TY - JOUR

T1 - CAPELLI OPERATORS FOR SPHERICAL SUPERHARMONICS AND THE DOUGALL–RAMANUJAN IDENTITY

AU - Sahi, Siddhartha

AU - Salmasian, Hadi

AU - Serganova, Vera

N1 - Funding Information: S.S. and H.S. thank Christoph Koutschan and Doron Zeilberger for helpful correspondences regarding Theorem E. They also thank the Fields Institute, the University of Ottawa, and the NSF (DMS-162350, DMS-1939600) for conference grants. The research of S.S. was partially supported by grants from the NSF (DMS-2001537) and the Simons foundation (509766), of H.S. by an NSERC Discovery Grant (RGPIN-2018-04044), and of V.S. by an NSF Grant (1701532). Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021

Y1 - 2021

N2 - 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 p(V) of superpolynomials on V is completely reducible, unless dim Vo ¯ and dim V1 ¯ are positive even integers and dim VO¯≤dimV1¯. When p(V) is not a completely reducible g-module, we construct a natural basis {D⋋}⋋∈T of “Capelli operators” for the algebra [InlineMediaObject not available: see fulltext.] (V) g of g -invariant superpolynomial superdifferential operators on V , where the index set I is the set of integer partitions of length at most two. We compute the action of the operators {D⋋}⋋∈T on maximal indecomposable components of [InlineMediaObject not available: see fulltext.] (V) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where [InlineMediaObject not available: see fulltext.] (V) is completely reducible, the eigenvalues of a subfamily of the {D⋋} are not given 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 category Rep (Ot). More precisely, we define categorical Capelli operators {Dt,⋋}⋋∈T that induce morphisms of indecomposable components of symmetric powers of Vt, where Vt is the generating object of Rep (Ot). We obtain formulas for the eigenvalue polynomials associated to the {Dt,⋋}⋋∈T that are analogous to our results for the operators {D⋋}⋋∈T.

AB - 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 p(V) of superpolynomials on V is completely reducible, unless dim Vo ¯ and dim V1 ¯ are positive even integers and dim VO¯≤dimV1¯. When p(V) is not a completely reducible g-module, we construct a natural basis {D⋋}⋋∈T of “Capelli operators” for the algebra [InlineMediaObject not available: see fulltext.] (V) g of g -invariant superpolynomial superdifferential operators on V , where the index set I is the set of integer partitions of length at most two. We compute the action of the operators {D⋋}⋋∈T on maximal indecomposable components of [InlineMediaObject not available: see fulltext.] (V) explicitly, in terms of Knop–Sahi interpolation polynomials. Our results show that, unlike the cases where [InlineMediaObject not available: see fulltext.] (V) is completely reducible, the eigenvalues of a subfamily of the {D⋋} are not given 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 category Rep (Ot). More precisely, we define categorical Capelli operators {Dt,⋋}⋋∈T that induce morphisms of indecomposable components of symmetric powers of Vt, where Vt is the generating object of Rep (Ot). We obtain formulas for the eigenvalue polynomials associated to the {Dt,⋋}⋋∈T that are analogous to our results for the operators {D⋋}⋋∈T.

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U2 - https://doi.org/10.1007/s00031-021-09655-y

DO - https://doi.org/10.1007/s00031-021-09655-y

M3 - Article

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

ER -