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

Let W be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer ZW (WI) of an arbitrary parabolic subgroup WI into the center of WI , a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. Moreover, by using our description of ZW (WI), we reveal a further...

0. Introduction. Let Qp be the field of p-adic numbers, and let Q∞ = R. Let Gp be a connected semisimpleQp-algebraic group. The unipotent radical of a proper parabolic Qp-subgroup of Gp is called a horospherical subgroup. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups. In [5] and [6], we studied discrete subgroups generated...

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 ...

Let G be a connected reductive group over an algebraically closed field k of characteristic not 2; let θ ∈ Aut(G) be an involution and K = Gθ ⊆ G the fixed point group of θ and let P ⊆ G be a parabolic subgroup. The set K\G/P of (K , P)-double cosets in G plays an important role in the study of Harish Chandra modules. In [BH00] we gave a description of the orbits of symmetric subgroups in a fla...

We solve the subgroup conjugacy problem for parabolic subgroups and Garside subgroups of a Garside group, and we present deterministic algorithms. This solution may be improved by using minimal simple elements. For standard parabolic subgroups of Garside groups we provide e ective algorithms for computing minimal simple elements.

Let G be a reductive algebraic group defined over an algebraically closed field k. Let H be a closed connected subgroup of G containing a maximal torus T of G. In [13] it was shown (at least in characteristic zero) that the parabolic subgroups of G can be characterized among all such subgroups H by a certain finiteness property of the induction functor (-)Iz and its derived functors Lk,G(-). Th...

A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL(2,C) consisting of the upper-triangular matrices. A description of the set ...

Let k be an algebraically closed field and let G be a reductive linear algebraic group over k. Let P be a parabolic subgroup of G, Pu its unipotent radical and pu the Lie algebra of Pu. A fundamental result of R. Richardson says that P acts on pu with a dense orbit (see [9]). The analogous result for the coadjoint action of P on pu is already known for char k = 0 (see [5]). In this note we prov...

A rational group of hermitian type is a Q-simple algebraic group G such that the symmetric space D of maximal compact subgroups of the real Lie group G(R) is a hermitian symmetric space of the non-compact type. There are two major classes of subgroups of G of importance to the geometry of D and to arithmetic quotients XΓ = Γ\D of D: parabolic subgroups and reductive subgroups. The former are co...