Twisted homology for the mirabolic nilradical

TL;DR

This paper proves the exactness of the twisted coinvariants functor related to the mirabolic nilradical, enabling the computation of derivatives of certain representations and advancing understanding of unitary representations.

Contribution

It establishes the exactness of the twisted coinvariants functor for the mirabolic nilradical and computes it for specific representations, extending derivative theory.

Findings

Proves exactness of the twisted coinvariants functor.

Computes the functor on a class of representations.

Enables calculation of derivatives for monomial representations.

Abstract

The notion of derivatives for smooth representations of $GL(n,\mathbb{Q}_p)$ was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the "adduced" representation. In [AGS] derivatives of all orders were defined for smooth admissible Frechet representations (of moderate growth). A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of representations. This implies exactness of the highest derivative functor, and allows to compute highest derivatives of all monomial representations. In [AGS] these results are applied to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations.