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

A. V. Petukhov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2012), pp. 51-55

Résumé

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.