Eigenvalues of supersymmetric Shimura operators and interpolation polynomials

TL;DR

This paper extends the theory of Shimura operators to a supersymmetric setting, showing their Harish-Chandra images relate to $BC$-supersymmetric interpolation polynomials, thus connecting invariant differential operators with supersymmetric polynomials.

Contribution

It introduces supersymmetric analogs of Shimura operators for a specific superpair and proves their Harish-Chandra images are specializations of Sergeev--Veselov's $BC$-supersymmetric interpolation polynomials.

Findings

Harish-Chandra images of supersymmetric Shimura operators are $BC$-supersymmetric interpolation polynomials

Established the supersymmetric generalization of Shimura operators for $(rak{g},rak{k})$

Connected invariant differential operators with Sergeev--Veselov polynomials

Abstract

The Shimura operators are a certain distinguished basis for invariant differential operators on a Hermitian symmetric space. Answering a question of Shimura, Sahi and Zhang showed that the Harish-Chandra images of these operators are specializations of certain $BC$-symmetric interpolation polynomials that were defined by Okounkov. We consider the analogs of Shimura operators for the Hermitian symmetric superpair $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}= \mathfrak{gl}(2p|2q)$ and $\mathfrak{k}= \mathfrak{gl}(p|q)\oplus \mathfrak{gl}(p|q)$ and we prove their Harish-Chandra images are specializations of certain $BC$-supersymmetric interpolation polynomials introduced by Sergeev--Veselov.