2010/07/08 by Alexey Petukhov, Petukhov, Alexey
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1007.1338
openalex publication_date 2010/07/08 · arxiv created 2011/03/15 · arxiv updated 2011/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrak g be a semisimple Lie algebra and \mathfrak k⊂\mathfrak g be a reductive in \mathfrak g subalgebra. A (\mathfrak g, \mathfrak k)-module is a \mathfrak g-module which after restriction to \mathfrak k becomes a direct sum of finite-dimensional \mathfrak k-modules. I.Penkov and V.Serganova introduce definition of bounded (\mathfrak g, \mathfrak k)-modules for reductive subalgebras \mathfrak k⊂\mathfrak g, i.e. (\mathfrak g, \mathfrak k)-modules whose \mathfrak k-multiplicities are uniformly bounded. A question arising in this context is, given \mathfrak g, to describe all reductive in \mathfrak g bounded subalgebras, i.e. reductive in \mathfrak g subalgebras \mathfrak k for which at least one infinite-dimensional bounded (\mathfrak g, \mathfrak k)-module exists. In the present paper we describe explicitly all reductive in \mathfraksln bounded subalgebras. 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 (\mathfrak g, \mathfrak k)-modules.