s_{A}).

Let

X_{dom} = {x ∈ X_{∗} | ⟨α, x⟩ ≥ 0, ∀α ∈ Π}
= {x ∈ X_{∗} | l(s_{α}a^{x}) = l(a^{x}) + 1, ∀α ∈ Π}.

Note that w ∈ W_{0}, x ∈ X_{dom} ⇒ l(wa^{x}) = l(w) + l(a^{x}).

Let Q̌ denote the subgroup of X_{∗} generated by Ř. Then the subgroup W_{a} = W_{0}Q̌ of W̃ is a Coxeter group with S_{a} the set of simple reflections, the length function being the restriction of l. This subgroup is normal in W̃ and admits a complement Ω = {w ∈ W̃ | w(Π̃) = Π̃} = {w ∈ W̃ | l(w) = 0}. It is known that Ω is an abelian group isomorphic to X_{∗}/Q̌.

We use ≤ to denote the Bruhat order on W̃. This is defined on the Coxeter group (W_{a}, S_{a}) as usual, and it is then extended to W̃ by declaring xτ ≤ x′τ′ (x, x′ ∈ W_{a}, τ, τ′ ∈ Ω) if x ≤ x′ and τ = τ′.

Definition 2.1. For μ ∈ X_{dom}, we say x ∈ W̃ is μ-admissible if x ≤ a^{w(μ)} for some w ∈ W_{0}. The set of all such x’s is called the μ-admissible set.

2.2. The Hecke Algebra. Let B be the group with generators T_{w}, (w ∈ W̃) and relations

T_{w}T_{w′} = T_{ww′} whenever l(ww′) = l(w) + l(w′).

We call B the braid group of W̃. For any x ∈ X_{∗} we define an element \overline{T}_{x} = T_{a^{x_{1}}} T_{a^{x_{2}}}^{−1} , where x = x_{1} − x_{2} and x_{1}, x_{2} ∈ X_{dom}. This is independent of the choice of the x_{i}.

Fix a parameter set L : S_{a} → ℕ for the root system. This means that L(s) = L(s′) whenever s , s′ ∈ S_{a} are conjugate in W̃. Equivalently, L is the restriction to S_{a} of a function L′ : W̃ → ℕ with L′(ww′) = L′(w) + L′(w′) whenever w , w′ ∈ W̃ satisfy l(ww′) = l(w) + l(w′). We denote both functions simply by L. We also use to L to denote the unique homomorphism L : B → ℤ such that L(T_{w}) = L(w).

Let v be an indeterminate (thought of as q^{1/2}), and let ℤ′ = ℤ[v, v^{−1}]. The Hecke algebra H is by definition the quotient of the group algebra of B (over ℤ′) by the two sided ideal generated by the elements

(T_{s} + 1)(T_{s} − v^{2L(s)}), s ∈ S_{a}.

The image of T_{w} (resp. \overline{T}_{x}) in H is still denoted by T_{w} (resp. \overline{T}_{x}).

Remark 2.2. Let G be a split connected reductive group over a p-adic field F, with root system (X^{∗}, X_{∗}, R, Ř, Π). Suppose q is the size of the residue field of F, and let v = q^{1/2}. Let L be the parameter set given by L(s) = 1 for every s ∈ S_{a}. Choose an Iwahori subgroup I ⊂ G(F) whose “reduction mod p” is the Borel corresponding to the choice of simple positive roots Π. Define convolution in the algebra C_{c}(I\G(F)/I) using the Haar measure on G(F) which gives I measure 1. Then there is a canonical isomorphism of ℤ[q^{1/2}, q^{−1/2}]-algebras

H ≅ C_{c}(I\G(F)/I).

Define for any w ∈ W̃ a renormalization T̃_{w} = v^{−L(w)}T_{w}. The elements T_{w} (resp. T̃_{w}) (w ∈ W̃) form a ℤ′-basis for H. For x ∈ X_{∗}, define

Θ_{x} = v^{−L(\overline{T}_{x})}\overline{T}_{x} = v^{−L(x_{1})+L(x_{2})}T_{a^{x_{1}}} T_{a^{x_{2}}}^{−1} = T̃_{a^{x_{1}}} T̃_{a^{x_{2}}}^{−1} ,

where x = x_{1} − x_{2}, x_{i} ∈ X_{dom}. It is known that the elements Θ_{x}T_{w} (x ∈ X_{∗}, w ∈ W_{0}) form a ℤ′-basis for H ([8], Prop. 3.7).

Definition 2.3. For each W_{0}-orbit M in X_{∗} define the Bernstein function attached to M by z_{M} = ∑_{λ∈M} Θ_{λ}.

When the W_{0}-orbit M contains the dominant element μ, this function will usually be denoted by z_{μ}.

The following theorem is due to Bernstein in the special case where L(s) is independent of s and the roots generate a direct summand of X^{∗}, and to Lusztig in general (see [8], Prop. 3.11):

Theorem 2.4. (Bernstein, Lusztig) Let Z(H) denote the center of H. Then Z(H) is the free ℤ′-module with basis z_{M}, where M runs over the W_{0}-orbits in X_{∗}.

Define for each s ∈ S_{a} an indeterminate Q_{s} = v^{−L(s)} − v^{L(s)}. With the normalizations above, the usual relations in H can be written simply as

T̃_{s}T̃_{w} = \begin{cases}T̃_{sw}, & if l(sw) = l(w) + 1, \\ −Q_{s}T̃_{w} + T̃_{sw}, & if l(sw) = l(w) − 1, \end{cases}

if w ∈ W̃ and s ∈ S_{a} (and similarly for T̃_{w}T̃_{s}). In particular, T̃_{s}^{−1} = T̃_{s} + Q_{s}.

