J. H. Yang Nagoya Math. J. Vol. 123 (1991), 103 117 HARMONIC ANALYSIS ON THE QUOTIENT SPACES OF HEISENBERG GROUPS JAE HYUN YANG A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisen berg group. For any positive integers g and h, we consider the Heisenberg group Hf h) : = {[(λ, μ), κ]\λ,μβ R (h > g) , ic e 2ϋ (Λ Λ) , κ + μ ι λ symmetric} endowed with the following multiplication law U μ\ κ] ο [(λ\ μ% Kf] = [(λ + λ',μ + μ% κ + κ' + λ'μ' μ </] . The mapping 9 [(λ, μ\ κ] defines an embedding of Η ( £ Λ) into the symplectic group Sp(g + h, R). We refer to [Ζ] for the motivation of the study of this Heisenberg group H%' h \ Hf' h) denotes the discrete subgroup of Η% >/ι) consisting of integral elements, and L\Hf h) \Hf h) ) is the L 2 space of the quotient space Hf h) \ H { £> h) with respect to the invariant measure d*n ' ' άλ Λ , 8 ιάλ^άμ η άμ κ ^_ χ άμ Κε άκ η άκ η dfc h _ l>h dtc hh . We have the natural unitary representation ρ on L 2 (H ( i ih) \H ( R fh) ) given by , μ'\ *'])Φ([(1 μ\ *]) = φ(Μ μ), fc] ο [(λ\ μ'), ^]). λ 0 0 0 Κ 0 0 0 μ 0 Received October 11, 1990. 103
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J.-H. YangNagoya Math. J.Vol. 123 (1991), 103-117
HARMONIC ANALYSIS ON THE QUOTIENT
SPACES OF HEISENBERG GROUPS
JAE-HYUN YANG
A certain nilpotent Lie group plays an important role in the study
of the foundations of quantum mechanics ([Wey]) and of the theory of
theta series (see [C], [I] and [Wei]). This work shows how theta series
are applied to decompose the natural unitary representation of a Heisen-
berg group.
For any positive integers g and h, we consider the Heisenberg group
Hfh) : = {[(λ, μ), κ]\λ,μβ R(h>g), ic e 2ϋ(Λ Λ ), κ + μ ιλ symmetr ic}
The Stone-von Neumann theorem says that an irreducible representationρ of H(g>h) is characterized by a real symmetric matrix c e Rih'h) (c ψ 0)such that
pXm 0), *]) = e x p {πίσ(οκ)}1 κ = ικβ R ^ h ) ,
where / denotes the identity mapping of the representation space. If c = 0,then it is characterized by a pair (k, m)eR{hg X R{hig) such that
PkM, μ), *]) = exp {2nia{k ιλ + m *μ)}1.
But only the irreducible representations ρ^ with Ji = lJi even integraland pkim (k, meZ{hig)) could occur in the right regular representation ρ in
In this article, we decompose the right regular representation p. Thereal analytic functions defined in (1.5) play an important role in decom-posing the right regular representation p.
NOTATIONS. We denote Ζ, R and C the ring of integers, the field ofreal numbers and the field of complex numbers respectively. F ( M ) denotesthe set of all k X I matrices with entries in a commutative ring F. Eg
denotes the identity matrix of degree g. σ(Α) denotes the trace of asquare matrix A.
Ζ§# = {J= (Jkl) e Z(^\Jkl > 0 for all /e, /},
J ± Ski = {J\U ' ' 9 Jkl ± 1> ' * * ί Jhg) 9
(λ + Ν+Α)< = (λη + Νη + Any« (Xhg + Nhg + Ahgy«.
§1. Theta series
Let Hg be the Siegel upper half plane of degree g. We fix an ele-ment Ω e Hg once and for all. Let J be a positive definite, symmetriceven integral matrix of degree h. A holomorphic function /: Cihig) -» Csatisfying the functional equation
for all X^eZ{Kg) is called a theta series of level Ji with respect to Ω.The set Τ^(Ω) of all theta series of level J( with respect to Ω is a vectorspace of dimension (det Ji)g with a basis consisting of theta series
where A runs over a complete system of representatives of the cosets
DEFINITION 1.1. A function φ: C(h'g) χ C(h>g)-+C is called an aux-iliary theta series of level Ji with respect to Ω if it satisfies the followingconditions (i) and (ii):
(i) <p(U, W) is a polynomial in W whose coefficients are entirefunctions,
(ii) <p(U + λ, W + λΩ + μ) = exp{- πΐ(^(λΩιλ + 2λ'Ψ))}φ(υ, W) forall (λ,μ)βΖ^^ χ Z^s\
The space θ ^ } of all auxiliary theta series of level Ji with respectto Ω has a basis consisting of the following functions:
(1.3) &/>\Α](Ω\χ,μ + Μ):= £ (λ + Ν + Α)<L U J
X exp {πίσ(^((Ν + Α)Ω ι(Ν + A) + (μ + λΩ) ι(Ν + A)))}.
where A (resp. J) runs over the cosets JC-lZ^^\Z^e) (resp. Z^g)).
DEFINITION 1.2. A real analytic function φ: Rih'g) χ R^g) -> C iscalled a mixed theta series of level Ji with respect to Ω if φ satisfies thefollowing conditions (1) and (2):
(1) φ{λ, μ) is a polynomial in λ whose coefficients are entire functionsin complex variables Ζ = μ + λΩ;
(2) φ(λ + I μ + β) = exp {- πΐσ(^(λΩ Γλ + 2(μ + λΩ) Γλ))}φ(λ, μ) for all(2,/0eZ (A"> X Ζ ( Λ '^ .
If AeJe-W^IZ*'* and JeZ^g\
(1.4) ^ f ^ l ( f l U ^ + >lO):= Σ tt + tf+A)'L O J 8L O J
X exp {iaa(Jt((N + Α)β 4(iV + A) + 2(μ + ίΛ) έ(Ν + A)))}
is a mixed theta series of level Ji.Now for a positive definite symmetric even integral matrix Ji of de-