Bounded reductive subalgebras of $ \mathfrak{s}{\mathfrak{l}_n} $

Abstract

Let g be a reductive Lie algebra and k ⊂ g be a reductive in g subalgebra. A (g, k)-module M is a g-module for which any element m ∈ M is contained in a finite-dimensional k-submodule of M . We say that a (g, k)-module M is bounded if there exists a constant C M such that the Jordan-Hölder multiplicities of any simple finitedimensional k-module in every finite-dimensional k-submodule of M are bounded by C M . In the present paper we describe explicitly all reductive in sln subalgebras k which admit a bounded simple infinite-dimensional (sln, k)-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded (g, k)-modules.

Spherical actions on flag varieties

“…We note that assertion (1 ′ ) was in fact proved in [Pet,Theorem 5.8] using the ideas presented in this paper.…”

“…Proposition 3.7 (see also [Pet,Theorem 5.8]). If the variety Fl a (V ) is K-spherical, then so is the variety P(V ).…”

Branching Rules Related to Spherical Actions on Flag Varieties

“…Remark 5.8. In fact, if a connected reductive subgroup H ⊂ SL n acts spherically on X I for some I = ∅ then H automatically acts spherically on P(F n ) ≃ X {n−1} (and hence on P((F n ) * ) ≃ X {1} by duality); see [Pet,Theorem 5.8] or [AvPe,Proposition 3.7].…”