For any y ∈ W̃, choose a reduced expression y = s_{1} ⋯ s_{r}τ, (s_{i} ∈ S_{a}, τ ∈ Ω). Then for any x ∈ W̃ we can define a polynomial expression R̃_{x,y}(Q_{S}) in variables Q_{S} = {Q_{s}, s ∈ S_{a}}, by the formula

T̃_{y^{−1}}^{−1} = ∑_{x∈W̃} R̃_{x,y}(Q_{S})T̃_{x}. (1)

The coefficient of T̃_{x} appearing in the above expression is a priori just an element of the ring ℤ′ = ℤ[v, v^{−1}]. However it is clear that we can also think of it as a polynomial expression in the indeterminates Q_{s} because of the identity

T̃_{y^{−1}}^{−1} = (T̃_{s_{1}} + Q_{s_{1}}) ⋯ (T̃_{s_{r}} + Q_{s_{r}})T̃_{τ} . (2)

The expression R̃_{x,y}(Q_{S}), viewed as an element of ℤ′, does not depend on the choice of reduced expression for y, although viewed formally as a polynomial in |S_{a}| variables, it does depend on the choice of reduced expression. The fact that the indeterminates Q_{s} are not independent (indeed not even distinct) variables will not affect any of the arguments we make using them. It will only be necessary sometimes to verify that some polynomial expressions in Q_{S} are not zero in the ring ℤ′.

These R̃-functions are analogous to the R-polynomials introduced in [4]. These are defined in the context of an affine Hecke algebra with trivial parameters, meaning that we take the parameter set L given by L(w) = l(w), (w ∈ W̃). In this case set v = q^{1/2} and Q = q^{−1/2} − q^{1/2} = Q_{s}, (s ∈ S_{a}). Following [4] define R_{x,y}(q) by the equation

T_{y^{−1}}^{−1} = ∑_{x} ε_{x}ε_{y}q^{−l(y)}R_{x,y}(q)T_{x},

where ε_{x} = (−1)^{l(x)}. Then we have

ε_{x}ε_{y}R_{x,y}(q) = q^{(l(y)−l(x))/2}R̃_{x,y}(Q).

It is easy to prove the following facts.

Lemma 2.5. For x, y ∈ W̃ and s ∈ S_{a} we have 1. sx < x, sy < y ⇒ R̃_{x,y}(Q_{S}) = R̃_{sx,sy}(Q_{S}),

2. x < sx, sy < y ⇒ R̃_{x,y}(Q_{S}) = Q_{s}R̃_{x,sy}(Q_{S}) + R̃_{sx,sy}(Q_{S}),

3. R̃_{x,y}(Q_{S}) ∈ ℤ_{+}[Q_{S}],

4. deg_{Q_{S}} R̃_{x,y}(Q_{S}) = l(y) − l(x) if x ≤ y,

5. R̃_{x,y}(Q_{S}) ≠ 0 in ℤ′ ⟺ x ≤ y.

Proof. The first two statements are consequences of the definition, and these immediately imply (3) and (4) by induction on l(y). Finally (5) (⇒) is easy using the equation (2) above. It remains to prove (5) (⇐). This is a consequence of (4) and the fact that no nontrivial polynomial expression in the indeterminates Q_{s} with nonnegative integer coefficients can be 0 in ℤ′ (multiply by a sufficiently high power of v to get a polynomial in v, and note that the leading coefficient is nonzero).

We write u < (v, w) < z for u, v, w, z ∈ W̃ when u < v < z and u < w < z (in the Bruhat order). The first two statements of the lemma easily yield the following:

Corollary 2.6. For x, z ∈ W̃ and s ∈ S_{a} we have 1. sxs < (sx, xs) < x, zs < (z, szs) < sz ⟹ R̃_{x,z}(Q_{S}) = R̃_{sxs,szs}(Q_{S}) − Q_{s}R̃_{xs,szs}(Q_{S}), 2. xs < (x, sxs) < sx, zs < (z, szs) < sz ⟹ R̃_{x,z}(Q_{S}) = R̃_{sxs,szs}(Q_{S}).

3. Computing Bernstein Functions for Minuscule Coweights

In this section we present an efficient method to compute the Bernstein function z_{μ} as a linear combination of the normalized basis elements T̃_{w} in the case where μ is minuscule, i.e., ⟨α, μ⟩ ∈ {−1, 0, 1}, ∀α ∈ R.

Throughout this section fix a dominant and minuscule coweight μ ∈ X_{∗}. Let λ denote an element in the W_{0}-orbit of μ. Let μ^{−} denote the unique antidominant coweight in W_{0}(μ). Because our root system is reduced we have λ ⪯ μ and so we can write μ − λ = ∑_{i=1}^{p} α̌_{i}, where α_{i} ranges over a subset of the simple roots (possibly with repeats). In the case λ = μ^{−} we see that p = l(a^{μ}) (let 2ρ be the sum of the positive roots, apply ⟨2ρ, ·⟩ to both sides, and use ⟨2ρ, α̌_{i}⟩ = 2). Thus for general λ we have p ≤ l(a^{μ}).

In what follows we will write s_{i} in place of s_{α_{i}}.

Lemma 3.1. For μ and λ as above, there exists a sequence of simple roots α_{1}, α_{2}, …, α_{p} such that

s_{1}(μ) = μ − α̌_{1},
s_{2}s_{1}(μ) = μ − α̌_{1} − α̌_{2},
·
·
λ = s_{p} ⋯ s_{1}(μ) = μ − α̌_{1} − ⋯ − α̌_{p}.

Proof. Use induction on p. If p = 0, then λ = μ and there is nothing to prove. If p = 1, write λ = μ − α̌_{1}. Then λ is not dominant, so there exists α ∈ Π such that ⟨α, λ⟩ = −1 (since λ is minuscule). Thus s_{α}(λ) = λ + α̌ and

α̌_{1} − α̌ = μ − s_{α}(λ).

Because the right hand side is either 0 or a sum of positive coroots and α_{1} is a simple root we must have α_{1} = α and μ = s_{α_{1}}(λ). It follows that λ = s_{1}(μ) = μ − α̌_{1} as desired.

