LATTICE REPRESENTATIONS OF HEISENBERG GROUPS

Jae-Hyun Yang

1 Introduction

For any positive integers g and h, we consider the Heisenberg group

H_{ℝ}^{(g,h)} := { (λ, μ, κ) | λ, μ ∈ ℝ^{(h,g)}, κ ∈ ℝ^{(h,h)}, κ + μ^{t}λ symmetric }

endowed with the following multiplication law

(λ, μ, κ) ∘ (λ′, μ′, κ′) := (λ + λ′, μ + μ′, κ + κ′ + λ^{t}μ′ − μ^{t}λ′).

Here ℝ^{(h,g)} ( resp. ℝ^{(h,h)} ) denotes the set of all h × g ( resp. h × h) real matrices.

The Heisenberg group H_{ℝ}^{(g,h)} is embedded in the symplectic group Sp(g + h, ℝ) via the mapping

H_{ℝ}^{(g,h)} ∋ (λ, μ, κ) ⟼ ( \begin{matrix}E_{g} & 0 & 0 & ^{t}μ \\ λ & E_{h} & μ & κ \\ 0 & 0 & E_{g} & −^{t}λ \\ 0 & 0 & 0 & E_{h} \end{matrix} ) ∈ Sp(g + h, ℝ).

This Heisenberg group is a 2-step nilpotent Lie group and is important in the study of toroidal compactification of Siegel moduli spaces. In fact, H_{ℝ}^{(g,h)} is obtained as the unipotent radical of the parabolic subgroup of Sp(g + h, ℝ) associated with the rational boundary component F_{g} ( cf. [N] p. 21 ). For the motivation of the study of this Heisenberg group we refer to [Y4]-[Y8] and [Z]. We also refer to [Y1]- [Y3] for more results on H_{ℝ}^{(g,h)}.

In [C], P. Cartier stated without proof that for h = 1, the lattice representation of H_{ℝ}^{(g,1)} associated to the lattice L is unitarily equivalent to the direct sum of [L^{∗} : L]^{\frac{1}{2}} copies of the Schrödinger representation of H_{ℝ}^{(g,1)}, where L^{∗} is the dual lattice of L with respect to a certain nondegenerate alternating bilinear form. R. Berndt proved the above fact for the case h = 1 in his lecture note [B]. In this paper, we give a complete proof of Cartier’s theorem for H_{ℝ}^{(g,h)}.

Typeset by AMS-TEX

This work was partially supported by the Max-Planck-Institute für Mathematik and TGRC-KOSEF.

Keywords : Heisenberg groups, Schrödinger representations, lattice representations, theta functions

Main Theorem. Let M be a positive definite, symmetric half-integral matrix of degree h and L be a self-dual lattice in ℂ^{(h,g)}. Then the lattice representation π_{M} of H_{ℝ}^{(g,h)} associated with L and M is unitarily equivalent to the direct sum of ( det 2M)^{g} copies of the Schrödinger representation of H_{ℝ}^{(g,h)}. For a precise detail, we refer to section 3.

This paper is organized as follows. In section 2, we review the Schrödinger representations of the Heisenberg group H_{ℝ}^{(g,h)}. In section 3, we prove the main theorem. In the final section, we provide a relation between lattice representations and theta functions.

Acknowledgement. This work was in part done during my stay at the Max-Planck-Institut für Mathematik in Bonn. I am very grateful to the institute for the hospitality and the financial support. I also would like to give my hearty thanks to Department of Mathematics, Harvard University for its hospitality during my short stay at Cambridge.

Notations: We denote by ℤ, ℝ and ℂ the ring of integers, the field of real numbers, and the field of complex numbers respectively. ℂ_{1}^{×} denotes the multiplicative group consisting of all complex numbers z with |z| = 1. Sp(g, ℝ) denotes the symplectic group of degree g. H_{g} denotes the Siegel upper half plane of degree g. The symbol “:=” means that the expression on the right is the definition of that on the left. We denote by ℤ^{+} the set of all positive integers. F^{(k,l)} denotes the set of all k × l matrices with entries in a commutative ring F. For any M ∈ F^{(k,l)}, ^{t}M denotes the transpose matrix of M. For A ∈ F^{(k,k)}, σ(A) denotes the trace of A. For A ∈ F^{(k,l)} and B ∈ F^{(k,k)}, we set B[A] = ^{t}ABA. E_{k} denotes the identity matrix of degree k. For a positive integer n, Symm (n, K) denotes the vector space consisting of all symmetric n × n matrices with entries in a field K.

2 Schrödinger Representations

First of all, we observe that H_{ℝ}^{(g,h)} is a 2-step nilpotent Lie group. It is easy to see that the inverse of an element (λ, μ, κ) ∈ H_{ℝ}^{(g,h)} is given by

(λ, μ, κ)^{−1} = (−λ, −μ, −κ + λ^{t}μ − μ^{t}λ).

Now we put

(2.1)  [λ, μ, κ] := (0, μ, κ) ∘ (λ, 0, 0) = (λ, μ, κ − μ^{t}λ).

