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

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 G₂. A feature of this article is the role played by cocharacters hc 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 hc. The cocharacters themselves appear to be closely related to Langlands parameters.