Now suppose that p > 1 and the result holds for p − 1. Since μ − λ is a sum of p simple coroots, λ is not dominant, so there exists α ∈ Π such that ⟨α, λ⟩ = −1. Thus s_{α}(λ) = λ + α̌ and

μ − λ = (μ − s_{α}(λ)) + α̌.

It follows that μ − s_{α}(λ) is a sum of p − 1 simple coroots, so the induction hypothesis applied to s_{α}(λ) yields a sequence α_{1}, …, α_{p−1} such that

s_{1}(μ) = μ − α̌_{1},
·
·
s_{α}(λ) = s_{p−1} ⋯ s_{1}(μ) = μ − α̌_{1} − ⋯ − α̌_{p−1}.

Now taking α_{p} = α easily yields the desired result.

Lemma 3.2. Let x ∈ X_{∗} be a (nonzero) coweight for which there exists a sequence of simple roots α_{1}, …, α_{p} (0 ≤ p ≤ r = l(a^{x})) such that

s_{1}(x) = x − α̌_{1},
·
·
s_{p} ⋯ s_{1}(x) = x − α̌_{1} − ⋯ − α̌_{p}.

Then there exists a reduced expression for a^{x} of the form

a^{x} = t_{1} ⋯ t_{r−p}(^{τ} s_{p}) ⋯ (^{τ} s_{1})τ,

where τ ∈ Ω is such that a^{x} ∈ W_{a}τ, and t_{j} ∈ S_{a}, for 1 ≤ j ≤ r − p.

Proof. We use induction on p. If p = 1, then ⟨α_{1}, x⟩ = 1, so that l(a^{x}s_{1}) < l(a^{x}). Then the Exchange property of the Coxeter group (W_{a}, S_{a}) shows that a^{x}τ^{−1} has a reduced expression ending with ^{τ} s_{1}, as desired.

Now suppose p > 1 and the result holds for p − 1. By the induction hypothesis s_{1}a^{x}s_{1} = a^{s_{1}(x)} has a reduced expression of the form

s_{1}a^{x}s_{1} = t_{1} ⋯ t_{r−p+1}(^{τ} s_{p}) ⋯ (^{τ} s_{2})τ,

and thus

a^{x} = s_{1}t_{1} ⋯ t_{r−p+1}(^{τ} s_{p}) ⋯ (^{τ} s_{2})(^{τ} s_{1})τ.

This last expression becomes reduced upon omitting exactly two letters. We must omit the first letter s_{1}, for otherwise s_{1}a^{x} < a^{x}, contrary to ⟨α_{1}, x⟩ = 1 > 0. If the other omitted letter is one of the t_{j}’s, we are done. We need to show therefore that the other omitted letter cannot be one of the ^{τ} s_{i}’s. Suppose it were. Then

a^{x} = t_{1} ⋯ t_{r−p+1}(^{τ} s_{p}) ⋯ ˆ (^{τ} s_{i}) ⋯ (^{τ} s_{1})τ.

Comparing with the expression for a^{s_{1}(x)} above it is easy to see

a^{s_{1}(x)} = a^{x}.s_{1} ⋯ s_{i}s_{i−1} ⋯ s_{2} ∈ X_{∗}W_{0}.

Since the s_{l} terms are in W_{0}, this implies s_{1}(x) = x, a contradiction.

We need the following lemma, which is due to Bernstein and Lusztig:

Lemma 3.3. (Bernstein, Lusztig) Let x ∈ X_{∗}, let α ∈ Π, and write s = s_{α}. Suppose that ⟨α, x⟩ = 1. Then

T̃_{s}^{−1}Θ_{x}T̃_{s}^{−1} = Θ_{s(x)}.

Proof. This is a consequence of (a dual version of) Proposition 3.6 of [8] when the parameter set L : S_{a} → ℕ is arbitrary (noting that ⟨α, x⟩ = 1 ⇒ α ∉ 2X^{∗}). It is due to Bernstein (see Lemma 4.4, [7]) in the case where L(s) = 1, ∀s ∈ S_{a}.

Proposition 3.4. Let μ be a dominant and minuscule coweight, and let τ ∈ Ω be the unique element such that a^{μ} ∈ W_{a}τ. Let λ ∈ W_{0}(μ). Suppose μ − λ is a sum of p simple coroots (0 ≤ p ≤ l(a^{μ}) = r). Then there exists a sequence of simple roots α_{1}, …, α_{p} such that the following hold (setting s_{i} = s_{α_{i}}):

1. ⟨ α_{i} , s_{i−1} ⋯ s_{1}(μ) ⟩ = 1, ∀1 ≤ i ≤ p,

2. There is a reduced expression for a^{μ} of the form a^{μ} = t_{1} ⋯ t_{r−p}(^{τ} s_{p}) ⋯ (^{τ} s_{1})τ,

3. There is a reduced expression for a^{λ} of the form a^{λ} = s_{p} ⋯ s_{1}t_{1} ⋯ t_{r−p}τ,

4. Θ_{λ} = T̃_{s_{p}}^{−1} ⋯ T̃_{s_{1}}^{−1}T̃_{t_{1}} ⋯ T̃_{t_{r−p}} T̃_{τ},

where t_{j} ∈ S_{a}, ∀j ∈ {1, 2, …, r − p}.

Proof. Lemma 3.1 and 3.2 applied to μ and λ immediately imply the existence of a sequence α_{1}, …, α_{p} of simple roots such that (1) and (2) hold. But (3) follows from (2) and the fact that s_{p} ⋯ s_{1}(μ) = λ. Note that (1) implies that the hypotheses of Lemma 3.3 are satisfied for x = s_{i−1} ⋯ s_{1}(μ) and α = α_{i}, ∀i. Therefore starting with the expression

Θ_{μ} = T̃_{t_{1}} ⋯ T̃_{t_{r−p}} T̃_{^{τ} s_{p}} ⋯ T̃_{^{τ} s_{1}} T̃_{τ}

and applying Lemma 3.3 repeatedly yields (4).

