Admissible Invariant Distributions on Reductive $P$-Adic Groups - v. 16

Harish-Chandra presented these lectures on admissible invariant distributions for $p$-adic groups at the Institute for Advanced Study in the early 1970s.

He published a short sketch of this material as his famous "Queen's Notes".

This book, which was prepared and edited by DeBacker and Sally, presents a faithful rendering of Harish-Chandra's original lecture notes.

The main purpose of Harish-Chandra's lectures was to show that the character of an irreducible admissible representation of a connected reductive $p$-adic group $G$ is represented by a locally summable function on $G$.

A key ingredient in this proof is the study of the Fourier transforms of distributions on $\mathfrak g$, the Lie algebra of $G$.

In particular, Harish-Chandra shows that if the support of a $G$-invariant distribution on $\mathfrak g$ is compactly generated, then its Fourier transform has an asymptotic expansion about any semisimple point of $\mathfrak g$.

Harish-Chandra's remarkable theorem on the local summability of characters for $p$-adic groups was a major result in representation theory that spawned many other significant results.

This book presents, for the first time in print, a complete account of Harish-Chandra's original lectures on this subject, including his extension and proof of Howe's Theorem.

In addition to the original Harish-Chandra notes, DeBacker and Sally provide a nice summary of developments in this area of mathematics since the lectures were originally delivered.

In particular, they discuss quantitative results related to the local character expansion.