Related papers: On exterior powers of reflection representations, …

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural…

Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between…

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad…

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that…

We continue our study of Hilbert space representations of the Reflection Equation Algebra, again focusing on the algebra constructed from the $R$-matrix associated to the $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We develop a form…

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this…

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural…

It is proved that certain types of modular cusp forms generate irreducible automorphic representation of the underlying algebraic group. Analogous archimedean and non-archimedean local statements are also given.

We give a survey of several models of irreducible complementary series representations and their limits, special representations, for the groups SU(n,1) and SO(n,1), including new ones. These groups, whose geometrical meaning is well known,…

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…

A representation $\rho$ of a compact group $\mathbb{G}$ selects eigenvalues if there is a continuous circle-valued map on $\mathbb{G}$ assigning an eigenvalue of $\rho(g)$ to every $g\in \mathbb{G}$. For every compact connected…

We announce here a number of results concerning representation theory of the algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra). We consider the question on 'classification' of finite-dimensional modules over the…

Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter…

We consider the energy representation for the gauge group. The gauge group is the set of (C^{\infty})-mappings from a compact Riemannian manifold to a semi-simple compact Lie group. In this paper, we obtain irreducibility of the energy…

We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0,…

A well-known theorem of Burnside says that if $\rho$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $\rho$.…

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. Such model can…

Let $G$ be a reflection group acting on a vector space $V$ (over a field with zero characteristic). We denote by $S(V^*)$ the coordinate ring of $V$, by $M$ a finite dimensional $G$-module and by $\chi$ a one-dimensional character of $G$.…

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer…

In this work we study representations of certain Coxeter groups to obtain some properties of the corresponding reflection groups.