Corollary 3.5. With the hypotheses above, a^{μ^{−}} ∈ W_{0}τ.

Proof. Take λ = μ^{−}, so that p = r. Then the result is obvious from the reduced expression for a^{μ^{−}} given in the Proposition.

In practice the sequence α_{1}, …, α_{p} for any λ is easily computed. The expansion of the resulting expression for Θ_{λ} in terms of the basis {T̃_{w}, w ∈ W̃} is also a straightforward matter, using the relation T̃_{s}^{−1} = T̃_{s} +Q_{s}. Summing over λ then yields the Bernstein function z_{μ} as a linear combination of the elements T̃_{w}. One can then use this expression to write z_{μ} in terms of the usual basis elements T_{w}.

Example: The Bernstein function for G = GSp_{6}, μ = (1, 1, 1, 0, 0, 0). Identifying X_{∗} ⊗ ℝ = ℝ^{6} with X^{∗} ⊗ ℝ using the standard inner product, we can write the simple roots and coroots as follows: α_{1} = (1/2, −1/2, 0, 0, 1/2, −1/2), α̌_{1} = (1, −1, 0, 0, 1, −1) α_{2} = (0, 1/2, −1/2, 1/2, −1/2, 0), α̌_{2} = (0, 1, −1, 1, −1, 0), α_{3} = α̌_{3} = (0, 0, 1, −1, 0, 0). Write s_{i} instead of s_{α_{i}}, and let s_{0} = (1, 0, 0, 0, 0, −1)(1 6) denote the simple affine reflection.

Note that a^{μ} = s_{0}s_{1}s_{0}s_{2}s_{1}s_{0}τ is a reduced expression, where τ ∈ Ω permutes the simple affine roots by interchanging the subscripts 0 ↔ 3 and 1 ↔ 2.

Consider the sequence of simple roots α_{3}, α_{2}, α_{1}, α_{3}, α_{2}, α_{3}. This satisfies (1) in Proposition 3.4, and we see that

Θ_{(1,1,1,0,0,0)} = T̃_{0}T̃_{1}T̃_{0}T̃_{2}T̃_{1}T̃_{0}T̃_{τ} ,
Θ_{(1,1,0,1,0,0)} = T̃_{3}^{−1}T̃_{0}T̃_{1}T̃_{0}T̃_{2}T̃_{1}T̃_{τ} ,
Θ_{(1,0,1,0,1,0)} = T̃_{2}^{−1}T̃_{3}^{−1}T̃_{0}T̃_{1}T̃_{0}T̃_{2}T̃_{τ} ,
Θ_{(0,1,1,0,0,1)} = T̃_{1}^{−1}T̃_{2}^{−1}T̃_{3}^{−1}T̃_{0}T̃_{1}T̃_{0}T̃_{τ} ,
Θ_{(0,1,0,1,0,1)} = T̃_{3}^{−1}T̃_{1}^{−1}T̃_{2}^{−1}T̃_{3}^{−1}T̃_{0}T̃_{1}T̃_{τ} ,
Θ_{(0,0,1,0,1,1)} = T̃_{2}^{−1}T̃_{3}^{−1}T̃_{1}^{−1}T̃_{2}^{−1}T̃_{3}^{−1}T̃_{0}T̃_{τ} ,
Θ_{(0,0,0,1,1,1)} = T̃_{3}^{−1}T̃_{2}^{−1}T̃_{3}^{−1}T̃_{1}^{−1}T̃_{2}^{−1}T̃_{3}^{−1}T̃_{τ} ,

where we write T̃_{i} instead of T̃_{s_{i}} for every s_{i} ∈ S_{a}. The missing term Θ_{(1,0,0,1,1,0)} can be computed using a slightly different sequence. Let z_{μ}(x) denote the coefficient of T_{x} in the expression for z_{μ}. We find that z_{μ}(x) is 0 unless x is in the μ-admissible set (a set with 79 elements) and that for μ-admissible x the coefficient is as follows:

q^{l(μ)/2}z_{μ}(x) = \begin{cases}(1 − q)^{2}(1 − q + q^{2} − q^{3} + q^{4}), & if l(x) = 0, \\ (1 − q)^{3}(1 + q^{2}), & if l(x) = 1, \\ (1 − q)^{2}(1 − q + q^{2}), & if l(x) = 2, \\ (1 − q)(1 − q + q^{2}), & if x ∈ S_{1}, \\ (1 − q)^{3}, & if x ∈ S_{2}, \\ (1 − q)^{l(μ)−l(x)}, & if l(x) > 3. \end{cases}

Here S_{1} and S_{2} are the following sets of μ-admissible elements of length three:

S_{1} = {s_{321}τ, s_{232}τ, s_{123}τ, s_{210}τ, s_{101}τ, s_{012}τ},
S_{2} = {s_{323}τ, s_{312}τ, s_{212}τ, s_{213}τ, s_{230}τ, s_{310}τ, s_{320}τ, s_{120}τ, s_{301}τ, s_{201}τ, s_{101}τ},

where the symbol s_{ijk} stands for the product s_{i}s_{j}s_{k}.

4. A Formula for Bernstein Functions in the Minuscule Case

In this section we present a formula for z_{μ} in the case where μ is minuscule. First we need some preparation.

