A notion of rank for unitary representations of reductive groups based on Kirillov's orbit method

Abstract

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe's theory of $N$-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors' works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov's theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S]