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2010.70: Geometric structure in the principal series of the p-adic group G_2

2010.70: Anne-Marie Aubert, Paul Baum and Roger Plymen (2010) Geometric structure in the principal series of the p-adic group G_2. Representation Theory.

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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 $G_2$.

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 correction, 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 correction is controlled by the cocharacters $h_c$. The cocharacters themselves appear to be closely related to Langlands parameters.

Item Type:Article
Additional Information:

To appear in Representation Theory, an electronic journal published by the American Mathematical Society

Uncontrolled Keywords:Geometric structure, principal series, cocharacters
Subjects:MSC 2000 > 20 Group theory and generalizations
MSC 2000 > 22 Topological groups, Lie groups
MIMS number:2010.70
Deposited By:Professor Roger Plymen
Deposited On:07 August 2010

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