Abstract
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 $\gamma$-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.
Highlights
Let K be a non-archimedean local field and let G be a connected reductive K-group
We will derive that from the following result, which says that one can determine the L-parameters of supercuspidal unipotent representations of a simple algebraic group by comparing formal degrees and adjoint γ-factors
The second bullet of Theorem 1 says that comparing formal degrees and adjoint γ-factors completely characterizes the L-parameters of supercuspidal unipotent representations of simple adjoint K-groups exhibited by Lusztig and Morris
Summary
Let K be a non-archimedean local field and let G be a connected reductive K-group. Roughly speaking, a representation of the reductive p-adic group G(K) is unipotent if it arises from a unipotent representation of a finite reductive group associated to a parahoric subgroup of G(K). We will derive that from the following result, which says that one can determine the L-parameters of supercuspidal unipotent representations of a simple algebraic group by comparing formal degrees and adjoint γ-factors. The second bullet of Theorem 1 says that comparing formal degrees and adjoint γ-factors completely characterizes the L-parameters of supercuspidal unipotent representations of simple adjoint K-groups exhibited by Lusztig and Morris. Our methods to generalize from simple adjoint to reductive groups are constructive, so that for any given supercuspidal unipotent representation one can in principle write down the enhanced L-parameter. Applying Hilbert’s theorem 90 to the maximal split torus, we obtain a corresponding decomposition of the group of K-rational points This enables us to reduce to the cases of tori (well-known) and of reductive K-groups with anisotropic connected center. Acknowledgements. — We thank the referees for their helpful comments and careful reading
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