Admissible invariant distributions on reductive

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.