Bounded reductive subalgebras of \mathfrak{s}{\mathfrak{l}_n} |
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Authors: | Alexey V Petukhov |
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Institution: | 1. Chair of High Algebra, Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992, GSP-2, Russia 2. Jacobs University Bremen, School of Science and Engineering, Bremen, D-28759, Germany
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Abstract: | Let
\mathfrakg \mathfrak{g} be a reductive Lie algebra and
\mathfrakk ì \mathfrakg \mathfrak{k} \subset \mathfrak{g} be a reductive in
\mathfrakg \mathfrak{g} subalgebra. A (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is a
\mathfrakg \mathfrak{g} -module for which any element m ∈ M is contained in a finite-dimensional
\mathfrakk \mathfrak{k} -submodule of M. We say that a (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is bounded if there exists a constant C
M
such that the Jordan-H?lder multiplicities of any simple finite-dimensional
\mathfrakk \mathfrak{k} -module in every finite-dimensional
\mathfrakk \mathfrak{k} -submodule of M are bounded by C
M
. In the present paper we describe explicitly all reductive in
\mathfraks\mathfrakln \mathfrak{s}{\mathfrak{l}_n} subalgebras
\mathfrakk \mathfrak{k} which admit a bounded simple infinite-dimensional (
\mathfraks\mathfrakln,\mathfrakk \mathfrak{s}{\mathfrak{l}_n},\mathfrak{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 (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-modules. |
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Keywords: | |
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