Related papers: A Note on the Peter-Weyl Theorem

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation…

With the aim of completing the previous study by A. Or{\l}owski and the author concerning intertwining maps between induced representations and conjugation representation, termed here weighted class operators, we compute the latter…

The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…

We deal with the Fourier-like analysis of functions on discrete grids in two-dimensional simplexes using $C-$ and $E-$ Weyl group orbit functions. For these cases we present the convolution theorem. We provide an example of application of…

We survey some aspects of the pseudo-differential Weyl calculus for irreducible unitary representations of nilpotent Lie groups, ranging from the classical ideas to recently obtained results. The classical Weyl-H\"ormander calculus is…

We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making…

We provide detailed calculations for the classification of representations of compact simple Lie groups with non-empty boundary in the orbit space, first announced in a previous paper [arXiv:2112.00513] by the same authors.

We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also…

We review various notions of correspondences for locally compact groupoids with Haar systems, in particular a recent definition due to R.D. Holkar. We give the construction of the representations induced by such a correspondence. Finally,…

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the second in a series of papers in…

Following the general idea of Schur--Weyl scheme and using two suitable symmetric groups (instead of one), we try to make more explicit the classical problem of decomposing tensor representations of finite and infinite symmetric groups into…

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those…

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable…

The paper is devoted to generalizations of actions of topological groups on manifolds. Instead of a topological group, we consider a local topological group generalizing the notion of a~germ or a~neighborhood in a topological group. The…

A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to…

To study induced representation of some class of groups, Mackey's theory is very useful. In this paper, we consider some generalization of Mackey's theory for locally profinite groups. In particular, we give conditions on groups under which…

Let $G$ be a reductive algebraic group over $\mathbb{Q}$ and $\Gamma\subset G(\mathbb{Q})$ an arithmetic subgroup. Let $K_\infty\subset G(\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions…

We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of L-functions. Comparison is made to the analogous…