For any x ∈ W̃, there is a unique expression x = w.a^{t(x)}, where w ∈ W_{0} and t(x) ∈ X_{∗}. It is obvious that if α ∈ Π and s = s_{α}, then t(sx) = t(x) and t(xs) = t(sxs) = s(t(x)). We need the following further properties of t(x):

Lemma 4.1. Let x = w.a^{t(x)} ∈ W̃ as above and let s = s_{α} for α ∈ Π. Then 1. xs < x ⇔ ⟨α, t(x)⟩ > 0 or (⟨α, t(x)⟩ = 0 & ws < w),

2. x < xs ⇔ ⟨α, t(x)⟩ < 0 or (⟨α, t(x)⟩ = 0 & w < ws),

3. sx < sxs ⇔ ⟨α, t(x)⟩ < 0 or (⟨α, t(x)⟩ = 0 & sw < sws),

4. sxs < sx ⇔ ⟨α, t(x)⟩ > 0 or (⟨α, t(x)⟩ = 0 & sws < sw).

Proof. This can be deduced from Proposition 1.28 of [3]. Alternatively, one notes that in the notation of §2.1, if A = (β, k) ∈ R̃^{+} and y ∈ W̃, we have y^{−1}(β, k) ∈ R̃^{−} ⇔ s_{A}y ≤ y. Apply this with A = (α, 0) and y = x^{−1} and use the definition of the action of W̃ on R̃ given in §2.1.

Lemma 4.2. Let μ be dominant and minuscule. Then any μ-admissible element x has the property that t(x) ∈ W_{0}(μ).

Proof. Recall that the Bruhat ordering on W̃ descends to give an order on W_{0}\W̃/W_{0} and that for λ and μ dominant we have

W_{0}a^{λ}W_{0} ≤ W_{0}a^{μ}W_{0} ⟺ λ ⪯ μ.

Now write t(x) = w(λ), where λ is dominant and w ∈ W_{0}. Then using the hypothesis on x we have W_{0}a^{λ}W_{0} = W_{0}a^{t(x)}W_{0} = W_{0}xW_{0} ≤ W_{0}a^{μ}W_{0}, so that λ ⪯ μ. But μ (being minuscule) is minimal with respect to ⪯, and so λ = μ. Therefore t(x) = w(μ) ∈ W_{0}(μ).

Theorem 4.3. Let μ be a dominant minuscule coweight. Then

z_{μ} = ∑_{x : x is μ−adm.} R̃_{x,a^{t(x)}}(Q_{S})T̃_{x}.

This will follow immediately if we establish that for λ ∈ W_{0}(μ), we have

Θ_{λ} = ∑_{x} R̃_{x,a^{λ}}(Q_{S})T̃_{x},

where x runs over elements in the μ-admissible set such that t(x) = λ (using Lemma 4.2, which asserts that each t(x) must be equal to some λ ∈ W_{0}(μ)). But the only elements x giving a nonzero contribution to the above sum are those for which x ≤ a^{λ} (by Lemma 2.5 (5)), and so the condition that x be μ-admissible is redundant. Therefore we need to prove the following proposition:

Proposition 4.4. Let μ be minuscule and dominant and suppose λ ∈ W_{0}(μ). Then

Θ_{λ} = ∑_{x : t(x)=λ} R̃_{x,a^{λ}}(Q_{S})T̃_{x}.

Proof. Suppose that μ − λ is a sum of p simple coroots (0 ≤ p ≤ r = l(a^{μ})). We proceed by downwards induction on p. First assume p = r. Then λ = μ^{−} and in view of Θ_{μ^{−}} = T̃_{a^{−μ^{−}}}^{−1} (since −μ^{−} is dominant) and equation (1), the statement to be proved is

∑_{x} R̃_{x,a^{μ^{−}}} (Q_{S})T̃_{x} = ∑_{x : t(x)=μ^{−}} R̃_{x,a^{μ^{−}}} (Q_{S})T̃_{x}.

Taking into account Lemma 2.5 (5) we need only show that x ≤ a^{μ^{−}} ⇒ t(x) = μ^{−}. But this is clear because a^{μ^{−}} ∈ W_{0}τ (Corollary 3.5) implies that x ∈ W_{0}τ = W_{0}a^{μ^{−}}, and so t(x) = μ^{−}.

Now assume p < r and the result holds for p + 1. Since λ is not antidominant, there exists s = s_{α} (α ∈ Π) such that ⟨α, λ⟩ = 1. Thus s(λ) = λ − α̌ ≺ λ and hence the induction hypothesis applies to s(λ); we multiply the resulting equality for s(λ) on both sides by T̃_{s} to get

T̃_{s}Θ_{s(λ)}T̃_{s} = ∑_{y : t(y)=s(λ)} R̃_{y,a^{s(λ)}}(Q_{S})T̃_{s}T̃_{y}T̃_{s}.

Since ⟨α, λ⟩ = 1, Lemma 3.3 implies that the left hand side is Θ_{λ}. Therefore we must show that the right hand side is

∑_{x : t(x)=λ} R̃_{x,a^{λ}}(Q_{S})T̃_{x}.

Now for every x such that t(x) = λ we have ⟨α, t(x)⟩ = ⟨α, λ⟩ = 1, hence by Lemma 4.1 we have xs < x and sxs < sx , so either xs < (x, sxs) < sx, or sxs < (xs, sx) < x. Similarly, for y such that t(y) = s(λ) we have ⟨α, t(y)⟩ = ⟨α, s(λ)⟩ = −1, hence by Lemma 4.1 we have y < ys and sy < sys, so either y < (ys, sy) < sys, or sy < (y, sys) < ys. Furthermore note that for such y we have

T̃_{s}T̃_{y}T̃_{s} = \begin{cases}T̃_{sys}, & if y < (ys, sy) < sys, \\ −Q_{s}T̃_{ys} + T̃_{sys}, & if sy < (y, sys) < ys. \end{cases}

