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Construction of generalized Harish-Chandra modules with arbitrary minimal \(\mathfrak k\)-type. (English) Zbl 1173.17010

This paper is a part of the project whose ultimate goal is the classification of simple generalized Harish-Chandra modules over a semisimple finite-dimensional complex Lie algebra \(\mathfrak g\) (i.e. infinite-dimensional modules having finite-dimensional isotypic components as a module over some reductive subalgebra \(\mathfrak k\)).
From the abstract: “For any simple finite dimensional \(\mathfrak k\)-module \(V\), we construct simple \((\mathfrak g, \mathfrak k)\)-modules \(M\) with finite dimensional \(\mathfrak k\)-isotypic components such that \(V\) is a \(\mathfrak k\)-submodule of \(M\) and the Vogan norm of any simple \(\mathfrak k\)-submodule \(V^\prime \subset M\), \(V^\prime \not\simeq V\), is greater than the Vogan norm of \(V\)”.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B55 Homological methods in Lie (super)algebras
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