Spherical supervarieties

Sherman, Alexander

doi:10.5802/aif.3421

Cited by 6 publications

(2 citation statements)

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“…Since A is an integral superdomain, if the reduced scheme X r = Spec (A) is smooth, then X is smooth as well. For this and other general results about smooth supervarieties, we refer to [4] and references therein.…”

Section: Preliminariesmentioning

confidence: 99%

Simplicity of the Lie superalgebra of vector fields

Rocha¹

2023

Preprint

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Section: Preliminariesmentioning

confidence: 99%

Simplicity of the Lie superalgebra of vector fields

Rocha¹

2023

Preprint

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“…We note that Conjecture 1.2 was shown to hold under an extra genericity hypothesis on λ in Sec. 6.4 of [18]. 1.6.…”

Section: Introductionmentioning

confidence: 99%

Ghost Distributions on Supersymmetric Spaces II: Basic Classical Superalgebras

Sherman

2023

International Mathematics Research Notices

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We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $({\mathfrak{g}},{\mathfrak{k}})$ and $({\mathfrak{g}},{\mathfrak{k}}^{\prime})$ admit Iwasawa decompositions. For such pairs, we define a ghost algebra, generalizing the subalgebra of ${\mathcal{U}}{\mathfrak{g}}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the “special” ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $({\mathfrak{g}},{\mathfrak{k}})$ is interlaced.

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Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

Sherman

2024

Transformation Groups

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Given a Lie superalgebra $$\mathfrak {g}$$ g , Gorelik defined the anticentre $$\mathcal {A}$$ A of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $$(\mathfrak {g},\mathfrak {k})$$ ( g , k ) , or more generally supersymmetric spaces G/K. We define certain invariant distributions on G/K, which we call ghost distributions, and which in some sense are induced from invariant distributions on $$G_0/K_0$$ G 0 / K 0 . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup $$K'$$ K ′ which is related (and sometimes conjugate) to K. We discuss the case of $$G\times G/G$$ G × G / G for an arbitrary quasireductive supergroup G, where our results prove the existence of a polynomial which determines projectivity of irreducible G-modules. Finally, a generalization of Gorelik’s ghost centre is defined which we call the full ghost centre, $$\mathcal {Z}_{full}$$ Z full . For type I basic Lie superalgebras $$\mathfrak {g}$$ g we fully describe $$\mathcal {Z}_{full}$$ Z full , and prove that if $$\mathfrak {g}$$ g contains an internal grading operator, $$\mathcal {Z}_{full}$$ Z full consists exactly of those elements in $$\mathcal {U}\mathfrak {g}$$ U g acting by $$\mathbb {Z}$$ Z -graded constants on every finite-dimensional irreducible representation.

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Spherical supervarieties

Cited by 6 publications

References 32 publications

Simplicity of the Lie superalgebra of vector fields

Simplicity of the Lie superalgebra of vector fields

Ghost Distributions on Supersymmetric Spaces II: Basic Classical Superalgebras

Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

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