On holomorphic representations of discrete Heisenberg groups

Abstract

We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations which are automorphic forms on the moduli space.

Representations of Higher Adelic Groups and Arithmetic

“…The morphism k defines an isogeny ϕ k : E → E ′ and the sheaf L is defined as (Id × ϕ k ) * P. By Mumford's theory [44], there exists a finite Heisenberg group Ker(ϕ k ), which is a central extension of the group Ker(ϕ k ). Then for all g = (m, p, c, k) ∈Ĝ(χ) the values of the characters Trπχ(g)χ −1 C (c)χ −1 A (k) are theta-functions for the bundle L. In addition, certain orthogonality relations are satisfied by the characters [63].…”

Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields

“…For any α ∈ C * the subspace β(Ψ α ) of the space Ψ isG -invariant. We have the traces Tr α of elements ofG in β(Ψ α ) with respect to a distinguished basis (g α ) l = β(∆(l, g α )) , where l ∈ Z (see discussion in the beginning of this section): The series l∈Zq l 2 η l is the classical Jacobi theta function (compare also with [11,Example]).…”

“…As it was shown in [12] and in the previous publications, the local fields will now be the two-dimensional local fields, and the corresponding discrete group is the simplest non-Abelian nilpotent group of class 2 : the Heisenberg group Heis(3, Z) of unipotent matrices of order 3 with integer entries. This group has a very non-trivial representation theory (see [11,12]).…”

Theta functions and arithmetic quotients of loop groups

“…It should not be too surprising to get some generalizations of part of this paper to more general framework, say, to the situation of higher local fields and adelic spaces, or the so-called C n -categories (see [6], [7], [8] and [10]). For instance, Parshin [9] considered Heisenberg groups associated with objects from C f.g. 0 , the category of finitely generated abelian groups. The loop Heisenberg groups introduced in Section 5 should be viewed as Heisenberg groups associated with objects from C f.g.…”