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WednesdayOct 06, 202116:30

Algebraic Geometry and Representation Theory Seminar

Speaker:Dmitry Gourevitch Title:Finite multiplicities beyond spherical pairsAbstract:opens in new windowin html pdfopens in new windowZoom at: https://weizmann.zoom.us/j/98304397425
Let G be a real reductive algebraic group, and let H be an algebraic subgroup of G.
Itis known that the action of G on the space of functions on G/H is "tame" if this space is spherical.
In particular, the multiplicities of the space of Schwartz functions on G/H are finite in this case.
I will talk about a recent joint work with A. Aizenbud in which we formulate and analyze a generalization of sphericity that implies finite multiplicities in the Schwartz space of G/H for small enough irreducible smooth representations of G.
In more detail, for every G-space X, and every closed G-invariant subset S of the nilpotent cone of the Lie algebra of G, we define when X is S-spherical, by means of a geometric condition involving dimensions of fibers of the moment map. We then show that if X is S-spherical, then every representation with annihilator variety lying in S has (at most) finite multiplicities in the Schwartz space of X.
For the case when S is the closure of the Richardson orbit of a parabolic subgroup P of G, we show that the condition is equivalent to P having finitely many orbits on X. We give applications of our results to branching problems.
Our main tool in bounding the multiplicity is the theory of holonomic D-modules. After formulating our main results, I will briefly recall the necessary aspects of this theory and sketch our proofs.
The talk is based on arXiv:2109.00204.