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

Abstract

Abstract. In this paper we study irreducible unitary representations of GLn (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 adduced representations, 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 GLn(R), which for unitarizable representations refines Howe's notion of rank.Our third result computes the adduced representations for (almost) all irreducible unitary representations in terms of the Vogan classification.We also indicate briefly the analogous results over complex and p-adic fields.

Derivatives for smooth representations of GL(n, ℝ) and GL(n, ℂ)

“…In this section we prove two results conjectured in [GS13a]: uniqueness of degenerate Whittaker models and computation of adduced representations for Speh complementary series. We also prove that for irreducible unitary representations, the three notions of highest derivative coincide and extend the notion of adduced representation.…”

“…The real case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89,Sah90,SaSt90,GS13a]. …”

“…We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89,Sah90,SaSt90,GS13a].…”

“…A (degenerate) Whittaker functional on a smooth representation of G n is an (N n , χ)-equivariant functional. Such functionals were studied in [GS13a]. In particular, [GS13a] associates a character χ π of N n to any irreducible representation π, using the annihilator variety of π and proves existence of the corresponding degenerate Whittaker functionals for unitarizable π.…”

“…Such functionals were studied in [GS13a]. In particular, [GS13a] associates a character χ π of N n to any irreducible representation π, using the annihilator variety of π and proves existence of the corresponding degenerate Whittaker functionals for unitarizable π. In this paper we deduce uniqueness of those functionals from Theorem A.…”