Derived Functors of Unitary Highest Weight Modules at Reduction Points

Abstract

The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the ${\mathbf {K}}$-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest ${\mathbf {K}}$-type are also easily computed.