The Capelli problem for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi mathvariant="fraktur">gl</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">|</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and the spectrum of invariant differential operators

Sahi, Siddhartha; Salmasian, Hadi

doi:10.1016/j.aim.2016.08.015

Cited by 14 publications

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“…We remark that it is possible to extend the results of the present paper and of Ref. [22] to the common setting of multiplicity-free actions on Jordan superalgebras. This will be established in a forthcoming paper [24].…”

Section: Introductionsupporting

confidence: 61%

“…Subsequently, supersymmetric analogs of the Knop-Sahi shifted Jack polynomials were constructed by Sergeev and Veselov in [28]. More recently, two of us (Sahi and Salmasian [22]) have extended this circle of ideas to the setting of the triples (l, m, V ) of the form (1.1) (gl(m|n) × gl(m|n), gl(m|n), Mat m|n (C)), (gl(m|2n), osp(m|2n), S 2 (C m|2n )).…”

Section: Introductionmentioning

confidence: 99%

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

Alldridge

Sahi

Salmasian

2018

Representation Theory and Harmonic Analysis on Symmetric Spaces

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Let l := q(n) × q(n), where q(n) denotes the queer Lie superalgebra. The associative superalgebra V of type Q(n) has a left and right action of q(n), and hence is equipped with a canonical l-module structure. We consider a distinguished basis {D λ } of the algebra of 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 λ , when it acts on the algebra P(V ) of super-polynomials on V , is given by the factorial Schur Qfunctions of Okounkov and Ivanov. As an application, we show that the radial projections of the spherical super-polynomials (corresponding to the diagonal symmetric pair (l, m), where m := q(n)) of irreducible l-submodules of P(V ) are the classical Schur Q-functions. As a further application, we compute the Harish-Chandra images of the Nazarov basis {C λ } of the centre of U(q(n)).

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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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

Alldridge

Sahi

Salmasian

2018

Representation Theory and Harmonic Analysis on Symmetric Spaces

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“…The study of the Capelli eigenvalue problem for Jordan superalgebras was initiated in [24], where it was solved in the cases J ∼ = gl(m, n) + and J ∼ = osp(n, 2m) + . These Jordan superalgebras correspond to symmetric pairs of types (gl × gl, gl) and (gl, osp), respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark 1.14. Theorem 1.13 is proved in [24] for Cases I-II, and in [1] for Case VII. We give a uniform proof for Cases I-IV and and VI in Sections 4 and 5 (see Proposition 5.2(iii)).…”

mentioning

confidence: 99%

The Capelli eigenvalue problem for Lie superalgebras

2019

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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 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}.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Spherical indecomposable representations of Lie superalgebras

Sherman

2020

Journal of Algebra

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We present a classification of all spherical indecomposable representations of classical and exceptional Lie superalgebras. We also include information about stabilizers, symmetric algebras, and Borels for which sphericity is achieved. In one such computation, the symmetric algebra of the standard module of osp(m|2n) is computed, which in particular gives the representation-theoretic structure of polynomials on the complex supersphere.Lemma 3.10. If g is basic, then an irreducible representation V is spherical if and only if V * is. If (V, g) is spherical, then (V * , g) is equivalent to it. 9

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The Capelli problem for gl(m|n) and the spectrum of invariant differential operators

Cited by 14 publications

References 22 publications

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

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

The Capelli eigenvalue problem for Lie superalgebras

Spherical indecomposable representations of Lie superalgebras

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