Then H_{ℝ}^{(g,h)} may be regarded as a group equipped with the following multiplication

(2.2)  [λ, μ, κ] ⋄ [λ_{0}, μ_{0}, κ_{0}] := [λ + λ_{0}, μ + μ_{0}, κ + κ_{0} + λ^{t}μ_{0} + μ_{0} ^{t}λ].

The inverse of [λ, μ, κ] ∈ H_{ℝ}^{(g,h)} is given by

[λ, μ, κ]^{−1} = [−λ, −μ, −κ + λ^{t}μ + μ^{t}λ].

We set

(2.3)  K := { [0, μ, κ] ∈ H_{ℝ}^{(g,h)} | μ ∈ ℝ^{(h,g)}, κ = ^{t}κ ∈ ℝ^{(h,h)} } .

Then K is a commutative normal subgroup of H_{ℝ}^{(g,h)}. Let K̂ be the Pontrajagin dual of K, i.e., the commutative group consisting of all unitary characters of K. Then K̂ is isomorphic to the additive group ℝ^{(h,g)} × Symm (h, ℝ) via

(2.4)  < a, â >:= e^{2πiσ(μ̂^{t}μ+κ̂κ)}, a = [0, μ, κ] ∈ K, â = (μ̂, κ̂) ∈ K̂.

We put

(2.5)  S := { [λ, 0, 0] ∈ H_{ℝ}^{(g,h)} | λ ∈ ℝ^{(h,g)} } ≅ ℝ^{(h,g)}.

Then S acts on K as follows:

(2.6)  α_{λ}([0, μ, κ]) := [0, μ, κ + λ^{t}μ + μ^{t}λ], [λ, 0, 0] ∈ S.

It is easy to see that the Heisenberg group (H_{ℝ}^{(g,h)}, ⋄) is isomorphic to the semidirect product S ⋉ K of S and K whose multiplication is given by

(λ, a) · (λ_{0}, a_{0}) := (λ + λ_{0}, a + α_{λ}(a_{0})), λ, λ_{0} ∈ S, a, a_{0} ∈ K.

On the other hand, S acts on K̂ by

(2.7)  α_{λ}^{∗}(â) := (μ̂ + 2κ̂λ, κ̂), [λ, 0, 0] ∈ S, a = (μ̂, κ̂) ∈ K̂.

Then we have the relation < α_{λ}(a), â >=< a, α_{λ}^{∗}(â) > for all a ∈ K and â ∈ K̂.

We have two types of S-orbits in K̂. Type I. Let κ̂ ∈ Symm (h, ℝ) with κ̂ ≠ 0. The S-orbit of â(κ̂) := (0, κ̂) ∈ K̂ is given by

(2.8)  Ôκ̂ := { (2κ̂λ, κ̂) ∈ K̂ | λ ∈ ℝ^{(h,g)} } ≅ ℝ^{(h,g)}.

Type II. Let ŷ ∈ ℝ^{(h,g)}. The S-orbit Ôŷ of â(ŷ) := (ŷ, 0) is given by

(2.9)  Ôŷ := { (ŷ, 0) } = â(ŷ).

We have

K̂ = ( ⋃_{κ̂∈Symm(h,ℝ)} Ôκ̂) ⋃ ( ⋃_{ŷ∈ℝ^{(h,g)}} Ôŷ)

as a set. The stabilizer Sκ̂ of S at â(κ̂) = (0, κ̂) is given by

(2.10)  Sκ̂ = {0}.

And the stabilizer Sŷ of S at â(ŷ) = (ŷ, 0) is given by

(2.11)  Sŷ = { [λ, 0, 0] | λ ∈ ℝ^{(h,g)} } = S ≅ ℝ^{(h,g)}.

From now on, we set G := H_{ℝ}^{(g,h)} for brevity. K is a closed, commutative normal subgroup of G. Since (λ, μ, κ) = (0, μ, κ + μ^{t}λ) ∘ (λ, 0, 0) for (λ, μ, κ) ∈ G, the homogeneous space X := K\G is identified with ℝ^{(h,g)} via

Kg = K ∘ (λ, 0, 0) ⟼ λ, g = (λ, μ, κ) ∈ G.

We observe that G acts on X by

(2.12)  (Kg) · g_{0} := K (λ + λ_{0}, 0, 0) = λ + λ_{0},

where g = (λ, μ, κ) ∈ G and g_{0} = (λ_{0}, μ_{0}, κ_{0}) ∈ G.

If g = (λ, μ, κ) ∈ G, we have

(2.13)  k_{g} = (0, μ, κ + μ^{t}λ), s_{g} = (λ, 0, 0)

in the Mackey decomposition of g = k_{g} ∘ s_{g} ( cf. [M] ). Thus if g_{0} = (λ_{0}, μ_{0}, κ_{0}) ∈ G, then we have

(2.14)  s_{g} ∘ g_{0} = (λ, 0, 0) ∘ (λ_{0}, μ_{0}, κ_{0}) = (λ + λ_{0}, μ_{0}, κ_{0} + λ^{t}μ_{0})

and so

(2.15)  k_{s_{g}∘g_{0}} = (0, μ_{0}, κ_{0} + μ_{0} ^{t}λ_{0} + λ^{t}μ_{0} + μ_{0} ^{t}λ).

