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Geometric structure in the principal series of the \(p\)-adic group \(\mathbf G_2\). (English) Zbl 1268.22015

Summary: 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 [the authors, C. R., Math., Acad. Sci. Paris 345, No. 10, 573–578 (2007; Zbl 1128.22009)], 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 the cocharacters \(h_c\). The cocharacters themselves appear to be closely related to Langlands parameters.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups

Citations:

Zbl 1128.22009
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References:

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