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Journal of Lie Theory 28 (2018), No. 4, 1137--1147
Copyright Heldermann Verlag 2018

On Annihilators of Bounded (g, k)-Modules

Alexey Petukhov
Institute for Information Transmission Problems, Bolshoy Karetniy 19-1, Moscow, 127994, Russia

Let g be a semisimple Lie algebra and k a reductive subalgebra. We say that a g-module M is a bounded (g, k)-module if M is a direct sum of simple finite-dimensional k-modules and the multiplicities of all simple k-modules in this direct sum are universally bounded.
The goal of this article is to show that the "boundedness" property for a simple (g, k)-module M is equivalent to a property of the associated variety of the annihilator of M (this is the closure of a nilpotent coadjoint orbit inside g* under the assumption that the main field is algebraically closed and of characteristic 0. In particular this implies that if M, M2 are simple (g, k)-modules such that M1 is bounded and the associated varieties of the annihilators of M1 and M2 coincide then M2 is also bounded. This statement is a geometric analogue of a purely algebraic fact due to I. Penkov and V. Serganova, and it was posed as a conjecture in my Ph. D. thesis.

Keywords: (g, k)-modules, spherical varieties, symplectic geometry.

MSC: 13A50, 14L24, 17B08, 17B63, 22E47.

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