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

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.