Explicit Formulas for Eigenvalues of Capelli Operators for the Lie Superalgebra osp(1 vertical bar 2n)

摘要

We define a natural basis for the algebra of gosp(1 vertical bar 2n)-invariant differential operators on the affine superspace C-1 vertical bar 2n Th . We prove that these operators lie in the image of the centre of the enveloping algebra of gosp(1 vertical bar 2n). Using this result, we compute explicit formulas for the eigenvalues of these operators on irreducible summands of P(C-1 vertical bar 2n) This settles the Capelli eigenvalue problem for orthosymplectic Lie superalgebras in the cases that were not addressed in recent papers by Sahi, Salmasian, and Serganova. Our main technique relies on an explicit calculation of a certain determinant with polynomial entries.