Shimura operators for certain Hermitian symmetric superpairs
Songhao Zhu
TL;DR
This work extends the Shimura–Sahi–Zhang eigenvalue framework to Hermitian symmetric superpairs by constructing Shimura operators in $ rak U( rak g)^{ rak k}$ for $( rak g, rak k)=( rak{gl}(2p|2q), rak{gl}(p|q)frak{gl}(p|q))$ and comparing their Harish-Chandra images to Type BC interpolation polynomials $I_ u$. Central to the result is a conjectured $ rak k$-sphericity condition for certain finite-dimensional $ rak g$-modules $V_ au$; the authors prove the conjecture for $p=q=1$ via quasi-spherical vectors in Kac modules and explicit coordinates. Assuming sphericity, they show $Gamma(D_ u)=c_ u I_ u$ with $c_ u eq 0$, thereby linking invariant differential operators on the superspace to Sergeev–Veselov interpolation polynomials. The paper develops a detailed realization of the symmetric superpair, the restricted root data of Type BC, and a generalized Verma-module framework to establish the vanishing properties that underpin these eigenvalue results. This approach provides a new algebraic avenue for the eigenstructure of invariant operators in the super setting and has potential extensions to broader Hermitian symmetric superpairs. $${}$
Abstract
We give a partial super analog of a result obtained by S. Sahi and G. Zhang relating Shimura operators and certain interpolation symmetric polynomials. In particular, we study the pair $(\mathfrak{gl}(2p|2q), \mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q))$, define the super Shimura operators in $\mathfrak{U}(\mathfrak{g})^\mathfrak{k}$, and using a new method, prove that their images under the Harish-Chandra homomorphism are proportional to Sergeev and Veselov's Type $BC$ interpolation supersymmetric polynomials, under the assumption that a family of irreducible $\mathfrak{g}$-modules are spherical. We prove this conjecture using the notion of quasi-sphericity for Kac modules when $p=q=1$, and give explicit coordinates of (quasi-)spherical vectors.