Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let p = LieP be the Lie algebra of P with Levi decomposition p = l⊕ u, where u is the Lie algebra of the unipotent radical of P and l is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson ([16]), P acts on u with an open dense orbit; this orbit is ...

Let [p] = {1, . . . , p} denote a totally ordered alphabet on p letters, and let α = (α1, . . . ,αm) ∈ [p1], β = (β1, . . . ,βm) ∈ [p2]. We say that α is order-isomorphic to β if for all 1≤ i< j ≤ m one has αi < α j if and only if βi < β j. For two permutations π ∈ Sn and τ ∈ Sk, an occurrence of τ in π is a subsequence 1 ≤ i1 < i2 < .. . < ik ≤ n such that (πi1 , . . . ,πik ) is order-isomorph...

We study those finite dimensional quotients of the rational Cherednik algebra at t = 0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain “cuspidal” quotient of a rational Cherednik algebra associated to a parabolic subgroup of W .

This paper discusses applications of LERF to tame covers of certain hyperbolic 3-manifolds. For example, if M = H/Γ is an orientable noncompact finite volume hyperbolic 3-manifold and Γ is LERF, then the cover of M corresponding to any subgroup of Γ generated by two parabolic elements is tame. 2000 Mathematics Subject Classification 57M50, 20H10

A subgroup H of a group G is conjugately dense in G if for each element g in G the intersection of H with the conjugates of g in G is nonempty. Conjugately dense subgroups deal with interesting open problems, related to parabolic groups. In the present paper we study them with respect to suitable coverings. 2000 Mathematics Subject Classification: 20E45, 20B35, 20F99.

We show that every admissible representation of a real reductive group has a canonical system of Sobolev norms parametrized by positive characters of a minimal parabolic subgroup. These norms are compatible with morphisms of representations. Similar statement also holds for representations of reductive p-adic groups. 1991 Mathematics Subject Classification: 22E46 43A70 46E39

Let G be a classical Lie group and P a maximal parabolic subgroup. We describe a quantum Pieri rule which holds in the small quantum cohomology ring of G/P . We also give a presentation of this ring in terms of special Schubert class generators and relations. This is a survey paper which reports on joint work with Anders S. Buch and Andrew Kresch.

Let F be a non-Archimedean locally compact field, let G be a split connected reductive group over F . For a parabolic subgroup Q ⊂ G and a ring L we consider the G-representation on the L-module (∗) C∞(G/Q, L)/ ∑