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.