For a real symmetric matrix c = ^{t}c ∈ ℝ^{(h,h)} with c ≠ 0, we consider the onedimensional unitary representation σ_{c} of K defined by

(2.16)  σ_{c} ((0, μ, κ)) := e^{2πiσ(cκ)} I, (0, μ, κ) ∈ K,

where I denotes the identity mapping. Then the induced representation U(σ_{c}) := Ind_{K}^{G} σ_{c} of G induced from σ_{c} is realized in the Hilbert space H_{σ_{c}} = L^{2}(X, dġ, ℂ) ≅ L^{2} (ℝ^{(h,g)}, dξ) as follows. If g_{0} = (λ_{0}, μ_{0}, κ_{0}) ∈ G and x = Kg ∈ X with g = (λ, μ, κ) ∈ G, we have

(2.17)  (U_{g_{0}}(σ_{c})f) (x) = σ_{c} (k_{s_{g}∘g_{0}}) (f(xg_{0})) , f ∈ H_{σ_{c}}.

It follows from (2.15) that

(2.18)  (U_{g_{0}}(σ_{c})f) (λ) = e^{2πiσ{c(κ_{0}+μ_{0} ^{t}λ_{0}+2λ^{t}μ_{0})}} f(λ + λ_{0}).

Here we identified x = Kg (resp. xg_{0} = Kgg_{0}) with λ (resp. λ + λ_{0}). The induced representation U(σ_{c}) is called the Schrödinger representation of G associated with σ_{c}. Thus U(σ_{c}) is a monomial representation.

Now we denote by H^{σ_{c}} the Hilbert space consisting of all functions ϕ : G ⟶ ℂ which satisfy the following conditions:

(1) ϕ(g) is measurable with respect to dg.

(2) ϕ ((0, μ, κ) ∘ g)) = e^{2πiσ(cκ)}ϕ(g) for all g ∈ G.

(3) ∥ ϕ ∥^{2}:= ∫_{X} |ϕ(g)|^{2} dġ < ∞, ġ = Kg,

where dg (resp. dġ) is a G-invariant measure on G (resp. X = K\G). The inner product ( , ) on H^{σ_{c}} is given by

(ϕ_{1}, ϕ_{2}) := ∫_{G} ϕ_{1}(g) \overline{ϕ_{2}(g)} dg, ϕ_{1}, ϕ_{2} ∈ H^{σ_{c}}.

We observe that the mapping Φ_{c} : H_{σ_{c}} ⟶ H^{σ_{c}} defined by

(2.19)  (Φ_{c}(f)) (g) := e^{2πiσ{c(κ+μ^{t}λ)}} f(λ), f ∈ H_{σ_{c}}, g = (λ, μ, κ) ∈ G

is an isomorphism of Hilbert spaces. The inverse Ψ_{c} : H^{σ_{c}} ⟶ H_{σ_{c}} of Φ_{c} is given by

(2.20)  (Ψ_{c}(ϕ)) (λ) := ϕ((λ, 0, 0)), ϕ ∈ H^{σ_{c}}, λ ∈ ℝ^{(h,g)}.

The Schrödinger representation U(σ_{c}) of G on H^{σ_{c}} is given by

(2.21)  (U_{g_{0}}(σ_{c})ϕ) (g) = e^{2πiσ{c(κ_{0}+μ_{0} ^{t}λ_{0}+λ^{t}μ_{0}−λ_{0} ^{t}μ)}} ϕ ((λ_{0}, 0, 0) ∘ g) ,

where g_{0} = (λ_{0}, μ_{0}, κ_{0}), g = (λ, μ, κ) ∈ G and ϕ ∈ H^{σ_{c}}. (2.21) can be expressed as follows.

(2.22)  ( U_{g_{0}}(σ_{c})ϕ) (g) = e^{2πiσ{c(κ_{0}+κ+μ_{0} ^{t}λ_{0}+μ^{t}λ+2λ^{t}μ_{0})}} ϕ((λ_{0} + λ, 0, 0)).

Theorem 2.1. Let c be a positive symmetric half-integral matrix of degree h. Then the Schrödinger representation U(σ_{c}) of G is irreducible.

Proof. The proof can be found in [Y1], Theorem 3. □

3 Proof of Main Theorem

Let L := ℤ^{(h,g)} × ℤ^{(h,g)} be the lattice in the vector space V ≅ ℂ^{(h,g)}. Let B be an alternating bilinear form on V such that B(L, L) ⊂ ℤ, that is, ℤ-valued on L × L. The dual L_{B}^{∗} of L with respect to B is defined by

L_{B}^{∗} := { v ∈ V | B(v, l) ∈ ℤ for all l ∈ L } .

