A Notion of Rank for Unitary Representations of General Linear Groups

Abstract

ABSTRACT. A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

“…Building on the work of Howe [How82], Scaramuzzi [Sca90] proved that µ π has "pure rank", i.e. there is some integer k = HR (π, a) such that µ π = µ π k .…”

Small principal series and exceptional duality for two simply laced exceptional groups

“…In this section we prove a result which provides a suitable adaptation of [Sca,Theorem II.1.1] and will be used in the proof of Proposition 5.4. The proof of Proposition 4.1 is lengthy, but easy, and could be omitted.…”

“…In fact, J is the subgroup of Q m given in [Sca,§I,Equation (18)]. (We advise the reader that in the notation of [Sca] our Q m is in fact denoted by Q 1 .…”

“…(The notation ρ tr is chosen to keep our presentation as close to [Sca,§II.2] as possible. In our situation the indeterminate k of [Sca,§II,Equation (39)] is equal to one. In such a simple case, the detailed definition of ρ tr in [Sca] is not needed.)…”

Associated varieties and Howe’sN-spectrum

“…In particular, Howe defined the notion of N n -rank for a unitary representation π to be the highest rank of the support of µ N n (π ) regarded as symmetric bilinear forms. Later, Howe's Z N krank was extended to all the type I classical groups by J.-S. Li [1989], to all the type II classical groups by R. Scaramuzzi [1990], and to the exceptional groups by H. Salmasian [2007]. This approach to studying Z N k -spectrum has lead to the classification of the "small" unitary representations for type I classical groups; see [Li 1989].…”