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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Geometric structure in the principal series of the $p$-adic group $\textrm {G}_2$
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by Anne-Marie Aubert, Paul Baum and Roger Plymen PDF
Represent. Theory 15 (2011), 126-169 Request permission

Abstract:

In the representation theory of reductive $p$-adic groups $G$, the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in (2007), that there exists a simple geometric structure underlying this intricate theory.

We will illustrate here the conjecture with some detailed computations in the principal series of $\textrm {G}_2$.

A feature of this article is the role played by cocharacters $h_{\mathbf {c}}$ attached to two-sided cells $\mathbf {c}$ in certain extended affine Weyl groups.

The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union $\mathfrak {A}(G)$ of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space $\mathfrak {A}(G)$ is a model of the smooth dual $\textrm {Irr}(G)$. In this respect, our programme is a conjectural refinement of the Bernstein programme.

The algebraic deformation is controlled by the cocharacters $h_{\mathbf {c}}$. The cocharacters themselves appear to be closely related to Langlands parameters.

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Additional Information
  • Anne-Marie Aubert
  • Affiliation: Institut de Mathématiques de Jussieu, U.M.R. 7586 du C.N.R.S. and U.P.M.C., 75005 Paris, France
  • MR Author ID: 256498
  • ORCID: 0000-0002-9613-9140
  • Email: aubert@math.jussieu.fr
  • Paul Baum
  • Affiliation: Pennsylvania State University, Mathematics Department, University Park, Pennsylvania 16802
  • MR Author ID: 32700
  • Email: baum@math.psu.edu
  • Roger Plymen
  • Affiliation: School of Mathematics, Alan Turing building, Manchester University, Manchester M13 9PL, England
  • Email: plymen@manchester.ac.uk
  • Received by editor(s): July 14, 2009
  • Received by editor(s) in revised form: May 27, 2010, and June 17, 2010
  • Published electronically: February 23, 2011
  • Additional Notes: The second author was partially supported by NSF grant DMS-0701184
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 15 (2011), 126-169
  • MSC (2010): Primary 20G05, 22E50
  • DOI: https://doi.org/10.1090/S1088-4165-2011-00392-7
  • MathSciNet review: 2772586