Therefore we need to prove that the expression

∑_{\begin{matrix}y : t(y)=s(λ) \\ y<(ys,sy)<sys \end{matrix}} R̃_{y,a^{s(λ)}}(Q_{S})T̃_{sys} + ∑_{\begin{matrix}y : t(y)=s(λ) \\ sy<(y,sys)<ys \end{matrix}} R̃_{y,a^{s(λ)}}(Q_{S})(−Q_{s}T̃_{ys}) + ∑_{\begin{matrix}y : t(y)=s(λ) \\ sy<(y,sys)<ys \end{matrix}} R̃_{y,a^{s(λ)}}(Q_{S})T̃_{sys} (3)

is equal to

∑_{\begin{matrix}x : t(x)=λ \\ sxs<(xs,sx)<x \end{matrix}} R̃_{x,a^{λ}}(Q_{S})T̃_{x} + ∑_{\begin{matrix}x : t(x)=λ \\ xs<(x,sxs)<sx \end{matrix}} R̃_{x,a^{λ}}(Q_{S})T̃_{x}. (4)

In the first and third sums in (3), replace y with sxs. In the second sum replace y with xs. Then (3) becomes

∑_{\begin{matrix}x : t(x)=λ \\ sxs<(sx,xs)<x \end{matrix}} R̃_{sxs,a^{s(λ)}}(Q_{S})T̃_{x} + ∑_{\begin{matrix}x : t(x)=λ \\ sxs<(xs,sx)<x \end{matrix}} R̃_{xs,a^{s(λ)}}(Q_{S})(−Q_{s}T̃_{x}) + ∑_{\begin{matrix}x : t(x)=λ \\ xs<(sxs,x)<sx \end{matrix}} R̃_{sxs,a^{s(λ)}}(Q_{S})T̃_{x} (5)

Note that ⟨α, λ⟩ = 1 implies a^{λ} satisfies a^{λ}s < (a^{λ}, sa^{λ}s) < sa^{λ}, by Lemma 4.1. Furthermore a^{s(λ)} = sa^{λ}s. Thus Corollary 2.6 with z = a^{λ} shows that (5) is indeed (4). The proposition follows, and thus Theorem 4.3 is proved.

Corollary 4.5. Let μ be dominant and minuscule, let s ∈ S_{a}, x ∈ W̃, and τ ∈ Ω. Then 1. If l(sxs) = l(x), and x is μ-admissible, then R̃_{x,a^{t(x)}}(Q_{S}) = R̃_{sxs,a^{t(sxs)}}(Q_{S}),

2. If l(sxs) = l(x) − 2 and x is μ-admissible, then R̃_{x,a^{t(x)}}(Q_{S}) = R̃_{sxs,a^{t(sxs)}}(Q_{S}) − Q_{s}R̃_{xs,a^{t(xs)}}(Q_{S}),

3. If x is μ-admissible, then R̃_{x,a^{t(x)}}(Q_{S}) = R̃_{τxτ^{−1},a^{t(τxτ^{−1})}}(Q_{S}).

Proof. These follow directly from the conditions on central elements of H proved in §3 of [2].

For ϕ = ∑_{x} a_{x}(Q_{S})T̃_{x} ∈ H define supp(ϕ) = {x | a_{x}(Q_{S}) ≠ 0}. It is obvious from Theorem 4.3 that supp(z_{μ}) is a subset of the μ-admissible set. We can now prove that these sets are in fact equal in this case.

Proposition 4.6. Let μ be minuscule and dominant. Then

supp(z_{μ}) = {x ∈ W̃ | x is μ-admissible}.

Proof. The left hand side is clearly contained in the right hand side, by Theorem 4.3. To prove the other inclusion it is enough to prove (by Lemma 2.5 (5)) that if x is μ-admissible, then x ≤ a^{t(x)}. By Lemma 4.2 it is enough to prove, for every λ ∈ W_{0}(μ), the statement

Hyp(λ): x is μ-admissible and t(x) = λ ⇒ x ≤ a^{λ}.

Suppose μ − λ is a sum of p simple coroots. We prove the statement Hyp(λ) by induction on p.

Suppose first that p = 0. Then λ = μ and it is enough to show that if x is μ-admissible and t(x) = μ, then x = a^{μ}. Write x = wa^{μ} for w ∈ W_{0}. Then l(x) = l(w) + l(a^{μ}) (see §2.1) and so x can be μ-admissible only if w = 1, i.e., x = a^{μ}.

Now suppose that p > 0 and that Hyp(λ′) is true for p − 1. Since λ is not dominant there exists s = s_{α} (α ∈ Π) such that ⟨α, λ⟩ = −1. Now suppose x is μ-admissible and t(x) = λ. Then ⟨α, t(x)⟩ < 0 so Lemma 4.1 implies x < xs and sx < sxs, so that either (I) x < (xs, sx) < sxs, or (II) sx < (x, sxs) < xs.

Consider case (I). It follows from Corollary 4.6 of [2] that sx and xs are both μ-admissible. But also t(xs) = s(λ) = λ + α̌ ≻ λ, so the induction hypothesis applied to s(λ) and xs shows that xs ≤ a^{t(xs)} and so R̃_{xs,a^{t(xs)}}(Q_{S}) ≠ 0 (Lemma 2.5 (5)). On the other hand, Corollary 4.5 (1) above (with xs for x) then implies R̃_{sx,a^{t(sx)}}(Q_{S}) ≠ 0, so again using Lemma 2.5 (5) we see x < sx ≤ a^{t(sx)} = a^{t(x)} = a^{λ}, as desired.

Finally consider (II). Since l(sxs) = l(x), sxs is μ-admissible (see Lemma 4.5 of [2]). Since t(sxs) = s(λ) the induction hypothesis applied to s(λ) and sxs yields sxs ≤ a^{s(λ)} = a^{t(sxs)}. The same argument as in Case (I) applies (using Corollary 4.5 (1) and Lemma 2.5 (5)) to give first R̃_{x,a^{t(x)}}(Q_{S}) ≠ 0 and then x ≤ a^{t(x)} = a^{λ} as desired.

The following answers a question of Rapoport affirmatively.

Proposition 4.7. If λ′ and λ are distinct elements of W_{0}(μ), then supp(Θ_{λ′})∩supp(Θ_{λ}) = ∅.

