Invariant holonomic systems for symmetric spaces

Toby Stafford (University of Manchester)

Wednesday 17th November, 2021 16:00-17:00 Maths 110


Fix a complex reductive Lie group G with Lie algebra g. Harish-Chandra's famous regularity theorem describes the equivariant eigendistributions on a real form of g. Algebraically this boils down to understanding the Harish-Chandra module N= D(V)/D(V)g+D(V)Sym(V)^G_+, over the ring of differential operators D(V) on V = g, where Sym(V) is identified with the constant coefficient differential operators. Harish-Chandra’s theorem then says that N has no delta-torsion factor module for the discriminant delta. Hotta and Kashiwara further showed that N is semi-simple, which is important for the geometric theory of g-representations.

We generalise these results to certain symmetric spaces V (and more generally). First, we show that there is a natural surjection from the invariant ring D(V )^G onto a spherical Cherednik algebra A. Analogues of the Harish-Chandra module N and delta are defined here and we prove:

(1) If A is simple, then N has no factor, nor submodule, that is delta-torsion.

(2) N is a semi-simple D(V )-module if and only if the Hecke algebra associated to A is semi-simple.

This has applications to eigendistributions on V and to admissible D-modules.

This work is joint with Bellamy, Levasseur and Nevin

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