A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are described completely for classes that meet a parabolic subgroup whose components are of type A or are dihedral, and for the class of Coxeter elements.

Let k be an algebraically closed field of positive characteristic, G a reductive group over k, and V a finite dimensional G-module. Let P be a parabolic subgroup of G, and UP its unipotent radical. We prove that if S = SymV has a good filtration, then SUP is strongly F -regular.

We define Eisenstein series twisted by modular symbols for the group SLn, generalizing a construction of the first author [12, 13]. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL2 and SL3.

Let q be a power of a prime and n a positive integer. Let P (q) be a parabolic subgroup of the finite general linear group GLn(q). We show that the number of P (q)conjugacy classes in GLn(q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin in [1].

In this paper, we show that the boundary ∂Σ(W, S) of a rightangled Coxeter system (W, S) is minimal if and only if W S̃ is irreducible, where W S̃ is the minimum parabolic subgroup of finite index in W . We also provide several applications and remarks.

We study the mirror symmetry on singular locus of Hitchin system at two levels. First, by covering it (supports of) (BBB)-branes, corresponding to Higgs bundles reducing their structure group Levi subgroup some parabolic P, whose conjectural dual (BAA)-branes we describe. Heuristically speaking, latter are given unipotent radical P. Second, when P is a Borel subgroup, able construct family hype...

Coxeter matroids are combinatorial objects associated with finite Coxeter groups; they can be viewed as subsets M of the factor set W/P of a Coxeter group W by a parabolic subgroup P which satisfy a certain maximality property with respect to a family of shifted Bruhat orders on W/P . The classical matroids of matroid theory are exactly the Coxeter matroids for the symmetric group Symn (which i...

Let $ G = \prod_{i 1}^{\mathsf r} G_i be a product of simple real algebraic groups rank one and \Gamma an Anosov subgroup with respect to minimal parabolic subgroup. For each \mathsf v in the interior positive Weyl chamber, let \mathscr R_ v\subset \Gamma\backslash denote Borel subset all points recurrent \exp (\mathbb R_+ v) $-orbits. maximal horospherical N $, we show that $-action on {\maths...

Let 1 → (K, K1) → (G, NG(K1)) → (Q, Q1) → 1 be a short exact sequence of pairs of finitely generated groups with K strongly hyperbolic relative to the subgroup K1. Let relative hyperbolic boundary of K with respect to K1 contains atleast three distinct parabolic end points and for all g ∈ G there exists k ∈ K such that gK1g −1 = kK1k , then we prove that there exists a quasi-isometric section s...

Let L be a Lie group and A a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L and G---A = L. Let a~G be such that Ada is semisimple and not contained in a compact subgroup of Aut (Lie(G)). Consider the expanding horospherical subgroup of G associated to a defined as U + = {g~ G:a-"ga"~ e as n ~ oo}. Let f2 be a non-empty open subset of U § and nf-* ~ be ...