Geometric structure in the principal series of the 𝑝-adic group 𝐺₂

Aubert, Anne-Marie; Baum, Paul; Plymen, Roger

doi:10.1090/S1088-4165-2011-00392-7

Geometric structure in the principal series of the $p$-adic group $\textrm {G}_2$
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by Anne-Marie Aubert, Paul Baum and Roger Plymen

Represent. Theory 15 (2011), 126-169

DOI: https://doi.org/10.1090/S1088-4165-2011-00392-7

Published electronically: February 23, 2011

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Abstract:

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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