Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika
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Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2012, Number 3, Pages 51–55 (Mi vmumm498)  

This article is cited in 1 scientific paper (total in 1 paper)

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Supports of $(\mathfrak g,\mathfrak k)$-modules of finite type

A. V. Petukhovab

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Jacobs University, Bremen

Abstract: Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak k$ be a reductive subalgebra in $\mathfrak g$. We say that a $\mathfrak g$-module $M$ is a $(\mathfrak g,\mathfrak k)$-module if $M$, considered as a $\mathfrak k$-module, is a direct sum of finite-dimensional $\mathfrak k$-modules. We say that a $(\mathfrak g,\mathfrak k)$-module $M$ is of finite type if all $\mathfrak k$-isotypic components of $M$ are finite-dimensional. In this article we prove that any simple $(\mathfrak g,\mathfrak k)$-module of finite type is holonomic. To a simple $\mathfrak g$-module $M$ one assigns invariants $\mathrm{V}(M)$, $\mathcal V(\operatorname{Loc}M)$ и $\mathrm{V}(M)$ reflecting the “directions of growth of $M$”. We also prove that, for a given pair $(\mathfrak g,\mathfrak k)$, the set of possible invariants is finite.

Key words: $(\mathfrak g,\mathfrak k)$-module, coadjoint orbit, null-cone.

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English version:
Moscow University Mathematics Bulletin, 2012, 67:3, 125–128

Bibliographic databases:

UDC: 512
Received: 20.04.2011

Citation: A. V. Petukhov, “Supports of $(\mathfrak g,\mathfrak k)$-modules of finite type”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2012, no. 3, 51–55; Moscow University Mathematics Bulletin, 67:3 (2012), 125–128

Citation in format AMSBIB
\Bibitem{Pet12}
\by A.~V.~Petukhov
\paper Supports of $(\mathfrak g,\mathfrak k)$-modules of finite type
\jour Vestnik Moskov. Univ. Ser.~1. Mat. Mekh.
\yr 2012
\issue 3
\pages 51--55
\mathnet{http://mi.mathnet.ru/vmumm498}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3026844}
\transl
\jour Moscow University Mathematics Bulletin
\yr 2012
\vol 67
\issue 3
\pages 125--128
\crossref{https://doi.org/10.3103/S0027132212030084}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84864813044}


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    This publication is cited in the following articles:
    1. Petukhov A., “On annihilators of bounded $(\mathfrak{g},\mathfrak{k})$-modules”, J. Lie Theory, 28:4 (2018), 1137–1147  mathscinet  zmath  isi
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