Geometric structure in the principal series of the p-adic group G_2

Aubert, Anne-Marie, Baum, Paul and Plymen, Roger (2011) Geometric structure in the principal series of the p-adic group G_2. Representation Theory, 15, 126-169. (doi:10.1090/S1088-4165-2011-00392-7).

Record type: Article

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 .

A feature of this article is the role played by cocharacters attached to two-sided cells 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 of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space is a model of the smooth dual . In this respect, our programme is a conjectural refinement of the Bernstein programme.

The algebraic deformation is controlled by the cocharacters . The cocharacters themselves appear to be closely related to Langlands parameters.

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