Journal of Lie Theory

no. 2
Shimura Operators for Certain Hermitian Symmetric Superpairs
Journal of Lie Theory, Volume 35 (2025) no. 2, pp. 377-410
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.
DOI: 10.5802/jolt.1388
Classification: 17B10, 17B60, 05E10, 81Q60
Keywords: Shimura operators, symmetric superpairs, Lie superalgebras, interpolation polynomials 
@article{JOLT_2025_35_2_a6,
     author = {S. Zhu},
     title = {Shimura {Operators} for {Certain} {Hermitian} {Symmetric} {Superpairs}},
     journal = {Journal of Lie Theory},
     pages = {377--410},
     year = {2025},
     volume = {35},
     number = {2},
     doi = {10.5802/jolt.1388},
     zbl = {08075077},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1388/}
}
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S. Zhu. Shimura Operators for Certain Hermitian Symmetric Superpairs. Journal of Lie Theory, Volume 35 (2025) no. 2, pp. 377-410. doi: 10.5802/jolt.1388

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