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

TL;DR

This paper extends the concept of derivatives for smooth representations of GL(n) over real and complex fields, establishing their properties and applications to unitary representations and Whittaker models.

Contribution

It defines derivatives of all orders for smooth admissible Fréchet representations in the archimedean case and proves their exactness, advancing the understanding of representation theory.

Findings

Proved exactness of the highest derivative functor

Computed highest derivatives of all monomial representations

Completed the classification of adduced representations for all irreducible unitary representations

Abstract

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean 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 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, GS12].