The Capelli eigenvalue problem for Lie superalgebras

Abstract

For a finite dimensional unital complex simple Jordan superalgebra J, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra g ♭ ∼ = g ♭ (−1) ⊕ g ♭ (0) ⊕ g ♭ (1), such that g ♭ (−1) ∼ = J. Set V := g ♭ (−1) * and g := g ♭ (0). In most cases, the space P(V ) of superpolynomials on V is a completely reducible and multiplicity-free representation of g, and there exists a direct sum decomposition P(V ) := λ∈Ω V λ , where (V λ ) λ∈Ω is a family of irreducible g-modules parametrized by a set of partitions Ω. In these cases, one can define a natural basis (D λ ) λ∈Ω of "Capelli operators" for the algebra PD(V ) g of g-invariant superpolynomial differential operators on V . In this paper we complete the solution to the Capelli eigenvalue problem, which asks for the determination of the scalar cµ(λ) by which Dµ acts on V λ .We associate a restricted root system Σ to the symmetric pair (g, k) that corresponds to J, which is either a deformed root system of type A(m, n) or a root system of type Q(n). We prove a necessary and sufficient condition on the structure of Σ for P(V ) to be completely reducible and multiplicityfree. When Σ satisfies the latter condition we obtain an explicit formula for the eigenvalue cµ(λ), in terms of Sergeev-Veselov's shifted super Jack polynomials when Σ is of type A(m, n), and Okounkov-Ivanov's factorial Schur Q-polynomials when Σ is of type Q(n). Along the way, we prove that the natural map from the centre of the enveloping algebra of g into PD(V ) g is surjective in all cases except when J ∼ = F , where F is the 10-dimensional exceptional Jordan superalgebra. . Science Foundation (DMS-162350), the Fields Institute, and the University of Ottawa for funding this workshop. 1 We remark that g ♭ is a slight modification of the simple Lie superalgebra that is constructed from J by the Kantor functor (see Remark A.3). (45)HC(j(z)) = res(HC(z)) for z ∈ Z(g). Recall that by Pwe denote the standard degree filtration of the polynomial algebra P J defined in (9). Let τ J : a * Ω → C n J and τ * J : P J → P (a * Ω ) be defined as in (10) and (14), respectively. Since τ J is a bijection, τ * J is an isomorphism of C-algebras. If J ∼ = F , then by Proposition 4.2, for every µ ∈ Ω d there exists an element z µ ∈ Z (d) (g) such that res(HC(z µ )) = τ * J (P J,µ ) , where P J,µ is as in Definition 1.11. Theorem 1.13 follows from Proposition 5.2(iii).Proposition 5.2. Assume that J ∼ = F . Then the following assertions hold. (i) j (z µ ) = D µ for all µ ∈ Ω. (ii) j(Z(g)) = PD (V ) g . (iii) HC Dµ (λ) = P J,µ (τ J (λ)) for all λ, µ ∈ Ω.Proof. (i) By a direct computation, from Theorem 1.8 and Theorem 1.10 it follows that P J,µ τ J (µ) = d! and P J,µ (τ J (λ)) = 0 for all λ ∈ d k=0 Ω k \{µ}.

Superspin chains and supersymmetric gauge theories

“…One can therefore deform this theory by two twisted masses u l , u r , so that the theory we discussed so far would correspond to the case u l + u r = 0. It is possible that such deformation also has an interesting Bethe/gauge dual (perhaps the generalized root systems of [55] would make an appearence, with κ/(1 − κ) = −u r /u l ). We propose another solidifier.…”

Spherical indecomposable representations of Lie superalgebras

“…Now consider the osp(2|4)-module V = L(δ 1 + δ 2 − ǫ 1 ). We express S d (V * ) as a osp(2|4) × C-module with respect to the Borel b (−ǫ)δδ , following the presentation given in [SSS18]. There it was shown that S • (V * ) is completely reducible with the lattice of highest weights given by Λ…”

“…The representation theory of the space of functions as well as the algebra of invariant differential operators has been studied in recent years (see e.g. [SS16], [All12], [AS15], [SV17], [SSS18]). In [SSS18], the authors associate to each simple Jordan superalgebra J a supersymmetric pair (g, k) via the TKK construction.…”

“…[SS16], [All12], [AS15], [SV17], [SSS18]). In [SSS18], the authors associate to each simple Jordan superalgebra J a supersymmetric pair (g, k) via the TKK construction. The vector superspace J inherits a g-action making it a simple spherical g-module, and the symmetric superspace G/K is realized as an open G-orbit.…”

Schur 𝑄-functions and the Capelli eigenvalue problem for the Lie superalgebra \ger𝑞(𝑛)

“…[22] to the common setting of multiplicity-free actions on Jordan superalgebras. This will be established in a forthcoming paper [24]. In addition, recently Sahi and Salmasian [23] constructed quadratic analogues of Capelli operators on Grassmannian manifolds by lifting the Capelli basis of [19] via a double fibration.…”