# Residue distributions and harmonic analysis on reductive groups

Thursday, November 15, 2018 - 9:30am - 10:30am

Keller 3-180

Eric Opdam (Universiteit van Amsterdam)

Let omega be a rational n-form on C^n whose singular locus is a finite affine real hyperplane arrangement, and let b denote a base point in R^n outside the singular locus. Given a Paley-Wiener function f on C^n we define I_b(f) as the integral of f times omega over b+iR^n. By Cauchy’s theorem this linear functional I_b on the space PW(C^n) of Paley-Wiener functions only depends on the connected component of the complement of the singular locus of omega in which b lies. In the study of Paley-Wiener Theorems in harmonic analysis on reductive groups over a local field one encounters such integrals I_b(f), where the base point b is typically a point deep in a chamber of the root system of the dual group.

It is a basic result that there exists a unique finite set C in R^n, and for each c in C a nonzero tempered distribution X^b_c on c+iR^n, characterised by the requirement that 1) I_b(f) is the sum over c in C of X^b_c evaluated at the restriction of f to c+iR^n, and 2) some natural condition on the support of X^b_c. In the context of reductive groups and more generally symmetric spaces, these residue distributions’’ X^b_c have been used to compute residual contributions to the spectral decomposition. In basic cases, they can be expressed in terms of Langlands parameters.

We discuss two applications. Firstly a proof of conjectures of Hiraga, Ichino and Ikeda in the special case of representations of unipotent reduction of a p-adic simple group of adjoint type, computing the Plancherel measure in terms of adjoint gamma factors. Secondly we show in a uniform way that certain residues of unramified Eisenstein series are square integrable, and compute their L2-norms. Previously similar results had been obtained in special cases by various authors (Jacquet, Langlands, Moeglin-Waldspurger, Moeglin, Kim, and Miller).

This talk is based in part on joint works with Ciubotaru, De Martino, Heckman, Heiermann, Feng and Solleveld.

It is a basic result that there exists a unique finite set C in R^n, and for each c in C a nonzero tempered distribution X^b_c on c+iR^n, characterised by the requirement that 1) I_b(f) is the sum over c in C of X^b_c evaluated at the restriction of f to c+iR^n, and 2) some natural condition on the support of X^b_c. In the context of reductive groups and more generally symmetric spaces, these residue distributions’’ X^b_c have been used to compute residual contributions to the spectral decomposition. In basic cases, they can be expressed in terms of Langlands parameters.

We discuss two applications. Firstly a proof of conjectures of Hiraga, Ichino and Ikeda in the special case of representations of unipotent reduction of a p-adic simple group of adjoint type, computing the Plancherel measure in terms of adjoint gamma factors. Secondly we show in a uniform way that certain residues of unramified Eisenstein series are square integrable, and compute their L2-norms. Previously similar results had been obtained in special cases by various authors (Jacquet, Langlands, Moeglin-Waldspurger, Moeglin, Kim, and Miller).

This talk is based in part on joint works with Ciubotaru, De Martino, Heckman, Heiermann, Feng and Solleveld.