Functions on the universal cover of the principal nilpotent orbit
type: Item
instance of: scholarly article
title: Functions on the universal cover of the principal nilpotent orbit
zbMATH Open document ID: 0781.22010
DOI: 10.1007/BF02100596
author: William Graham
published in: Inventiones Mathematicae
publication date: 1993-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143985
review text: Let \(G\) be a complex simply connected semisimple Lie group, \(T\) a maximal torus, \(\mathcal O\) the principal nilpotent orbit in the Lie algebra \(\mathfrak g\) of \(G\), and \(\widetilde{\mathcal O}\) the universal cover of \(\mathcal O\). This paper studies the ring \(\widetilde{R}\) of regular functions on \(\widetilde{\mathcal O}\), proving a formula for its \(G\)-module structure which had earlier been conjectured by the reviewer and \textit{R. Brylinski} [J. Am. Math. Soc. 2, 517-533 (1989; Zbl 0729.17005); Invent. Math. 97, 209-217 (1989; Zbl 0648.22004)]. The formula states that \(\widetilde{R}\) is obtained as a \(G\)-module by algebraically inducing the sum of the one- dimensional \(T\)-modules corresponding to minimal dominant weights (including 0) to \(G\). (By contrast, an old result of Kostant states that the ring \(R\) of regular functions on \(\mathcal O\) itself agrees with the \(G\)- module induced from the trivial \(T\)-module.) Thus the ``smallest'' nontrivial \(G\)-submodules of \(\widetilde{R}\) in some sense are those with minimal highest weight, together with the unique copy of the adjoint representation in lowest degree. The author makes this last assertion precise by introducing a \(G\)-equivariant grading on \(\widetilde{R}\) such that the augmentation ideal \(\mathfrak m\) of \(\widetilde{R}\), modulo its square, is spanned by exactly these submodules (if \(\mathfrak g\) has no simple component of type \(A_ 1\)). This description of \({\mathfrak m}/{\mathfrak m}^ 2\) is particularly interesting in the light of a recent result of \textit{R. Brylinski} and \textit{B. Kostant} [Bull. Am. Math. Soc., New Ser. 26, 269-275 (1992; Zbl 0762.22008)] stating that this quotient has a natural Lie algebra structure extending that of \(\mathfrak g\). The author also points out an important mistake in [\textit{H. H. Andersen} and \textit{J. C. Jantzen}, Math. Ann. 269, 487-525 (1984; Zbl 0529.20027)], which invalidates Proposition 5.4 in that paper,…
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 149684
zbMATH Keywords: equivariant grading
zbMATH Keywords: complex simply connected semisimple Lie group
zbMATH Keywords: maximal torus
zbMATH Keywords: principal nilpotent orbit
zbMATH Keywords: Lie algebra
zbMATH Keywords: regular functions
zbMATH Keywords: dominant weights
zbMATH Keywords: highest weight
zbMATH Keywords: adjoint representation
zbMATH Keywords: augmentation ideal
reviewed by: William M. McGovern
MaRDI profile type: Publication
cites work: Cohomology of induced representations for algebraic groups
cites work: Limits of Weight Spaces, Lusztig's q-Analogs, and Fiberings of Adjoint Orbits
cites work: Nilpotent orbits, normality,\\ and Hamiltonian group actions
cites work: The normality of closures of orbits in a Lie algebra
cites work: Introduction to Lie Algebras and Representation Theory
cites work: The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group
cites work: Lie Group Representations on Polynomial Rings
cites work: Q3031132
cites work: Rings of regular functions on nilpotent orbits and their covers
OpenAlex ID: W1964359544
description: scientific article; zbMATH DE number 149684