In the context of Hecke algebras complex reflection groups, we prove that generalized normalizers parabolic subgroups are semidirect products, under suitable conditions on parameters involved in their definition.

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I • be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjec...

Definition 1. Let G be a permutation group on a set  and x be an element of . Then Gx  g ∈ G ∣ gx  x is called the stabilizer of x and consists of all the permutations of G that produce group fixed points in x. Definition 2. A vector subspace S ⊂ V is isotropic if for any v,w ∈ S, the symmetric bilinear form satisfies: Bv,w  0 Definition 3. A maximal parabolic subgroup in an orthogon...

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on induced from unramified cusp forms finite-dimensional Levi subgroups of maximal parabolic subgroups. Under natural condition subgroups, we prove that the cuspidal are entire full complex plane.

Let P+(n) be the Siegel parabolic subgroup of O(n, n), and P−(n) be the Siegel parabolic subgroup of Sp2n(R). In this paper, we study the coadjoint orbits of P±(n). We establish a one-to-one correspondence between the real coadjoint orbits of Sp2n(R) and the principal coadjoint orbits of P+(2n), and a one-to-one correspondence between the real coadjoint orbits of O(p, n− p) with p ∈ [0, n] and ...

We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the non-reflection part of a reflection centralizer is fre...

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical Pu, and A a closed connected unipotent subgroup of Pu which is normalized by P. We show that P acts on A with nitely many orbits provided A is abelian. This generalizes a well-known niteness result, namely the case when A is central in Pu. We also obtain an analogous result for the adjoint action of P on in...

We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results h...

We prove that any standard parabolic subgroup of any Artin group is convex with respect to the standard generating set.

Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form ...