Manchester Algebra Seminar - Toby Stafford - University of Manchester
|Dates:||25 May 2021|
|Times:||13:00 - 14:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
|Who is it for:||University staff, Current University students|
Speaker: Toby Stafford - University of Manchester
Title: Invariant holonomic systems for symmetric spaces and quiver representations.
Abstract: This work is joint with Bellamy, Levasseur and Nevins.
Fix a complex reductive Lie group G with Lie algebra g. Harish-Chandra’s
famous regularity theorem describes the G-equivariant eigendistributions on a real
form of g. Algebraically this boils down to understanding a particular module, the
Harish-Chandra module N, over the ring of differential operators D(V) on V = g.
Harish-Chandra’s theorem then says that N has no factor that is ?-torsion, where
? is the discriminant. Hotta and Kashiwara further showed that N is semi-simple
and this has significant applications to the geometric theory of g-representations.
We will be interested in generalising these results to the case where V is a symmetric space (and to more general polar representations), where we also generalise
results of Sekiguchi and others. First, we show that there is a natural surjective
map from the invariant ring D(V)^G to a spherical Cherednik algebra A. This algebra A is a deformation of the fixed ring D(h)^W for an analogue of the Cartan
subalgebra h with Weyl group W. Analogues of the Harish-Chandra module N and
? are defined here and we prove: If A is a simple algebra then N has no factor, nor
submodule, that is ?-torsion. Secondly we prove: N is a semi-simple D(V )-module
if and only if the Hecke algebra associated to A is a semi-simple algebra. If time
permits we will give various applications.
Travel and Contact Information
Zoom - online
Alan Turing Building