## 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
- Represent. Theory
**15**(2011), 126-169 - DOI: https://doi.org/10.1090/S1088-4165-2011-00392-7
- Published electronically: February 23, 2011
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## 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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## Bibliographic 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