## 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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