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Generalized Harish-Chandra modules with generic minimal \({\mathfrak k}\)-type. (English) Zbl 1079.17004

Let \(\mathfrak g\) be a complex semisimple Lie algebra and \(\mathfrak k\) an algebraic subalgebra of \(\mathfrak g\) which is reductive in \(\mathfrak g\). A generalized Harish-Chandra \((\mathfrak g,\mathfrak k)\)-module \(M\) is a \(\mathfrak g\)-module such that \(M\) as the \(\mathfrak k\)-module is a direct sum of finite-dimensional isotypic components \(M[\mu ]\neq 0\), where \(\mu \) is the highest weight of an irreducible summand \(V(\mu )\) of \(M[\mu ]\) (\(V(\mu )\) is called a \(\mathfrak k\)-type) and \(\dim M[\mu ]\neq \infty \). The classical theory of Harish-Chandra modules deals with the symmetric pair \((\mathfrak g,\mathfrak k)\), that is \(\mathfrak k\) is a subalgebra consisting of fixed elements with respect to an involutive automorphism of the semisimple Lie algebra \(\mathfrak g\).
In the paper for the pair \((\mathfrak g,\mathfrak k)\) the minimal compatible reductive subalgebra \(\mathfrak p\) is defined. Using coinduction from \(\mathfrak p\)-modules, the functor of \(\mathfrak k\)-locally finite vectors on the category of \((\mathfrak g,\mathfrak t)\)-modules where \(\mathfrak t\) is a fixed Cartan subalgebra in \(\mathfrak k\), and cohomological induction the authors construct the fundamental series of generalized Harish-Chandra modules such that any module of the series has the unique minimal generic \(\mathfrak k\)-type \(V(\mu)\) and the corresponding isotypic component generates the unique simple \(\mathfrak g\)-submodule. It is proved that any simple generalized Harish-Chandra \((\mathfrak g, \mathfrak k)\)-module with minimal generic \(\mathfrak k\)-type \(V(\mu )\) is canonically isomorphic to the unique simple \((\mathfrak g,\mathfrak k)\)-submodule of an appropriate module of the fundamental series. Moreover, the analogue of Harish-Chandra’s admissibility theorem for simple \((\mathfrak g,\mathfrak k)\)-modules with a generic minimal \(\mathfrak k\)-type under the assumption that the Cartan subalgebra of \(\mathfrak k\) contains a regular element of \(\mathfrak g\) is proved. As an illustration at the end of the paper the case when \(\mathfrak k=sl(2)\) is considered.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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