Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

Abstract

Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F , in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F . This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143-2207].More generally for G connected reductive (still assumed to be split over an unramified extension of F ), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op. cit.Date: 2018-07-30. 2000 Mathematics Subject Classification. Primary 20C08; Secondary 22D25, 43A30. Key words and phrases. Cuspidal unipotent representation, formal degree, discrete unramified Langlands parameter.It is a pleasure to thank Maarten Solleveld for many useful comments. It is also a pleasure to acknowledge the excellent comments and suggestions of the participants of the workshop "Representation Theory of Reductive Groups Over Local Fields and Applications to Automorphic forms" at the Weizmann institute, and of the workshop "Representation theory of p-adic groups" at IISER, Pune.

On Formal Degrees of Unipotent Representations

“…Finally, with different methods, the third author constructed a local Langlands correspondence for all unipotent representations of reductive groups over K [49]. In Theorem 3.1 we show that the approaches in [39] and [49] agree, and we derive some extra properties of these instances of a local Langlands correspondence. (Meanwhile, all this has been generalised to ramified groups [51].)…”

“…For every M ∈ Lev(G) we choose a bijection ( 14) which satisfies all the requirements from [39]. In this way we obtain…”

“…The classification has also been worked out in several papers when G splits over an unramified extension of K. We have exhibited a local Langlands correspondence for supercuspidal unipotent representations of reductive groups over K in [16,17]. Then the second author generalised this to a Langlands parametrisation of all tempered unipotent representations in [39]. Finally, with different methods, the third author constructed a local Langlands correspondence for all unipotent representations of reductive groups over K [49].…”

“…In the remainder of the introduction, we assume that G splits over an unramified field extension. The conjectures by Hiraga, Ichino and Ikeda ('HII-conjectures') were proven for supercuspidal unipotent representations in [43,16,15,17], for unipotent representations of simple adjoint groups in [38] and for tempered unipotent representations in [39]. However, in the last case the method only sufficed to establish the desired formulas up to a constant.…”

“…In Section 7.2 we establish Theorem 2(b) for any Levi subgroup M ⊂ G. This involves a translation to Plancherel densities for affine Hecke algebras, via the aforementioned types. In the final stage we use the fact that Theorem 2 was already known up to constants [39].…”