The universal Harish-Chandra j-function

Abstract: The Harish–Chandra μ-function plays a central role in the explicit Plancherel formula for a p-adic group G. It arises as the normalising factor for the Plancherel measure on the unitary dual of G, and is defined through the theory of intertwining operators.

In this talk, we show how to extend the construction of the μ-function—or more precisely its inverse, the j-function—to all finitely generated representations, and over general coefficient rings such as Z[1/p]. This leads to a universal j-function with values in the Bernstein centre, which specialises to the classical j-function. Beyond its role in harmonic analysis, the universal j-function also encodes arithmetic information: it reflects aspects of the local Langlands correspondence for classical groups, via Mœglin’s criterion and its connection to reducibility points of parabolically induced representations. Time permitting, we will illustrate how this perspective applies to the study of the local Langlands correspondence in families. This is joint work with Gil Moss.

number theoryrepresentation theory

Audience: researchers in the topic

University of Utah Representation Theory / Number Theory Seminar