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.