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1 November 2004 On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type
Ivan Penkov, Vera Serganova, Gregg Zuckerman
Duke Math. J. 125(2): 329-349 (1 November 2004). DOI: 10.1215/S0012-7094-04-12525-4

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $\mathfrak{k}$ be a subalgebra reductive in $\mathfrak{g}$. We prove that $\mathfrak{g}$ admits an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M which has finite $\mathfrak{k}$-multiplicities and which is not a ($\mathfrak{g}$,$\mathfrak{k}$′)-module for any proper inclusion of reductive subalgebras $\mathfrak{k}$⊂$\mathfrak{k}$′⊂$\mathfrak{g}$ if and only if $\mathfrak{k}$ contains its centralizer in $\mathfrak{g}$. The main point of the proof is a geometric construction of ($\mathfrak{g}$,$\mathfrak{k}$)-modules which is analogous to cohomological induction. For $\mathfrak{g}=\mathfrak{g}\mathfrak{l}$(n) we show that whenever $\mathfrak{k}$ contains its centralizer, there is an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M of finite type over $\mathfrak{k}$ such that $\mathfrak{k}$ coincides with the subalgebra of all $g∈\mathfrak{g}$ which act locally finitely on M. Finally, for a root subalgebra $\mathfrak{k}⊂\mathfrak{g}\mathfrak{l}(n)$, we describe all possibilities for the subalgebra $\mathfrak{l}$⊃$\mathfrak{k}$ of all elements acting locally finitely on some M.

Citation

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Ivan Penkov. Vera Serganova. Gregg Zuckerman. "On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type." Duke Math. J. 125 (2) 329 - 349, 1 November 2004. https://doi.org/10.1215/S0012-7094-04-12525-4

Information

Published: 1 November 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1097.17007
MathSciNet: MR2096676
Digital Object Identifier: 10.1215/S0012-7094-04-12525-4

Subjects:
Primary: 17B10
Secondary: 22E46

Rights: Copyright © 2004 Duke University Press

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Vol.125 • No. 2 • 1 November 2004
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