Duke Mathematical Journal

On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type

Ivan Penkov, Vera Serganova, and Gregg Zuckerman

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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.

Article information

Source
Duke Math. J., Volume 125, Number 2 (2004), 329-349.

Dates
First available in Project Euclid: 27 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1098892272

Digital Object Identifier
doi:10.1215/S0012-7094-04-12525-4

Mathematical Reviews number (MathSciNet)
MR2096676

Zentralblatt MATH identifier
1097.17007

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 22E46: Semisimple Lie groups and their representations

Citation

Penkov, Ivan; Serganova, Vera; Zuckerman, Gregg. On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type. Duke Math. J. 125 (2004), no. 2, 329--349. doi:10.1215/S0012-7094-04-12525-4. https://projecteuclid.org/euclid.dmj/1098892272


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