Shimura Operators for Certain Hermitian Symmetric Superpairs
Journal of Lie Theory, Tome 35 (2025) no. 2, pp. 377-410
Voir la notice de l'article provenant de la source Heldermann Verlag
We give a partial super analog of a result obtained by Sahi and 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 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.
Classification :
17B10, 17B60, 05E10, 81Q60
Mots-clés : Shimura operators, symmetric superpairs, Lie superalgebras, interpolation polynomials
Mots-clés : Shimura operators, symmetric superpairs, Lie superalgebras, interpolation polynomials
Affiliations des auteurs :
Songhao Zhu 1
Songhao Zhu. Shimura Operators for Certain Hermitian Symmetric Superpairs. Journal of Lie Theory, Tome 35 (2025) no. 2, pp. 377-410. http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a6/
@article{JOLT_2025_35_2_a6,
author = {Songhao Zhu},
title = {Shimura {Operators} for {Certain} {Hermitian} {Symmetric} {Superpairs}},
journal = {Journal of Lie Theory},
pages = {377--410},
year = {2025},
volume = {35},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a6/}
}