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

A. V. Petukhov<sup>ab

a</sup> 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.

UDC: 512

Received: 20.04.2011

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

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