A symmetric space analogue of Casselman’s criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space.

for a parabolic subgroup P ⊂ G. In Part I [Na 2], when g is classical, we have proved that any two symplectic resolutions of Ō are connected by a sequence of Mukai flops of type A or of type D. In this paper (Part II), we shall improve and generalize all arguments in Part I so that the exceptional Lie algebras can be dealt with. We shall replace all arguments of [Na 2] which use flags, by those...

Let W be a Coxeter group. We generalise the Knuth plactic relations to W and we show that their equivalence classes decompose the Kazhdan-Lusztig cells. In the case of symmetric groups, we show that the set of elements of minimal length in a double sided cell is the set of elements of maximal length in conjugated parabolic (i.e. Young) subgroups. We also give an interpretation of this set with ...

This is an expanded version of a series of lectures delivered at the 25th Winter School “Geometry and Physics” in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the equivalence between parabolic geometries and underlying geometric structures. The second part of the paper is devoted to constructions which relate parabolic geometr...

For the class of locally compact groups P that can be written as the semidirect product of a locally compact subgroup P0 and a oneparameter group R+ of automorphisms of P0, a new model of representations of the current groups P is constructed. The construction is applied to the maximal parabolic subgroups of all simple groups of rank 1. In the case of the groups G = SO(n, 1) and G = SU(n, 1), a...

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperb...

In [12], I. Kapovich and P. Schupp showed that certain 2-dimensional Artin groups (those with all relator indices at least 7) are hyperbolic relative (in the sense of Farb) to their non-free rank 2 parabolic subgroups. This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of the associated Deligne complex. We prove a relative version of the Mi...

— We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show fo...