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Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in…

Let G be the fundamental group of the complement of a K(G,1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group (as defined in the paper). The subgroup of elements in the complex…

Let $B$ be a ring, not necessarily commutative, having an involution $*$ and let ${\mathrm U}_{2m}(B)$ be the unitary group of rank $2m$ associated to a hermitian or skew hermitian form relative to $*$. When $B$ is finite, we construct a…

The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent…

In this paper we study higher level Deligne--Lusztig representations of reductive groups over discrete valuation rings, with finite residue field $\mathbb{F}_q$. In previous work we proved that, at even levels, these geometrically…

In this paper we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations…

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…

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that…

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…

We consider the finite Weyl groups of classical type -- $W(A_{r})$ for $r \geq 1$, $W(B_{r}) = W(C_{r})$ for $r \geq 2$, and $W(D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the…

For any finite partially ordered set $P$, the $P$-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of $P$, and is closely related to the order polynomial of $P$ arising in the theory of…

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a…

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…

We consider the semisimple orbits of a Vinberg $\theta$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one…

We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation $U$. The group $G$ is an arbitrary semidirect product whose normal factor $A$ is abelian and whose homogeneous factor…

Let ${\mathcal O}$ be an involutive discrete valuation ring with residue field of characteristic not 2. Let $A$ be a quotient of ${\mathcal O}$ by a nonzero power of its maximal ideal and let $*$ be the involution that $A$ inherits from…

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality…

We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of…

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a…

We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex…