Shimura operators for certain Hermitian symmetric superpairs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916666815807488 |
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| author | Zhu, Songhao |
| author_facet | Zhu, Songhao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09249 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shimura operators for certain Hermitian symmetric superpairs Zhu, Songhao Representation Theory Combinatorics 17B10, 17B60, 05E10, 81Q60 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. |
| title | Shimura operators for certain Hermitian symmetric superpairs |
| topic | Representation Theory Combinatorics 17B10, 17B60, 05E10, 81Q60 |
| url | https://arxiv.org/abs/2212.09249 |