Related papers: Matrix balls, radial analysis of Berezin kernels, …

Let $G$ be a connected simple Lie group of real rank one and finite center, and let $K$ be a maximal compact subgroup. We study the families of spherical, ball, and uniform averages $(\sigma_t)_{t>0}$, $(\beta_t)_{t>0}$, and $(\mu_t)_{t>0}$…

Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on…

We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible *-representations. Then we study their behaviour under the…

Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in…

We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion)…

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$…

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$…

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig…

For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For…

We argue that modular classes of Q-manifolds provide an efficient method for addressing the existence of supersymmetric Berezin volumes in the supergeometric representation theory of the $\mathcal{N}=2$ $d=1$ supertranslation algebra. We…

In this paper, we study noncommutative varieties in polydomains in $B(H)^n$. The goal is to understand the structure of these varieties, determine their elements and classify them up to unitary equivalence. Using noncommutative Berezin…

We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking…

For a Hermitian Lie group $G$, we study the family of representations induced from a character of the maximal parabolic subgroup $P=MAN$ whose unipotent radical $N$ is a Heisenberg group. Realizing these representations in the non-compact…

We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of…

The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U(1), U(2),.... This novel object is a commutative graded algebra R with infinite-dimensional homogeneous…

Linear operators $R$ are introduced on tensor products of evaluation modules of $U'_q\bigl(\widehat{sl}(2)\bigr)$ obtained from the complementary and strange series representations. The operators $R$ satisfy the intertwining condition on…

For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how…

The main objective of this paper is twofold. One is to classify and construct $SL(3,\mathbb{R})$-intertwining differential operators between vector bundles over the real projective space $\mathbb{RP}^2$. It turns out that two kinds of…

We give a fundament for Berezin's analytic $\Psi$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{\'c} spaces. We deduce basic continuity results for such $\Psi$do, especially when the operator kernels are in…

Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in the high-energy physics literature. These are finite dimensional noncommutative approximations of the algebra of functions on a classical space. The most well…