Spherical representations of Lie supergroups

Alldridge, Alexander; Schmittner, Sebastian E.

doi:10.1016/j.jfa.2014.11.018

Cited by 11 publications

(22 citation statements)

References 29 publications

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“…The goal of our paper is to extend this circle of ideas to the Lie superalgebra setting. The Jack polynomials for θ = 1, 1 2 , 2 are spherical polynomials of the symmetric pairs (gl(n) × gl(n), gl(n)), (gl(n), o(n)), (gl(2n), sp(2n)).…”

Section: Introductionmentioning

confidence: 99%

“…We prove in Proposition 4.6 and Remark 4.7 that every irreducible gl(m|2n)-submodule of S(S 2 (C m|2n )) or P(S 2 (C m|2n )) has a unique (up to scalar) nonzero osp(m|2n)-fixed vector. It is worth mentioning that Proposition 4.6 does not follow from the work of Alldridge and Schmittner [1], since they need to assume that the highest weight is "high enough" in some sense. In Section 5, we prove Theorem 5.8 and Theorem 5.9 (see the second goal above).…”

Section: Introductionmentioning

confidence: 99%

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

Sahi

Salmasian

2016

Advances in Mathematics

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The "Capelli problem" for the symmetric pairs (gl × gl, gl) (gl, o), and (gl, sp) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [12], [15], [14], [18]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (g, k) := (gl(m|2n), osp(m|2n)) and (gl(m|n) × gl(m|n), gl(m|n)), acting on W := S 2 (C m|2n ) and C m|n ⊗ (C m|n ) * .To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g maps surjectively onto the algebra PD(W ) g of g-invariant differential operators on the superspace W , thereby providing an affirmative answer to the "abstract" Capelli problem for W . Our proof works more generally for gl(m|n) acting on S 2 (C m|n ) and is new even for the "ordinary" cases (m = 0 or n = 0) considered by Howe and Umeda in [9].We next describe a natural basis {D λ } of PD(W ) g , that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [12], [15], [20] by showing that the spectrum of D λ is given by a polynomial c λ , which is characterized uniquely by certain vanishing and symmetry properties.We further show that the top homogeneous parts of the eigenvalue polynomials c λ coincide with the spherical polynomials d λ , which arise as radial parts of k-spherical vectors of finite dimensional g-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14].Finally, we make a precise connection between the polynomials c λ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the "Frobenius transform".

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Sahi

Salmasian

2016

Advances in Mathematics

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“…. , y ±1 m ] be the subalgebra consisting of S n × S m -invariant Laurent polynomials f ∈ C(a), satisfying the quasi-invariance conditions (2).…”

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confidence: 99%

Symmetric Lie superalgebras and deformed quantum Calogero–Moser problems

Сергеев¹,

Веселов²

2017

Advances in Mathematics

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The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g = gl(n, 2m), k = osp(n, 2m) we establish a natural bijection between projective covers of spherically typical irreducible g-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals Dn,m acting on the corresponding Laurent quasi-invariants An,m.

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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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Spherical representations of Lie supergroups

Cited by 11 publications

References 29 publications

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

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

Symmetric Lie superalgebras and deformed quantum Calogero–Moser problems

Spherical indecomposable representations of Lie superalgebras

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