Explicit formulas for Eigenvalues of Capelli operators for the Lie\n superalgebra frakosp(1|2n)

2020/11/16 by Dene Lepine, Lepine, Dene
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2011.08357

openalex publication_date 2020/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a natural basis for the algebra of frakgosp(1|2n)-invariant\ndifferential operators on the affine superspace \ℂ^1|2n. We prove\nthat these operators lie in the image of the centre of the enveloping algebra\nof frakgosp(1|2n). Using this result, we compute explicit formulas for the\neigenvalues of these operators on irreducible summands of\n\P(\ℂ^1|2n). This settles the Capelli eigenvalue problem\nfor orthosymplectic Lie superalgebras in the cases that were not addressed in\nSahi-Salmasian-Serganova. Our main technique relies on an explicit calculation\nof a certain determinant with polynomial entries.\n

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