Related papers: Heisenberg Groups, Theta Functions and the Weil Re…

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of…

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…

We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics…

The aim of this paper is to study the classification of generic coherent state representations of a Lie group and its connection to the structure theory of K\"{a}hler algebras. The group we deal with is the holomorphic automorphism group of…

A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary…

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d, R), for d >= 3. These are natural generalizations of the wellknown classical Selberg zeta function and L-function…

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

We prove that the global geometric theta-lifting functor for the pair (H, G) is compatible with the Whittaker normalization, where (H,G) is one of the pairs (SO_{2n}, Sp_{2n}), (Sp_{2n}, SO_{2n+2}) or (GL_{n},GL_{n+1}). That is, the…

We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces $Z=G/H$ attached to a real reductive Lie group $G$. A special emphasis is made to the case where $Z$ is real spherical.

In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{\Gamma}$ where $\Gamma$ is a product of general linear groups over a field…

We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important…

In this paper we categorify the Heisenberg action on the Fock space via the category O of cyclotomic rational double affine Hecke algebras. This permits us to relate the filtration by the support on the Grothendieck group of O to a…

The aim of the present article is to reveal a structure shared by two basic zeta-functions in their fourth power moments through the view point of representation theory of Lie groups, relying specifically upon the Kirillov model. It might…

This paper presents the square integrable representations of generalized Weyl-Heisenberg group. We investigate the quasi regular representation of generalized Weyl-Heisenberg group. Moreover, we obtain a concrete from for admissible vector…

A variation of multiple $L$-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function…

Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of…

We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent…

The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions graphically because the…

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary…

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta…