Eigenvalues of supersymmetric Shimura operators and interpolation 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.

23 pages. Title changed to better reflect the results. Main updates include a new proof (leading to Thm B) independent of prior work in the literature, a new appendix, remarks on improvements over an earlier paper, and an example on difficulties with other "gl pairs". Minor updates include corrections of typos and language imprecision, and better notation