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Lie Group Quantum Mathematics Seminar

Eigenvalues of Shimura Operators for Lie Superalgebras

Songhao Zhu, Rutgers University

Location:  Hill Center Room 705
Date & time: Friday, 10 March 2023 at 12:10AM - 1:10PM

Abstract A recent work by Siddhartha Sahi and Genkai Zhang relates the Shimura operators defined for Hermitian symmetric pairs (g, k) and symmetric interpolation polynomials. Specifically, these operators, which are realized as elements in the universal enveloping algebra, have eigenvalues equal to the Okounkov interpolation symmetric polynomials of Type BC.

We obtained partial generalization to the super case. In particular, I will discuss super Shimura operators defined for a Hermitian symmetric superpair (g, k), the Harish-Chandra isomorphism, the Weyl groupoid invariance, and how these puzzle pieces are put together in the very specific Type A scenario. Of particular interest and challenge are the spherical representations which are the key ingredient in our proof. With the help of Kac modules, we propose an algebraic way to answer the question of when is an irreducible highest weight g-module spherical.

No prior knowledge of Lie superalgebras will be assumed.

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