Then L ⊂ L_{B}^{∗}. If B is nondegenerate, L_{B}^{∗} is also a lattice in V, called the dual lattice of L. In case B is nondegenerate, there exist a ℤ-basis { ξ_{11}, ξ_{12}, ⋯, ξ_{hg}, η_{11}, η_{12}, ⋯, η_{hg} } of L and a set { e_{11}, e_{12}, ⋯, e_{hg} } of positive integers with e_{11}|e_{12}, e_{12}|e_{13}, ⋯, e_{h,g−1}|e_{hg} such that

( \begin{matrix}B(ξ_{ka}, ξ_{lb}) & B(ξ_{ka}, η_{lb}) \\ B(η_{ka}, ξ_{lb}) & B(η_{ka}, η_{lb} \end{matrix} ) = ( \begin{matrix}0 & e \\ −e & 0 \end{matrix} ) ,

where 1 ≤ k, l ≤ h, 1 ≤ a, b ≤ g and e := diag (e_{11}, e_{12}, ⋯, e_{hg}) is the diagonal matrix of degree hg with entries e_{11}, e_{12}, ⋯, e_{hg}. It is well known that [L_{B}^{∗} : L] = ( det e)^{2} = (e_{11}e_{12} ⋯ e_{hg})^{2} (cf. [I] p. 72). The number det e is called the Pfaffian of B.

Now we consider the following subgroups of G:

(3.1)  Γ_{L} := { (λ, μ, κ) ∈ G | (λ, μ) ∈ L, κ ∈ ℝ^{(h,h)} }

and

(3.2)  Γ_{L_{B}^{∗}} := { (λ, μ, κ) ∈ G | (λ, μ) ∈ L_{B}^{∗}, κ ∈ ℝ^{(h,h)} } .

Then both Γ_{L} and Γ_{L_{B}^{∗}} are the normal subgroups of G. We put

(3.3)  Z_{0} := { (0, 0, κ) ∈ G | κ = ^{t}κ ∈ ℤ^{(h,h)} integral } .

It is easy to show that

Γ_{L_{B}^{∗}} = { g ∈ G | gγg^{−1}γ^{−1} ∈ Z_{0} for all γ ∈ Γ_{L} } .

We define

(3.4)  Y_{L} := { φ ∈ Hom (Γ_{L}, ℂ_{1}^{×}) | φ is trivial on Z_{0} }

and

(3.5)  Y_{L,S} := { φ ∈ Y_{L} | φ(κ) = e^{2πiσ(Sκ)} for all κ = ^{t}κ ∈ ℝ^{(h,h)} }

for each symmetric real matrix S of degree h. We observe that if S is not halfintegral, then Y_{L} = ∅ and so Y_{L,S} = ∅. It is clear that if S is symmetric half-integral, then Y_{L,S} is not empty. Thus we have

(3.6)  Y_{L} = ∪_{M} Y_{L,M},

where M runs through the set of all symmetric half-integral matrices of degree h.

Lemma 3.1. Let M be a symmetric half-integral matrix of degree h with M ≠ 0. Then any element φ of Y_{L,M} is of the form φ_{M,q}. Here φ_{M,q} is the character of Γ_{L} defined by

(3.7)  φ_{M,q}((l, κ)) := e^{2πiσ(Mκ)} · e^{πiq(l)}, (l, κ) ∈ Γ_{L},

where q : L ⟶ ℝ/2ℤ ≅ [0, 2) is a function on L satisfying the following condition:

(3.8)  q(l_{0} + l_{1}) ≡ q(l_{0}) + q(l_{1}) − 2σ{M(λ_{0} ^{t}μ_{1} − μ_{0} ^{t}λ_{1})} ( mod 2 )

for all l_{0} = (λ_{0}, μ_{0}) ∈ L and l_{1} = (λ_{1}, μ_{1}) ∈ L.

Proof. (3.8) follows immediately from the fact that φ_{M,q} is a character of Γ_{L}. It is obvious that any element of Y_{L,M} is of the form φ_{M,q}. □

Lemma 3.2. An element of Y_{L,0} is of the form φ_{k,l} (k, l ∈ ℝ^{(h,g)}). Here φ_{k,l} is the character of Γ_{L} defined by

(3.9)  φ_{k,l}(γ) := e^{2πiσ(k^{t}λ+l^{t}μ)}, γ = (λ, μ, κ) ∈ Γ_{L}.

Proof. It is easy to prove it and so we omit the proof. □

Lemma 3.3. Let M be a nonsingular symmetric half-integral matrix of degree h. Let φ_{M,q_{1}} and φ_{M,q_{2}} be the characters of Γ_{L} defined by (3.7). The character φ of Γ_{L} defined by φ := φ_{M,q_{1}} · φ_{M,q_{2}}^{−1} is an element of Y_{L,0}.

Proof. It follows from the fact that there exists an element g = (M^{−1}λ, M^{−1}μ, 0) ∈ G with (λ, μ) ∈ V such that

φ_{M,q_{1}}(γ) = φ_{M,q_{2}}(gγg^{−1}) for all γ ∈ Γ_{L}.
□

For a unitary character φ_{M,q} of Γ_{L} defined by (3.7), we let

(3.10)  π_{M,q} := Ind_{Γ_{L}}^{G} φ_{M,q}

be the representation of G induced from φ_{M,q}. Let H_{M,q} be the Hilbert space consisting of all measurable functions ϕ : G ⟶ ℂ satisfying

(L1) ϕ(γg) = φ_{M,q}(γ) ϕ(g) for all γ ∈ Γ_{L} and g ∈ G.

(L2) ∥ϕ∥_{M,q}^{2} := ∫_{Γ_{L}\G} |ϕ(ḡ)|^{2} dḡ < ∞, ḡ = Γ_{L}g.

