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Ghost distributions on supersymmetric spaces I: Koszul induced superspaces, branching, and the full ghost centre
Representation Theory · Submitted 27 Jan 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation.
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The abstract, sentence by sentence
Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ 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 $(\mathfrak{g},\mathfrak{k})$, 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 $G_0/K_0$. Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from $G$ to a symmetric subgroup $K'$ which is related (and sometimes conjugate) to $K$. We discuss the case of $G\times G/G$ 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 called the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ of its enveloping algebra, which consists of certain elements that square to the center. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s1 | We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $(\mathfrak{g},\mathfrak{k})$, or more generally supersymmetric spaces $G/K$. | contribution 0.65 | 0.37 | 0.04 | asserted 0.67 |
| s2 | We define certain invariant distributions on $G/K$, which we call ghost distributions, and which in some sense are induced from invariant distributions on $G_0/K_0$. | contribution 0.99 | 0.93 | 0.10 | asserted 0.98 |
| s3 | Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from $G$ to a symmetric subgroup $K'$ which is related (and sometimes conjugate) to $K$. | interpretation 0.63 | 0.67 | 0.13 | asserted 0.73 |
| s4 | We discuss the case of $G\times G/G$ for an arbitrary quasireductive supergroup $G$, where our results prove the existence of a polynomial which determines projectivity of irreducible $G$-modules. | result 0.90 | 0.92 | 0.19 | asserted 1.00 |
| s5 | Finally, a generalization of Gorelik's ghost centre is defined called the full ghost centre, $\mathcal{Z}_{full}$. | contribution 0.99 | 0.93 | 0.14 | asserted 1.00 |
| s6 | For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation. | result 0.98 | 0.94 | 0.35 | asserted 1.00 |
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The paper
Original abstract
Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ 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 $(\mathfrak{g},\mathfrak{k})$, 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 $G_0/K_0$. Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from $G$ to a symmetric subgroup $K'$ which is related (and sometimes conjugate) to $K$. We discuss the case of $G\times G/G$ 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 called the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation.
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How the literature uses it
1 arXiv paper cite this in 20 sentences
Papers citing it · 16 sentences without a confident label
Uses its method or tool 1 paper
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We consider the filtration on $(G/K,eK)$ defined in Sec. 8.6 of [ref] , except that we `halve' the indexing; i.e. our filtration on $(G/K,eK)$ will be indexed by half-integers, and if an element previously lied in the $r$th part of the filtration, then it now lies in the $r/2$ part of the filtration by definition.
Ghost distributions on supersymmetric spaces II: basic classical superalgebras · Injectivity of the Harish-Chandra Homomorphism · 0.82
Cites it as background 1 paper
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Recall from Section 10 of [ref] the full ghost center $_full$ of $$.
Ghost distributions on supersymmetric spaces II: basic classical superalgebras · Map from full ghost center of $$ · 0.98
What it cites · 5 sentences without a confident label
Cites as background 3 papers
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We will use the symbols $X,Y,…$ for supervarieties with even subschemes $X_0,Y_0,….$ We will be considering supervarieties in the sense of [ref] , however all spaces of interest will be smooth and affine.
Spherical Supervarieties · Algebraic supergeometry notation · 0.98
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For technical aspects of such spaces see [ref] and [ref] .
Quotient sheaves of algebraic supergroups are superschemes · A local isomorphism for certain homogeneous supervarieties · 0.97
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For technical aspects of such spaces see [ref] and [ref] .
Geometric construction of quotients $G/H$ in supersymmetry · A local isomorphism for certain homogeneous supervarieties · 0.97
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