Twisted homology for the mirabolic nilradical

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.

28 pages. v2: minor corrections, version to appear in the Israel Journal of Mathematics

References in corpus (2)

• Schwartz functions on Nash manifolds
• Derivatives for smooth representations of GL(n,R) and GL(n,C)

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• Doubling Constructions: the complete L-function for coverings of the symplectic group
• Composition series for degenerate principal series of GL(n)
• Doubling Constructions and Tensor Product $L$-Functions: the linear case
• The generalized doubling method: $(k,c)$ models
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