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Let V be the associative superalgebra of type Q(n). Then V carries a two sided action of the queer Lie superalgebra q(n) and hence an action of l:= q(n)× q(n). We consider a distinguished basis {D_λ} of the algebra of l-invariant polynomial super-differential operators on V, which is indexed by strict partitions of length at most n. We show that the spectrum of D_λ, on the algebra P(V) of super-polynomials on V, is given by the factorial Schur Q-function of Okounkov and Ivanov.

This generalizes a result of Nazarov. As a further application we show that the radial projections of the spherical polynomials of the symmetric pair (q(n) x q(n),q(n)) are the classical Schur Q-functions.

This is joint work with Alexander Alldridge and Hadi Salmasian and available at https://arxiv.org/abs/1701.03401