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

Abstract

In the representation theory of reductive p p -adic groups G 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 G 2 \textrm {G}_2 . A feature of this article is the role played by cocharacters h c h_{\mathbf {c}} attached to two-sided cells c \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 A ( G ) \mathfrak {A}(G) of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space A ( G ) \mathfrak {A}(G) is a model of the smooth dual Irr ( G ) \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 c h_{\mathbf {c}} . The cocharacters themselves appear to be closely related to Langlands parameters.