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

TL;DR

This paper explores a conjectural geometric structure underlying the reducibility of induced representations in the principal series of the p-adic group G_2, using extended quotients and cocharacters related to Langlands parameters.

Contribution

It introduces a conjectural geometric model for the smooth dual of G_2 using extended quotients and cocharacters, refining the Bernstein programme.

Findings

Detailed computations in the principal series of G_2 support the conjecture.

Extended quotient varieties form a model for the smooth dual after algebraic deformation.

Cocharacters linked to two-sided cells relate to Langlands parameters.

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 [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 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 cocharacters $h_c$, one for each two-sided cell $c$ in certain extended affine Weyl groups. The cocharacters themselves appear to be closely related to Langlands parameters.