The induced representation π_{M,q} is realized in H_{M,q} as follows:

(3.11)  ( π_{M,q}(g_{0})ϕ )(g) := ϕ(gg_{0}), g_{0}, g ∈ G, ϕ ∈ H_{M,q}.

π_{M,q} is called the lattice representation of G associated with the lattice L.

Main Theorem. Let M be a positive definite, symmetric half integral matrix of degree h. Let φ_{M} be the character of Γ_{L} defined by φ_{M}((λ, μ, κ)) := e^{2πiσ(Mκ)} for all (λ, μ, κ) ∈ Γ_{L}. Then the lattice representation

π_{M} := Ind_{Γ_{L}}^{G} φ_{M}

induced from the character φ_{M} is unitarily equivalent to the direct sum

⨁ U(σ_{M}) := ⨁ Ind_{K}^{G} σ_{M} ( ( det 2M)^{g}-copies )

of the Schrödinger representation Ind_{K}^{G} σ_{M}.

Proof. We first recall that the induced representation π_{M} is realized in the Hilbert space H_{M} consisting of all measurable functions ϕ : G ⟶ ℂ satisfying the conditions

(3.13)  ϕ((λ_{0}, μ_{0}, κ_{0}) ∘ g) = e^{2πiσ(Mκ_{0})} ϕ(g), (λ_{0}, μ_{0}, κ_{0}) ∈ Γ_{L}, g ∈ G

and

(3.14)  ∥ϕ∥_{π,M}^{2} := ∫_{Γ_{L}\G} |ϕ(ḡ)|^{2} dḡ < ∞, ḡ = Γ_{L} ∘ g.

Now we write

g_{0} = [λ_{0}, μ_{0}, κ_{0}] ∈ Γ_{L} and g = [λ, μ, κ] ∈ G.

For ϕ ∈ H_{M}, we have

(3.15)  ϕ(g_{0} ⋄ g) = ϕ([λ_{0} + λ, μ_{0} + μ, κ_{0} + κ + λ_{0} ^{t}μ + μ^{t}λ_{0}]).

On the other hand, we get

ϕ(g_{0} ⋄ g) = ϕ((λ_{0}, μ_{0}, κ_{0} − μ_{0} ^{t}λ_{0}) ∘ g)
= e^{2πiσ{M(κ_{0}−μ_{0} ^{t}λ_{0})}} ϕ(g)
= e^{2πiσ(Mκ_{0})} ϕ(g) ( because σ(Mμ_{0} ^{t}λ_{0}) ∈ ℤ)

Thus putting κ′ := κ_{0} + λ_{0} ^{t}μ + μ^{t}λ_{0}, we get

(3.16)  ϕ([λ_{0} + λ, μ_{0} + μ, κ + κ′]) = e^{2πiσ(Mκ′)} · e^{−4πiσ(Mλ_{0} ^{t}μ)} ϕ([λ, μ, κ]).

Putting λ_{0} = κ_{0} = 0 in (3.16), we have

(3.17)  ϕ([λ, μ + μ_{0}, κ]) = ϕ([λ, μ, κ]) for all μ_{0} ∈ ℤ^{(h,g)} and [λ, μ, κ] ∈ G.

Therefore if we fix λ and κ, ϕ is periodic in μ with respect to the lattice ℤ^{(h,g)} in ℝ^{(h,g)}. We note that

ϕ([λ, μ, κ]) = ϕ([0, 0, κ] ⋄ [λ, μ, 0]) = e^{2πiσ(Mκ)} ϕ([λ, μ, 0])

for [λ, μ, κ] ∈ G. Hence ϕ admits a Fourier expansion in μ :

(3.18)  ϕ([λ, μ, κ]) = e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{N}(λ) e^{2πiσ(N ^{t}μ)}.

If λ_{0} ∈ ℤ^{(h,g)}, then we have

ϕ([λ + λ_{0}, μ, κ]) = e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{N}(λ + λ_{0}) e^{2πiσ(N ^{t}μ)}

= e^{−4πiσ(Mλ_{0} ^{t}μ)} ϕ([λ, μ, κ]) ( by (3.16) )

= e^{−4πiσ(Mλ_{0} ^{t}μ)} e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{N}(λ) e^{2πiσ(N ^{t}μ)},

= e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{N}(λ) e^{2πiσ{(N−2Mλ_{0})^{t}μ}}. ( by (3.18) )

So we get

∑_{N∈ℤ^{(h,g)}} c_{N}(λ + λ_{0}) e^{2πiσ(N ^{t}μ)}

= ∑_{N∈ℤ^{(h,g)}} c_{N}(λ) e^{2πiσ{(N−2Mλ_{0})^{t}μ}}
= ∑_{N∈ℤ^{(h,g)}} c_{N+2Mλ_{0}}(λ) e^{2πiσ(N ^{t}μ)}.

Hence we get

(3.19)  c_{N}(λ + λ_{0}) = c_{N+2Mλ_{0}}(λ) for all λ_{0} ∈ ℤ^{(h,g)} and λ ∈ ℝ^{(h,g)}.

