Classes of unitarizable derived functor modules.

For a real semisimple Lie group G, the description of the unitary dual remains an elusive question. One of the difficulties has been the lack of technique for constructing unitary representations. Unitary induction from parabolic subgroups of G yields unitary representations by the very definition of these representations. However, not all unitary irreducible representations of G are obtained by this type of induction. In addition, we need derived functor parabolic induction [ef. Vogan, D. (1981) Representations of Real Reductive Lie Groups (Birkhäuser, Boston)] to describe all irreducible representations of G. For this second type of induction, the obvious analogues from parabolic subgroup induction regarding unitarity are false. In this announcement, we describe a setting where derived functor parabolic induction yields unitary representations of G. These results include proofs of unitarity for some of the representations conjectured to be unitary by Vogan and Zuckerman [(1983) Invent. Math., in press] and also proofs of unitarity for some which lie outside the domain described in those conjectures.

Minimal immersions of spheres into spheres.

In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the m-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is known that for n = 2 each admissible r determines a unique element of the above class. The main result here is that for each n >/= 3 and each admissible r >/= [unk]8 there exists a continuum of distinct such submanifolds.