Multiplicity formulas for finite dimensional and generalized principal series representations
The author obtains multiplicity formulas for finite dimensional representations of a reductive Lie algebra \({\mathfrak a}\) over \(\mathbb{C}\) and for generalized principal series representations of a real semisimple Lie group \(G\). These formulas generalize results due to Kostant, and Enright and Wallach, respectively. Let \({\mathfrak b}\) be a reductive subalgebra of \({\mathfrak a}\). Suppose \({\mathfrak s}\) is a Cartan subalgebra of \({\mathfrak a}\), and \({\mathfrak t}\) is a Cartan subalgebra of \({\mathfrak b}\) such that \({\mathfrak t}\subset{\mathfrak s}\). Then the author obtains a formula for the multiplicity with which a given finite dimensional irreducible \({\mathfrak b}\)-module occurs in a given finite dimensional irreducible \({\mathfrak a}\)-module. Kostant proved the formula under the assumption that \({\mathfrak t}\) has an \(({\mathfrak a}, {\mathfrak s})\)-regular element. Let \(K\) be a maximal compact subgroup of \(G\) and \(Q=MAN\) a cuspidal parabolic subgroup of \(G\). Using the above result the author proves a Blattner type formula for the multiplicity with which a finite dimensional irreducible representation of \(K\) occurs in \(\text{Ind}^G_{Q^0}(\sigma\otimes e^\nu\otimes 1)\), where \(\sigma\) is a discrete series or the limit of a discrete series of \(M^0\) and \(\nu\in {\mathfrak a}^*\). \(Q^0\) (resp. \(M^0)\) denotes the connected component of the identity of \(Q\) (resp. \(M)\), and \({\mathfrak a}^*\) the dual space of the Lie algebra of \(A\). Enright and Wallach proved the formula in the fundamental case. As the author quotes the referee's comments in the Introduction, Vogan's results yield the main theorem, provided one makes a transition from cohomological induction to classical parabolic induction, and Zuckerman's cohomological construction yields the construction of modules. However, his aim is to see if one can adjust Enright's machinery to the nonfundamental case, and his construction is different.
scientific article; zbMATH DE number 1043988