Consequently, it is enough to know only the coefficients c_{α}(λ) for the representatives α in ℤ^{(h,g)} modulo 2M. It is obvious that the number of all such representatives α’s is (det 2M)^{g}. We denote by J a complete system of such representatives α’s in ℤ^{(h,g)} modulo 2M. Then we have

ϕ([λ, μ, κ])
= e^{2πiσ(Mκ)} { ∑_{N∈ℤ^{(h,g)}} c_{α+2MN}(λ) e^{2πiσ{(α+2MN)^{t}μ}}

+ ∑_{N∈ℤ^{(h,g)}} c_{β+2MN}(λ) e^{2πiσ{(β+2MN)^{t}μ}}

· ·

+ ∑_{N∈ℤ^{(h,g)}} c_{γ+2MN}(λ) e^{2πi{(γ+2MN)^{t}μ}} },

where { α, β, ⋯, γ } denotes the complete system J .

For each α ∈ J , we denote by H_{M,α} the Hilbert space consisting of Fourier expansions

e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{α+2MN}(λ) e^{2πiσ{ (α+2MN)^{t}μ}}, (λ, μ, κ) ∈ G,

where c_{N}(λ) denotes the coefficients of the Fourier expansion (3.18) of ϕ ∈ H_{M} and ϕ runs over the set { ϕ ∈ π_{M} }. It is easy to see that H_{M,α} is invariant under π_{M}. We denote the restriction of π_{M} to H_{M,α} by π_{M,α}. Then we have

(3.20)  π_{M} = ⨁_{α∈J} π_{M,α}.

Let ϕ_{α} ∈ π_{M,α}. Then for [λ, μ, κ] ∈ G, we get

(3.21)  ϕ_{α}([λ, μ, κ]) = e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} c_{α+2MN}(λ) e^{2πiσ{(α+2MN)^{t}μ}}.

10 JAE-HYUN YANG

We put

I_{λ} := \overbrace{[0, 1] × [0, 1] × ⋯ × [0, 1]}^{(h×g)-times} ⊂ \{ [λ, 0, 0] | λ ∈ ℝ^{(h,g)} \}

and

I_{μ} := \overbrace{[0, 1] × [0, 1] × ⋯ × [0, 1]}^{(h×g)-times} ⊂ \{ [0, μ, 0] | μ ∈ ℝ^{(h,g)}\} .

Then we obtain

(3.22)  ∫_{I_{μ}} ϕ_{α}([λ, μ, κ]) e^{−2πiσ(α^{t}μ)} dμ = e^{2πiσ(Mκ)} c_{α}(λ), α ∈ J .

Since Γ_{L}\G ≅ I_{λ} × I_{μ}, we get

∥ϕ_{α}∥_{π,M,α}^{2} : = ∥ϕ_{α}∥_{π,M}^{2} = ∫_{Γ_{L}\G} |ϕ_{α}(ḡ)|^{2} dḡ

= ∫_{I_{λ}} ∫_{I_{μ}} |ϕ_{α}(ḡ)|^{2} dλdμ

= ∫_{I_{λ}×I_{μ}} | ∑_{N∈ℤ^{(h,g)}} c_{α+2MN}(λ) e^{2πiσ{(α+2MN)^{t}μ}} |^{2} dλdμ

= ∫_{I_{λ}} ∑_{N∈ℤ^{(h,g)}} |c_{α+2MN}(λ)|^{2} dλ

= ∫_{I_{λ}} ∑_{N∈ℤ^{(h,g)}} |c_{α}(λ + N)|^{2} dλ ( by (3.19) )

= ∫_{ℝ^{(h,g)}} |c_{α}(λ)|^{2} dλ.

Since ϕ_{α} ∈ π_{M,α}, ∥ϕ_{α}∥_{π,M,α} < ∞ and so c_{α}(λ) ∈ L^{2} (ℝ^{(h,g)}, dξ) for all α ∈ J .

For each α ∈ J , we define the mapping ϑ_{M,α} on L^{2} (ℝ^{(h,g)}, dξ) by

(3.23) (ϑ_{M,α}f)([λ, μ, κ]) := e^{2πiσ(Mκ)} ∑_{N∈ℤ^{(h,g)}} f(λ + N) e^{2πiσ{(α+2MN)^{t}μ}},

where f ∈ L^{2} (ℝ^{(h,g)}, dξ) and [λ, μ, κ] ∈ G. Lemma 3.4. For each α ∈ J , the image of L^{2} (ℝ^{(h,g)}, dξ) under ϑ_{M,α} is contained in H_{M,α}. Moreover, the mapping ϑ_{M,α} is a one-to-one unitary operator of L^{2} (ℝ^{(h,g)}, dξ) onto H_{M,α} preserving the norms. In other words, the mapping

ϑ_{M,α} : L^{2} (ℝ^{(h,g)}, dξ) ⟶ H_{M,α}

is an isometry.

Proof. We already showed that ϑ_{M,α} preserves the norms. First we observe that if (λ_{0}, μ_{0}, κ_{0}) ∈ Γ_{L} and g = [λ, μ, κ] ∈ G,

(λ_{0}, μ_{0}, κ_{0}) ∘ g = [λ_{0}, μ_{0}, κ_{0} + μ_{0} ^{t}λ_{0}] ⋄ [λ, μ, κ]
= [λ_{0} + λ, μ_{0} + μ, κ + κ_{0} + μ_{0} ^{t}λ_{0} + λ_{0} ^{t}μ + μ^{t}λ_{0}].

