Summary: Geometric structure in the principal series of
the p-adic group G2
Anne-Marie Aubert, Paul Baum and Roger Plymen
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 , that there exists
a simple geometric structure underlying this intricate theory.
We will illustrate here the conjecture with some detailed compu-
tations in the principal series of G2.
A feature of this article is the role played by cocharacters hc at-
tached 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