Abstracts for MPI-Oberseminar

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. In my talk, I am going to speak about actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We show that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite- dimensional $G$-module $V$, the subspace of fixed vectors $V^{U'}$ is a cyclic $U/U'$-module.

We construct some representations of braid groups, arising from the representations of quantum groups.

Thin buildings (in the sense of Tits) arise as coset geometries of Coxeter groups, most of the spherical buildings as coset geometries of groups with a BN-pair. We generalize the notion of a group to ``generalized groups" in such a way that arbitrary (not necessarily thin or spherical) buildings arise as coset geometries of ``generalized Coxeter groups". Each set of right cosets of a subgroup of a given group turns out to be a generalized group. These generalized groups are called ``schurian".

In my talk I will present several topics related to Algebraic K-theory of number fields. I will discuss the connection of divisible elements in even K-groups to the conjectures of Kummer-Vandiver, Iwasawa, Coates-Sinnott and Quillen-Lichtenbaum. I will show the construction of the Stickelberger's splitting map to the boundary map in the Quillen localization sequence and its relation with Coates-Sinnott conjecture and the divisible elements.