Shimura operators for certain Hermitian symmetric superpairs

2022/12/19 by Zhu, Songhao
#05E10 #17B10 #17B60 #81Q60 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2212.09249

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 (\mathfrakgl(2p|2q), \mathfrakgl(p|q)⊕\mathfrakgl(p|q)), define the super Shimura operators in \mathfrakU(\mathfrakg)^\mathfrakk, 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 \mathfrakg-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.

Related

• Eigenvalues of supersymmetric Shimura operators and interpolation polynomials
• Supersingular loci of some Shimura varieties for CM unitary groups
• The supersingular locus of unitary Shimura varieties with exotic good reduction
• Schur Q-functions and the Capelli eigenvalue problem for the Lie superalgebra \mathfrak q(n)
• Simultaneous supersingular reductions of abelian varieties