Thus we get

(ϑ_{M,α}f)((λ_{0}, μ_{0}, κ_{0}) ∘ g)

= e^{2πiσ{M(κ+κ_{0}+μ_{0} ^{t}λ_{0}+λ_{0} ^{t}μ+μ^{t}λ_{0})}} ∑_{N∈ℤ^{(h,g)}} f(λ + λ_{0} + N) e^{2πi{(α+2MN)^{t}(μ_{0}+μ)}}

= e^{2πiσ(Mκ_{0})} · e^{2πiσ(Mκ)} · e^{2πiσ(α^{t}μ_{0})} ∑_{N∈ℤ^{(h,g)}} f(λ + N) e^{2πiσ{(α+2MN)^{t}μ}}

= e^{2πiσ(Mκ_{0})} (ϑ_{M,α}f)(g).

Here in the above equalities we used the facts that 2σ(MN ^{t}μ_{0}) ∈ ℤ and α^{t}μ_{0} ∈ ℤ. It is easy to show that

∫_{Γ_{L}\G} |ϑ_{M,α}f(ḡ)|^{2} dḡ = ∫_{ℝ^{(h,g)}} |f(λ)|^{2} dλ = ∥f∥_{2}^{2} < ∞.

This completes the proof of lemma 3.4. //

Finally it is easy to show that for each α ∈ J , the mapping ϑ_{M,α} intertwines the Schrödinger representation (U(σ_{M}), L^{2}(ℝ^{(h,g)}, dξ)) and the representation (π_{M,α}, H_{M,α}). Therefore, by lemma 3.4, for each α ∈ J , π_{M,α} is unitarily equivalent to U(σ_{M}) and so π_{M,α} is an irreducible unitary representation of G. According to (3.20), the induced representation π_{M} is unitarily equivalent to

⨁ U(σ_{M}) ( ( det 2M)^{g}-copies ).

This completes the proof of Main Theorem. □

4 Relation of Lattice Representations to Theta Functions

In this section, we state the connection between the lattice representations and theta functions. As before, we write V = ℝ^{(h,g)}×ℝ^{(h,g)} ≅ ℂ^{(h,g)}, L = ℤ^{(h,g)}×ℤ^{(h,g)} and M is a positive symmetric half-integral matrix of degree h. The function q_{M} : L ⟶ ℝ/2ℤ = [0, 2) defined by

(4.1)  q_{M}((ξ, η)) := 2 σ(Mξ^{t}η), (ξ, η) ∈ L

satisfies the condition (3.8). We let φ_{M,q_{M}} : Γ_{L} ⟶ ℂ_{1}^{×} be the character of Γ_{L} defined by

φ_{M,q_{M}}((l, κ)) = e^{2πi σ(Mκ)} e^{πi q_{M}(l)} , (l, κ) ∈ Γ_{L}.

We denote by H_{M,q_{M}} the Hilbert space consisting of measurable functions ϕ : G ⟶ ℂ which satisfy the conditions (4.2) and (4.3):

(4.2) ϕ((l, κ) ∘ g) = φ_{M,q_{M}}((l, κ)) ϕ(g) for all (l, κ) ∈ Γ_{L} and g ∈ G.

(4.3)  ∫_{Γ_{L}\G} ∥ϕ(ġ)∥^{2} dġ < ∞, ġ = Γ_{L} ∘ g.

Then the lattice representation

π_{M,q_{M}} := Ind_{Γ_{L}}^{G} φ_{M,q_{M}}

of G induced from the character φ_{M,q_{M}} is realized in H_{M,q_{M}} as

( π_{M,q_{M}}(g_{0}) ϕ) (g) = ϕ(gg_{0}), g_{0}, g ∈ G, ϕ ∈ H_{M,q_{M}}.

Let H_{M,q_{M}} be the vector space consisting of measurable functions F : V ⟶ ℂ satisfying the conditions (4.4) and (4.5).

(4.4)  F (λ + ξ, μ + η) = e^{2πi σ{M(ξ^{t}η+λ^{t}η−μ^{t}ξ)}} F (λ, μ)

for all (λ, μ) ∈ V and (ξ, η) ∈ L.

(4.5)  ∫_{L\V} ∥F (v̇)∥^{2} dv̇ = ∫_{I_{λ}×I_{μ}} ∥F (λ, μ)∥^{2} dλdμ < ∞.

Given ϕ ∈ H_{M,q_{M}} and a fixed element Ω ∈ H_{g}, we put

(4.6) E_{ϕ}(λ, μ) := ϕ((λ, μ, 0)), λ, μ ∈ ℝ^{(h,g)}, (4.7) F_{ϕ}(λ, μ) := ϕ([λ, μ, 0]), λ, μ ∈ ℝ^{(h,g)},

(4.8) F_{Ω,ϕ}(λ, μ) := e^{−2πi σ(MλΩ^{t}λ)} F_{ϕ}(λ, μ), λ, μ ∈ ℝ^{(h,g)}.

