Lie Groups and Representation Theory

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Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

2020/07/14

17:30-18:30   Room ## (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology. Online.
Takayuki Okuda (Hiroshima University)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (Japanese)
[ Abstract ]
Let G be a Lie group and X a homogeneous G-space.
A discrete subgroup of G acting on X properly is called a discontinuous group for X.
We are interested in constructions and classifications of discontinuous groups for a given X.

It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly.
However, the cases where the isotropies are non-compact, the same claim does not hold in general.

Let us consider the case where G is a linear reductive.
In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)] gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.

In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces.
As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry.
In particular, for a torsion-free discrete subgroup of G, the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.