Related papers: Higher Lie characters and root enumeration in clas…

Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that, in type $D_r$ with $r$ odd, their number is $2^r-1$. We prove a stronger structural statement. For…

With the introduction of special roots, we show the existence of some special weights with quite interesting properties for finite Lie algebras. We propose and discuss two statements which lead us to an explicit construction of these…

Using a generating function for the Wigner's $D$-matrix elements of $SU(3)$ Weyl's character formula for $SU(3)$ is derived using Schwinger's technique.

We correct the proof of the main result of an earlier paper, parametrizing orbital varieties in a complex simple Lie algebra of type $B$ or $C$ in terms of domino tableaux and showing how to compute the orbital variety attached to an…

We compute the rational Betti numbers of the real toric varieties associated to Weyl chambers of types $E_7$ and $E_8$, completing the computations for all types of root systems.

In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation in terms of the same geometric data attached to it. When specialized to the case of a compact Lie…

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal…

We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard…

We generalize I. Frenkel's orbital theory for non twisted affine Lie algebras to the case of twisted affine Lie algebras using a character formula for certain non-connected compact Lie groups.

The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

For integral weights $\lambda$ and $\mu$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $\mu$ in the irreducible representation with highest weight $\lambda$,…

In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the…

Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to…

We describe algorithms to represent and compute groups of Hecke characters. We make use of an id{\`e}lic point of view and obtain the whole family of such characters, including transcendental ones. We also show how to isolate the algebraic…

The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or…

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $\alpha \not= 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that…

In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of [FT] and [FHT3]. In…

The action of a Weyl group on the associated weight lattice induces an additive action on the symmetric algebra and a multiplicative action on the group algebra of the lattice. We show that the coinvariant space of the multiplicative action…

We show that the irreducible representation of the asymptotic Hecke algebra corresponding to a special representation of a Weyl group admits a basis with strong positivity properties.