In addition, we put for W = λΩ + μ ∈ ℂ^{(h,g)},

(4.9)  ϑ_{Ω,ϕ}(W) := ϑ_{Ω,ϕ}(λΩ + μ) := F_{Ω,ϕ}(λ, μ).

We observe that E_{ϕ}, F_{ϕ}, F_{Ω,ϕ} are functions defined on V and ϑ_{Ω,ϕ} is a function defined on ℂ^{(h,g)}.

Proposition 4.1. If ϕ ∈ H_{M,q_{M}}, (ξ, η) ∈ L and (λ, μ) ∈ V, then we have the formulas

(4.10)  E_{ϕ}(λ + ξ, μ + η) = e^{2πi σ{M(ξ^{t}η+λ^{t}η−μ^{t}ξ)}} E_{ϕ}(λ, μ).
(4.11)  F_{ϕ}(λ + ξ, μ + η) = e^{−4πi σ(Mξ^{t}μ)} F_{ϕ}(λ, μ).

(4.12)  F_{Ω,ϕ}(λ + ξ, μ + η) = e^{−2πi σ{M(ξΩ^{t}ξ+2λΩ^{t}ξ+2μ^{t}ξ)}} F_{Ω,ϕ}(λ, μ).

If W = λΩ + η ∈ ℂ^{(h,g)}, then we have

(4.13)  ϑ_{Ω,ϕ}(W + ξΩ + η) = e^{−2πi σ{M(ξΩ^{t}ξ+2W ^{t}ξ)}} ϑ_{Ω,ϕ}(W).

Moreover, F_{ϕ} is an element of H_{M,q_{M}}.

Proof. We note that

(λ + ξ, μ + η, 0) = (ξ, η, −ξ^{t}μ + η^{t}λ) ∘ (λ, μ, 0).

Thus we have

E_{ϕ}(λ + ξ, μ + η) = ϕ((λ + ξ, μ + η, 0))
= ϕ((ξ, η, −ξ^{t}μ + η^{t}λ) ∘ (λ, μ, 0))
= e^{2πiσ{M(ξ^{t}η+λ^{t}η−μ^{t}ξ)}} ϕ((λ, μ, 0))
= e^{2πiσ{M(ξ^{t}η+λ^{t}η−μ^{t}ξ)}} E_{ϕ}(λ, μ).

This proves the formula (4.10). We observe that

[λ + ξ, μ + η, 0] = (ξ, η, −ξ^{t}μ − μ^{t}ξ − η^{t}ξ) ∘ [λ, μ, 0].

Thus we have

F_{ϕ}(λ + ξ, μ + η) = ϕ([λ + ξ, μ + η, 0])
= e^{−2πi σ{M(ξ^{t}μ+μ^{t}ξ+η^{t}ξ)}}
× e^{2πi σ(Mξ^{t}η)} ϕ([λ, μ, 0])
= e^{−4πi σ(Mξ^{t}μ)} ϕ([λ, μ, 0])
= e^{−4πi σ(Mξ^{t}μ)} F_{ϕ}(λ, μ).

This proves the formula (4.11). According to (4.11), we have

F_{Ω,ϕ}(λ + ξ, μ + η) = e^{−2πi σ{M(λ+ξ)Ω^{t}(λ+ξ)}} F_{ϕ}(λ + ξ, μ + η)
= e^{−2πi σ{M(λ+ξ)Ω^{t}(λ+ξ)}}
× e^{−4πi σ(Mξ^{t}μ)} F_{ϕ}(λ, μ)
= e^{−2πi σ{M(ξΩ^{t}ξ+2λΩ^{t}ξ+2μ^{t}ξ)}}
× e^{−2πi σ(MλΩ^{t}λ)} F_{ϕ}(λ, μ)
= e^{−2πi σ{M(ξΩ^{t}ξ+2λΩ^{t}ξ+2μ^{t}ξ)}} F_{Ω,ϕ}(λ, μ).

This proves the formula (4.12). The formula (4.13) follows immediately from the formula (4.12). Indeed, if W = λΩ + μ with λ, μ ∈ ℝ^{(h,g)}, we have

ϑ_{Ω,ϕ}(W + ξΩ + η) = F_{Ω,ϕ}(λ + ξ, μ + η)
= e^{−2πi σ{M(ξΩ^{t}ξ+2(λΩ+μ)^{t}ξ)}} F_{Ω,ϕ}(λ, μ)
= e^{−2πi σ{M(ξΩ^{t}ξ+2W ^{t}ξ)}} ϑ_{Ω,ϕ}(W).
□

Remark 4.2. The function ϑ_{Ω,ϕ}(W) is a theta function of level 2M with respect to Ω if ϑ_{Ω,ϕ} is holomorphic. For any ϕ ∈ H_{M,q_{M}}, the function ϑ_{Ω,ϕ} satisfies the well known transformation law of a theta function. In this sense, the lattice representation ( π_{M,q_{M}}, H_{M,q_{M}} ) is closely related to theta functions.

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Department of Mathematics Harvard University Cambridge, MA 02138 U. S. A.

and

Max-Planck Institut für Mathematik Gottfried-Claren-Strasse 26 53225 Bonn Germany

email : yang@mpim-bonn. mpg.de or yang@math.harvard.edu