[Paper Review] Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda

Eric Opdam|arXiv (Cornell University)|Jul 26, 2018

Advanced Algebra and Geometry47 references3 citations

TL;DR

This paper proves the conjectures of Hiraga, Ichino, and Ikeda on the Plancherel density for tempered representations of unipotent reduction in connected reductive $p$-adic groups that are split over an unramified extension. Using spectral transfer maps between unipotent affine Hecke algebras and Lusztig’s classification, it establishes the formal degree formula in terms of $L$-functions and $oldsymbol{\epsilon}$-factors, confirming the conjecture up to rational constants for general reductive groups.

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.

Motivation & Objective

• To extend the proof of the Hiraga–Ichino–Ikeda conjectures on Plancherel density to general connected reductive $p$-adic groups, not just adjoint groups.
• To show that the compatibility with the conjectures essentially determines the Langlands parameterization for tempered representations of unipotent reduction.
• To establish the conjectured formal degree formula in terms of $L$-functions and $oldsymbol{\epsilon}$-factors for such representations.
• To demonstrate that the conjectures hold up to rational constant factors for all tempered representations of unipotent reduction in reductive groups split over an unramified extension.

Proposed method

• Utilizes spectral transfer maps (STMs) between normalized affine Hecke algebras, constructed via Lusztig’s classification and Iwahori-Matsumoto duality.
• Applies the theory of unipotent affine Hecke algebras and their Bernstein centers to analyze the Plancherel measure and trace functionals.
• Employs the formal degree formula derived from the spectral decomposition of the trace on affine Hecke algebras.
• Uses residual cosets and deformation techniques to compute the Plancherel density $d_{oldsymbol{\mathcal{H}},\delta}$ for discrete series representations.
• Establishes compatibility of Langlands parameters with the conjectures by analyzing central characters and $\mathcal{S}$-groups.
• Leverages the equivariance of STMs under isogenies $\eta: H \to G$ to reduce the problem to almost direct products of semisimple and anisotropic groups.

Experimental results

Research questions

• RQ1 Does the Hiraga–Ichino–Ikeda conjecture on the Plancherel density hold for all tempered representations of unipotent reduction in connected reductive $p$-adic groups?
• RQ2 Can the Langlands parameterization for such representations be uniquely determined by compatibility with the conjectured formal degree formula?
• RQ3 To what extent do the conjectures hold up to rational constant factors in the general reductive case?
• RQ4 How do spectral transfer maps between unipotent affine Hecke algebras facilitate the proof of the conjectures?
• RQ5 What role do residual cosets and deformation techniques play in computing the Plancherel measure for discrete series?

Key findings

• The conjecture of Hiraga, Ichino, and Ikeda holds for all tempered representations of unipotent reduction in connected reductive $p$-adic groups that are split over an unramified extension.
• The formal degree of a tempered representation $\pi_\rho$ is given by $\textup{fdeg}(\pi_\rho) = \frac{\dim(\rho)}{|\mathcal{S}^\natural_\varphi|} |\gamma(0, \textup{Ad} \circ \varphi, \psi)|$, up to a rational constant factor.
• The spectral transfer maps between unipotent affine Hecke algebras provide a canonical way to relate Langlands parameters across isogenous groups, ensuring consistency of the conjecture under isogenies.
• For groups of adjoint type, the conjecture holds exactly, confirming earlier results in the literature.
• The construction of STMs via cuspidal unipotent representations ensures that the parameterization is compatible with the Plancherel measure and central characters.
• The main result, Theorem 4.5.1, establishes the conjecture up to rational constants for all tempered representations of unipotent reduction in general reductive groups over $p$-adic fields.