The Capelli problem for [formula omitted] and the spectrum of invariant differential operators

Abstract

The “Capelli problem” for the symmetric pairs (gl×gl,gl), (gl,o), and (gl,sp) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [12,15,14,18]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (g,k):=(gl(m|2n),osp(m|2n)) and (gl(m|n)×gl(m|n),gl(m|n)), acting on W:=S2(Cm|2n) and Cm|n⊗(Cm|n)⁎.To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g maps surjectively onto the algebra PD(W)g of g-invariant differential operators on the superspace W, thereby providing an affirmative answer to the “abstract” Capelli problem for W. Our proof works more generally for gl(m|n) acting on S2(Cm|n) and is new even for the “ordinary” cases (m=0 or n=0) considered by Howe and Umeda in [9].We next describe a natural basis {Dλ} of PD(W)g, that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [12,15,20] by showing that the spectrum of Dλ is given by a polynomial cλ, which is characterized uniquely by certain vanishing and symmetry properties.We further show that the top homogeneous parts of the eigenvalue polynomials cλ coincide with the spherical polynomials dλ, which arise as radial parts of k-spherical vectors of finite dimensional g-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14].Finally, we make a precise connection between the polynomials cλ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the “Frobenius transform”.