Analytic Continuation Abstract

Anthony W. Knapp

Abstract of "Analytic continuation of nonholomorphic discrete series for classical groups"

The question of unitarity of representations in the analytic continuation of discrete series from a Borel--de Siebenthal chamber is considered for those linear equal-rank classical simple Lie groups G that have not been treated fully before. Groups treated earlier by other authors include those for which G has real rank one or has a symmetric space with an invariant complex structure. Thus the groups in question are locally isomorphic to SO(2m,n)_0 with m \geq 2 and n \geq 3, or to Sp(m,n) with m \geq 2 and n \geq 2.

The representations under study are obtained from cohomological induction. One starts from a finite-dimensional irreducible representation of a compact subgroup L of G associated to a Borel--de Siebenthal chamber, forms a generalized Verma-like module, applies a derived Bernstein functor, and passes to a specific irreducible quotient. Enright, Parthasarthy, Wallach, and Wolf had previously identified all cases where the representation of L is 1-dimensional and the generalized Verma-like module is irreducible; for these cases they proved that unitarity is automatic. B.~Gross and Wallach had proved unitarity for additional cases for a restricted class of groups when the representation of L is 1-dimensional.

The present work gives results for all groups and allows higher-dimensional representations of L. In the case of 1-dimensional representations of L, the results address unitarity and nonunitarity and are conveniently summarized in a table that indicates how close the results are to best possible. In the case of higher-dimensional representations of L, the method addresses only unitarity and in effect proceeds by reducing matters to what happens for a 1-dimensional representation of L and a lower-dimensional group G.