Proof. It follows from Proposition 4.4 that supp(Θ_{λ}) ⊂ {x | t(x) = λ} for any λ ∈ W_{0}(μ). This immediately implies the result.

Remark 4.8. One can show using the example of Gl_{3}, μ = (2, 1, 0) that this “Disjointness Property” is not true if one removes the hypothesis that μ be minuscule.

5. The Bernstein Function in the Drinfeld Case

In this section we explain how to use Theorem 4.3 to prove Theorem 1.2. Roughly speaking the method is to compare the formula in Theorem 4.3 for GL_{d} and μ = (1, 0^{d−1}) with Rapoport’s formula for the trace of Frobenius on nearby cycles in the Drinfeld case (see Proposition 5.1).

First we recall the setup. Fix a positive integer d > 2 and a prime number p. Let E be an imaginary quadratic extension of ℚ such that p decomposes as 𝔭_{1}𝔭_{2} in O_{E}. We fix embeddings σ : E ↪ \overline{ℚ} ⊂ ℂ and ϕ : \overline{ℚ} ↪ \overline{ℚ}_{p} such that ϕ ∘ σ determines the place 𝔭_{1} of E. Let (D, ∗) be a central simple algebra over E of dimension d^{2} together with a positive involution * which induces the nontrivial automorphism on E. (Thus ∗ is necessarily of the second type since E is imaginary). Fix an isomporphism D ⊗_{ℚ} ℝ ⟶^{∽} M_{d}(ℂ) such that * is carried over to the standard involution A ↦ \overline{A}^{t} on M_{d}(ℂ). Let G be the ℚ-group whose points in any commutative ℚ-algebra R are given by

G(R) = {x ∈ D ⊗_{ℚ} R | xx^{∗} ∈ R^{×}}.

Choose i = \sqrt{−1} ∈ ℂ and let h_{0} : ℂ → D ⊗_{ℚ} ℝ = M_{d}(ℂ) be given by

h_{0}(a + ib) = diag(a + ib, a − ib, …, a − ib).

The restriction of h_{0} to ℂ^{×} gives a homomorphism

h : R_{ℂ/ℝ}(𝔾_{m}) → G_{ℝ}.

Let X denote the G(ℝ)-conjugacy class of h. Let O_{D} be a ∗-stable order in D which is a maximal order at p. Let K = K^{p}K_{p} be a compact open subgroup which leaves O_{D} ⊗ ℤ̂ invariant (acting by multiplication on the right), where K^{p} is a sufficiently small subgroup of G(𝔸_{f}^{p}) and where K_{p} is an Iwahori subgroup of G(ℚ_{p}). Note that if D_{𝔭_{i}} is a division algebra (i = 1, 2), then K_{p} is the unique maximal compact subgroup. The triple (G, X, K) is a Shimura datum with reflex field E (see [6]) giving rise to a quasi-projective scheme S_{K} over O_{E,𝔭_{1}} (see [1]). The homomorphism h gives rise to a G(\overline{ℚ})-conjugacy class \{μ\} of cocharacters μ : (𝔾_{m})\overline{ℚ} → G\overline{ℚ}, and \{μ\} is defined over the field E.

Next we want to study the group G at the prime p. We will henceforth denote the group G_{ℚ_{p}} simply by G. Moreover from now on we will view \{μ\} as a G(\overline{ℚ}_{p})-conjugacy class of cocharacters of G_{\overline{ℚ}_{p}}, via the choice of embedding ϕ : \overline{ℚ} ↪ \overline{ℚ}_{p} made above. Note that

D ⊗ ℚ_{p} = D_{𝔭_{1}} × D_{𝔭_{2}} and that D_{𝔭_{1}} ⟶^{∽} D_{𝔭_{2}}^{op} via ∗. For any algebra R over the field ℚ_{p} = E_{𝔭_{1}} we can therefore identify G(R) with the group

{(x_{1}, x_{2}) ∈ (D_{𝔭_{1}} ⊗ R)^{×} × (D_{𝔭_{2}} ⊗ R)^{×} | x_{1} = cx_{2}^{−1}, for some c ∈ R^{×}}.

Therefore there is an isomorphism of ℚ_{p}-groups G ≅ D_{𝔭_{1}}^{×} × 𝔾_{m} given by (x_{1}, x_{2}) ↦ (x_{1}, c). Let H denote the ℚ_{p}-group D_{𝔭_{1}}^{×}.

Now fix an unramified extension F = ℚ_{p^{j}} of E_{𝔭_{1}} = ℚ_{p} such that j·(inv(D_{𝔭_{1}})) ≡ 0 (mod d). Then H_{F} = Gl_{d} and G_{F} = Gl_{d} × 𝔾_{m}. Under these identifications we get isomorphisms on the level of affine Weyl groups: W̃(G_{F}) = W̃(Gl_{d}) × W̃(𝔾_{m}), where W̃(Gl_{d}) = ℤ^{d} ⋊ S_{d} , W_{0}(G_{F}) = S_{d}, and W̃(𝔾_{m}) = X_{∗}(𝔾_{m}). Furthermore we can identify the conjugacy class of μ with the S_{d}-orbit of (((1, 0^{d−1}), 1), t) ∈ (ℤ^{d} ⋊ S_{d}) ⋊ t^{ℤ}. Here t denotes the element x ↦ x in X_{∗}(𝔾_{m}). We will abuse notation and denote the element (1, 0^{d−1})t by μ.

Let M^{loc} denote the local model attached to the datum (G, X, K) as in [10]. The following theorem follows from an explicit calculation of nearby cycles on M^{loc}:

Proposition 5.1. (Rapoport) Let q = p^{j} be such that j · (inv(D_{𝔭_{1}})) ≡ 0 (mod d). Let y ∈ M^{loc}(𝔽_{q}). Then

tr(F r_{q}; RΨ_{y}^{I}(\overline{ℚ}_{l})) = (1 − q)^{|S_{y}|−1},

where S_{y} denotes the set of strata of M_{\overline{𝔽}_{p}}^{loc} that contain the point y.

