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 Lparameters 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 γ-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.
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Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation π can be expressed in terms of the adjoint γ-factor of the enhanced L-parameter of π. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint γ-factors. Contents 24 6.1. Normalization of densities 24 6.2. Parabolic induction and Plancherel densities 27 Appendix A. Adjoint γ-factors 29 A.1. Independence of the nilpotent operator 30 A.2. Relation with µ-functions 32 References 38
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation $\pi $ can be expressed in terms of the adjoint $\gamma $ -factor of the enhanced L-parameter of $\pi $ . A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $\gamma $ -factors.
The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig [Lus3]), is a rational function of q evaluated at q = q, the cardinality of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of M. Reeder [R3], and for the remaining exceptional cases this is verified in [Fe2]. In the present paper we treat the classical families.The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in [Opd2]) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [Opd3] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.
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