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ISSN 1088-4165

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

Author(s): Anne-Marie Aubert; Paul Baum; Roger Plymen
Journal: Represent. Theory 15 (2011), 126-169.
MSC (2010): Primary 20G05, 22E50
Posted: February 23, 2011
MathSciNet review: 2772586
Retrieve article in: PDF

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Abstract: In the representation theory of reductive -adic groups , 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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