1,433 research outputs found
A Note on the Spectral Transfer Morphisms for Affine Hecke Algebras
E. Opdam introduced the tool of spectral transfer morphism (STM) of affine
Hecke algebras to study the formal degrees of unipotent discrete series
representations. He established a uniqueness property of STM for the affine
Hecke algebras associated of unipotent discrete series representations. Based
on this result, Opdam gave an explanation for Lusztig's arithmetic/geometric
correspondence (in Lusztig's classification of unipotent representations of
-adic adjoint simple groups) in terms of harmonic analysis, and partitioned
the unipotent discrete series representations into -packets based on the
Lusztig-Langlands parameters. The present paper provides some omitted details
for the argument of the uniqueness property of STM. In the last section, we
prove that three finite morphisms of algebraic tori are spectral transfer
morphisms, and hence complete the proof of the uniqueness property.Comment: title changed; irrelevant materials delete
Supercuspidal unipotent representations: L-packets and formal degrees
Let K be a non-archimedean local field and let G be a connected reductive
K-group which splits over an unramified extension of K. We investigate
supercuspidal unipotent representations of the group G(K). We establish a
bijection between the set of irreducible G(K)-representations of this kind and
the set of cuspidal enhanced L-parameters for G(K), which are trivial on the
inertia subgroup of the Weil group of K. The bijection is characterized by a
few simple equivariance properties and a comparison of formal degrees of
representations with adjoint -factors of L-parameters.
This can be regarded as a local Langlands correspondence for all
supercuspidal unipotent representations. We count the ensueing L-packets, in
terms of data from the affine Dynkin diagram of G. Finally, we prove that our
bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal
degrees of the representations.Comment: Version 2: minor corrections and additions, mainly in section 1
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