AG Lie-Gruppen: L. Seco, Universidade de Brasília: Counting Geodesics in Compact Symmetric Spaces: The Centralizer Modulo the Lattice

Counting Geodesics in Compact Symmetric Spaces: The Centralizer
Modulo the Lattice – Vortragender: Lucas Seco, Universidade de Brasília – Einladender: K.-H. Neeb

Abstract: We describe the
inverse image of the Riemannian exponential map of a compact symmetric space as
the disjoint union of orbits through the maximal torus: These are orbits of a
subgroup of the isotropy group and we show how their dimension (infinitesimal
data) and connected components (topological data) are encoded in the diagram,
Weyl group and lattice of the symmetric space: This is precisely what we mean
by counting geodesics.

A central object that
appears is the centralizer of an element H in the maximal torus by the Weyl
group modulo the lattice, it is isomorphic to the centralizer of H in the
affine Weyl group and well known for simply connected spaces: We discuss some
properties that we are currently investigating for non-simply connected spaces.

References: Seco, L., Patrão, M.: Counting geodesics on compact
symmetric spaces, Monatsh. Math. (2024) Seco, L., San Martin, L.A.B.: Counting
geodesics on compact Lie groups, Diff. Geom. Appl. 56 (2018)