Monday, February 15, 2021
02/15/2021 - 10:15am
Abstract. We answer a question of Margulis by proving the following: Let G be a higher rank simple Lie group and let Λ ≤ G be a discrete subgroup of infinite covolume, then the locally symmetric space Λ\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan’s property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank groups and a stationary variant of the Stuck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.
Joint work with Mikolaj Fraczyk
02/15/2021 - 4:30pm
Zhu proved a duality theorem between level one affine Demazure modules and function rings of torus fixed point subschemes of affine Schubert varieties in affine Grassmannian. Using his methods and results, we prove a similar duality theorem between level one twisted affine Demazure modules and twisted affine Schubert varieties for absolutely special parahoric group schemes. As a consequence, we determine the smooth locus of all twisted affine Schubert varieties for many types of parahoric group scheme. This confirms a conjecture of Haines and Richarz for these types of group schemes. If time permits, I will also talk about how this duality theorem is related to the Frenkel-Kac isomorphism for twisted affine Lie algebras, and also the fusion product for twisted affine Demazure modules. This is a joint work with Marc Besson.