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

Aubert, Anne-Marie and Baum, Paul and Plymen, Roger (2009) Geometric structure in the principal series of the p-adic group G_2. [MIMS Preprint]

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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 [3], 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 the exceptional group $G_2$. A feature of this article is the role played by cocharacters $h_c$ attached to two-sided cells $c$ 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 $A(G)$ of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space $A(G)$ is a model of the smooth dual $Irr(G)$. In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by the cocharacters $h_c$. The cocharacters themselves appear to be closely related to Langlands parameters.

Item Type: MIMS Preprint
Uncontrolled Keywords: Representation theory, p-adic groups, exceptional group G_2, Hecke algebra, asymptotic Hecke algebra
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 22 Topological groups, Lie groups
Depositing User: Professor Roger Plymen
Date Deposited: 29 Oct 2009
Last Modified: 08 Nov 2017 18:18

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