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Twisted homology for the mirabolic nilradical

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Abstract

The notion of derivatives for smooth representations of GL(n, 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 Fréchet 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.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot, 76100, Israel

    Avraham Aizenbud & Dmitry Gourevitch

  2. Department of Mathematics, Rutgers University, Hill Center — Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Siddhartha Sahi

Authors
  1. Avraham Aizenbud
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  2. Dmitry Gourevitch
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  3. Siddhartha Sahi
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Correspondence to Avraham Aizenbud.

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Aizenbud, A., Gourevitch, D. & Sahi, S. Twisted homology for the mirabolic nilradical. Isr. J. Math. 206, 39–88 (2015). https://doi.org/10.1007/s11856-014-1150-3

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  • DOI: https://doi.org/10.1007/s11856-014-1150-3

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