# Manchester Algebra Seminar - Toby Stafford - University of Manchester

Dates: | 25 May 2021 |
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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.

Time: 1pm Place: Zoom