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Register a new userThe Capelli eigenvalue problem for Lie superalgebras
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For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $mathfrak g_flatcong mathfrak g_flat(-1)oplusmathfrak g_flat(0)oplusmathfrak g_flat(1)$, such that $mathfrak g_flat(-1)cong J$. Set $V:=mathfrak g_flat(-1)^*$ and $mathfrak g:=mathfrak g_flat(0)$. In most cases, the space $mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $mathfrak g$, with a decomposition $mathcal P(V):=bigoplus_{lambdainOmega}V_lambda$, where $left(V_lambdaright)_{lambdainOmega}$ is a family of irreducible $mathfrak g$-modules parametrized by a set of partitions $Omega$. In these cases, one can define a natural basis $left(D_lambdaright)_{lambdainOmega}$ of Capelli operators for the algebra $mathcal{PD}(V)^{mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_mu(lambda)$ by which $D_mu$ acts on $V_lambda$. We associate a restricted root system $mathit{Sigma}$ to the symmetric pair $(mathfrak g,mathfrak k)$ that corresponds to $J$, which is either a deformed root system of type $mathsf{A}(m,n)$ or a root system of type $mathsf{Q}(n)$. We prove a necessary and sufficient condition on the structure of $mathit{Sigma}$ for $mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $mathit{Sigma}$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_mu(lambda)$, in terms of Sergeev-Veselovs shifted super Jack polynomials when $mathit{Sigma}$ is of type $mathsf{A}(m,n)$, and Okounkov-Ivanovs factorial Schur $Q$-polynomials when $mathit{Sigma}$ is of type $mathsf{Q}(n)$.
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