Construction of Generalized Harish-Chandra Modules with Arbitrary Minimal -Type
Abstract Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $\mathfrak{k}\,\subset \mathfrak{g}$ be any algebraic subalgebra reductive in $\mathfrak{g}$ . For any simple finite dimensional $\mathfrak{k}$ -module $V$ , we construct simple $\left( \mathfrak{g},\mathfrak{k} \right)$ -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 $V\prime \subset M,V\prime \ne \,V$ , is greater than the Vogan norm of $V$ . The $\left( \mathfrak{g},\mathfrak{k} \right)$ -modules $M$ are subquotients of the fundamental series of $\left( \mathfrak{g},\mathfrak{k} \right)$ -modules.