Annihilator varieties, highest derivatives, Whittaker functionals, and rank for unitary representations of GL(n,R)

Abstract

In this paper we study irreducible unitary representations of GL(n,R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, for both smooth and K-finite vectors, thereby generalizing results of Kostant, Matumoto and others. Our second result relates the annihilator to the sequence of highest derivatives, as defined in this setting by one of the authors. Based on those results, we suggest a new notion of rank of a smooth admissible representation of GL(n,R), which for unitarizable representations refines Howe's notion of rank. Our third result computes the highest derivatives for (almost) all unitary representations in terms of the Vogan classification. We also indicate briefly the analogous results over complex and p-adic fields.

21 pages, no figures. v2: minor changes in the exposition, version accepted to Selecta Mathematica

References in corpus (6)

• Multiplicity one theorem for (GL(n+1,R),GL(n,R))
• Derivatives for smooth representations of GL(n,R) and GL(n,C)
• Existence of Klyachko Models for GL(n;R) and GL(n;C)
• Generalized Jacquet modules of parabolic induction
• The SL(2)-type and Base Change
• Kirillov's conjecture and $\CaD$-modules

Cited by in corpus (8)

• Eisenstein series and automorphic representations
• Generalized and degenerate Whittaker models
• Derivatives for smooth representations of GL(n,R) and GL(n,C)
• Existence of Klyachko Models for GL(n;R) and GL(n;C)
• Intertwining operators between line bundles on Grassmannians
• Composition series for degenerate principal series of GL(n)
• Distinction inside L-packets of SL(n)
• The generalized doubling method: $(k,c)$ models