CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY
CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY
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球面超谐波的 CAPELLI 算子和 DOUGALL-RAMANUJAN 恒等式
DOI：
10.1007/s00031-021-09655-y
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发表时间：
2021
影响因子：
0.7
通讯作者：
SERGANOVA, VERA
中科院分区：
文献类型：
--
作者：
SAHI, SIDDHARTHA;SALMASIAN, HADI;SERGANOVA, VERA
Let (V, ω) be an orthosymplectic ℤ2-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V) of superpolynomials onVis completely reducible, unless dimand dimare positive even integers and dim. When (V) isnota completely reducible 𝔤-module, we construct a natural basis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} of “Capelli operators” for the algebra (V)𝔤of 𝔤 -invariant superpolynomial superdifferential operators onV, where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} on maximal indecomposable components of (V) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V) is completely reducible, the eigenvalues of a subfamily of the {D⋋} arenotgiven by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall–Ramanujan hypergeometric identity.We also transcend our results on the eigenvalues of Capelli operators to the Deligne categoryRep(Ot). More precisely, we define categorical Capelli operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} that induce morphisms of indecomposable components of symmetric powers ofVt, whereVtis the generating object ofRep(Ot). We obtain formulas for the eigenvalue polynomials associated to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} that are analogous to our results for the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document}.
Let (V, ω) be an orthosymplectic ℤ2-graded vector space and let 𝔤:= 𝔤𝔬𝔰𝔭 (V, ω) denote the Lie superalgebra of similitudes of (V, ω). It is known that as a 𝔤-module, the space (V) of superpolynomials onVis completely reducible, unless dimand dimare positive even integers and dim. When (V) isnota completely reducible 𝔤-module, we construct a natural basis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} of “Capelli operators” for the algebra (V)𝔤of 𝔤 -invariant superpolynomial superdifferential operators onV, where the index set 𝒯 is the set of integer partitions of length at most two. We compute the action of the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} on maximal indecomposable components of (V) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where (V) is completely reducible, the eigenvalues of a subfamily of the {D⋋} arenotgiven by specializing the Knop–Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. This is in contrast with what occurs for previously studied Capelli operators. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall–Ramanujan hypergeometric identity.We also transcend our results on the eigenvalues of Capelli operators to the Deligne categoryRep(Ot). More precisely, we define categorical Capelli operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} that induce morphisms of indecomposable components of symmetric powers ofVt, whereVtis the generating object ofRep(Ot). We obtain formulas for the eigenvalue polynomials associated to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{t,\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document} that are analogous to our results for the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{{D}_{\leftthreetimes}\right\}}_{\leftthreetimes \in \mathcal{T}} $$\end{document}.
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