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...

In symmetric groups, viewed as Coxeter 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 tableau, using the RobinsonSchensted correspondance. We show also that the set of elements of maximal length in a two sided cell is the set...

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under developmen...

In symmetric groups, viewed as Coxeter 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 these sets with tableau, using the RobinsonSchensted correspondence. We show also that the set of elements of maximal length in a two sided cell is the s...