Proof. For the case where D is a central division algebra over E, this is proved in Theorem 3.12 of [10]. The proof there works in the case we consider as well.

As a consequence of Theorem 4.3 (or Theorem 1.1) we also have the following explicit formula for the Bernstein function for Gl_{d} and the coweight (1, 0^{d−1}).

Proposition 5.2. Let H = Gl_{d}(F) and ν = (1, 0^{d−1}). Let z_{ν} denote the corresponding Bernstein function. For x ∈ W̃(H) let z_{ν}(x) denote the coefficient of T_{x} in the expression for element z_{ν}. Then

q^{l(a^{ν})/2}z_{ν}(x) = \begin{cases}0, & if x is not ν-admissible, \\ (1 − q)^{l(a^{ν})−l(x)}, & if x is ν-admissible. \end{cases}

Proof. Because H = Gl_{d} is split, the parameters of the corresponding affine Hecke algebra are trivial: L(s) = 1, for every s ∈ S_{a} (cf. §2). Therefore for each s ∈ S_{a} we have Q_{s} = Q, where Q = q^{−1/2} − q^{1/2}. Now fix x ∈ W̃(H), which we assume is ν-admissible (the other case being trivial). Note that l(a^{t(x)}) = l(a^{ν}) by Lemma 4.2. Using the identity q^{1/2}Q = 1 − q and recalling that T̃_{x} = q^{−l(x)/2}T_{x}, we see by Theorem 4.3 that it is enough to show

Q^{l(a^{t(x)})−l(x)} = R̃_{x,a^{t(x)}}(Q).

Now if one numbers the simple affine reflections for W_{a}(Gl_{d}) in the standard way (s_{0} = (1, 0, …, 0, −1)(1d), s_{1} = (12), …, s_{d−1} = (d − 1, d)), then it is easy to show that a^{ν} = s_{0}s_{d−1} ⋯ s_{2}τ, where τ = (1, 0^{d−1})c ∈ W̃(H) and c = (12 … d) ∈ S_{d} = W_{0}. In particular the simple reflections in any reduced expression for a^{ν} are pairwise distinct. Since Int(τ) acts transitively on W_{0}(ν), the same is true of a^{t(x)} = t_{1} ⋯ t_{d−1}τ. We have

T̃_{(a^{t(x)})^{−1}}^{−1} = ∑_{y∈W̃} R̃_{y,a^{t(x)}}(Q)T̃_{y}

= (T̃_{t_{1}} + Q) ⋯ (T̃_{t_{d−1}} + Q)T̃_{τ} .

Because the t_{j}’s are pairwise distinct, any two different subsets {j_{1}, … j_{r}} and {j′_{1}, … j′_{r′}} yield distinct elements t_{j_{1}} ⋯ t_{j_{r}}τ and t_{j′_{1}} ⋯ t_{j′_{r′}} τ, and moreover these expressions are reduced. Taking into account these remarks, the proposition follows.

We conclude with the proof of Kottwitz’ conjecture in the Drinfeld case, for those q = p^{j} where j is such that j · (inv(D_{𝔭_{1}})) ≡ 0 (mod d).

Proof of Theorem 1.2: We first make the following remarks:

1. Let F = ℚ_{p^{j}}. Then G_{F} = Gl_{d} × 𝔾_{m} and the cocharacter μ = (1, 0^{d−1})t = νt of G_{F} gives rise via Bernstein’s construction (§2) to the function z_{μ} = z_{ν} ⊗T_{t} ∈ Z(H(Gl_{d}))⊗ℤ[T_{t}^{±}]. Note that z_{μ} (resp. z_{ν}) can be viewed as an element of the Iwahori-Hecke algebra of G(F) (resp. Gl_{d}(F)), by Remark 2.2. By definition of the Bruhat order (§2), an element y ∈ W̃(G_{F}) is in the μ-admissible set if and only if it is of the form y = xt, where x is an ν-admissible element of W̃(Gl_{d}). Furthermore we have l(y) = l(x) and z_{μ}(y) = z_{ν}(x) if y = xt.

2. The strata of M^{loc} in the Drinfeld case are indexed by elements of the form y = xt ∈ W̃(G_{F}) = W̃(Gl_{d}) ⋊ t^{ℤ}, where y (resp. x) is μ-admissible (resp. ν-admissible).

3. If y = xt is a μ-admissible, then

|S_{y}| − 1 = l(a^{μ}) − l(y) = l(a^{ν}) − l(x),

in the notation of Proposition 5.1. The equality in Theorem 1.2 now follows easily by combining these three remarks with Propositions 5.1 and 5.2.

6. Acknowledgements

This paper was written during a visit to the Max-Planck-Institut fuer Mathematik in Bonn. I thank the Institut for their support and for providing an excellent working environment. I am especially grateful to Robert Kottwitz and Michael Rapoport for many interesting conversations concerning this material, and for asking thought-provoking questions. I am indebted to Michael Rapoport for explaining to me the generalization here (Proposition 5.1) of his calculations in [10]. The final version of this paper was written during a visit to the Mathematisches Institut at the University of Köln, which I thank for the hospitality and support.

References

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[9] M. Rapoport and T. Zink, Über die lokale Zetafunktion von Shimuravarietäten, Monodromiefiltration und verschwindende Zyklen in umgleicher Charakteristik, Invent. math. 68, 21-101 (1980).

[10] M. Rapoport, On the Bad Reduction of Shimura Varieties, Automorphic Forms, Shimura Varieties and L-functions, part II, Perspectives in Mathematics, vol. 11, Academic Press, San Diego, CA, 1990, 253-321.

University of Toronto Department of Mathematics 100 St. George Street Toronto, ON M5S 1A1, CANADA email: haines@math.toronto.edu