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Geometric structure in the principal series of the $p$-adic group $\textrm {G}_2$

Authors: Anne-Marie Aubert, Paul Baum and Roger Plymen
Journal: Represent. Theory 15 (2011), 126-169
MSC (2010): Primary 20G05, 22E50
Published electronically: February 23, 2011
MathSciNet review: 2772586
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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

Paul Baum
Affiliation: Pennsylvania State University, Mathematics Department, University Park, Pennsylvania 16802
MR Author ID: 32700

Roger Plymen
Affiliation: School of Mathematics, Alan Turing building, Manchester University, Manchester M13 9PL, England

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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.