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Register a new userThe Capelli problem for $mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators
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The Capelli problem for the symmetric pairs $(mathfrak{gl}times mathfrak{gl},mathfrak{gl})$ $(mathfrak{gl},mathfrak{o})$, and $(mathfrak{gl},mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(mathfrak{g},mathfrak{g}):=(mathfrak{gl}(m|2n),mathfrak{osp}(m|2n))$ and $(mathfrak{gl}(m|n)timesmathfrak{gl}(m|n),mathfrak{gl}(m|n))$, acting on $W:=S^2(mathbb C^{m|2n})$ and $mathbb C^{m|n}otimes(mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.
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Let $mathfrak l:= mathfrak q(n)timesmathfrak q(n)$, where $mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $mathfrak q(n)$, and hence is equipped with a canonical $mathfrak l$-module structure. We consider a distinguished basis ${D_lambda}$ of the algebra of $mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_lambda$, when it acts on the algebra $mathscr P(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_lambda$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(mathfrak l,mathfrak m)$, where $mathfrak m:=mathfrak q(n)$, of irreducible $mathfrak l$-submodules of $mathscr P(V)$ are the classical Schur $Q$-functions.
In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $widehat{mathfrak{gl}_{m|n}}(mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $mu$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $mu$ is nonzero.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. In this paper we study the {em Whittaker coinvariants functor}, which is an exact functor from category $mathcal O$ for $mathfrak{gl}_{m|n}(mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergels functor $mathbb V$ in the setting of category $mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $mathcal O$ up to Morita/derived equivalence.
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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