Harish-Chandra

This volume is to be regarded as the fifth in the series of Harish-Chandra’s collected papers, continuing the four volumes already published by Springer-Verlag. Because of manifold illnesses in the last ten years of his life, a large part of Harish-Chandra’s work remained unpublished. The present volume deals with those unpublished manuscripts involving real groups, and includes only those pertaining to the theorems which Harish-Chandra had announced without proofs.An attempt has been made by the volume editors to bring out this material in a more coherent form than in the handwritten manuscripts, although nothing essentially new has been added and editorial comments are kept to a minimum. The papers deal with several topics: characters on non-connected real groups, Fourier transforms of orbital integrals, Whittaker theory, and supertempered characters. The generality of Harish-Chandra’s results in these papers far exceeds anything in print. The volume will be of great interest to all mathematicians interested in Lie groups, and all who have an interest in the opus of a twentieth century giant. Harish-Chandra was a great mathematician, perhaps one of the greatest of the second half of the twentieth century.

From the Preface by V. S. VARADARAJAN: “These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them.” The papers are arranged chronologically, Volume III collects his articles written between 1959 and 1968.?

From the Preface by V. S. VARADARAJAN: “These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them.” The papers are arranged chronologically, Volume II collects his articles written between 1955 and 1958.

From the Preface by V. S. VARADARAJAN: “These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them.” The papers are arranged chronologically, Volume IV collects Harish-Chandra's articles written between 1970 and 1983.

From the Preface by V. S. VARADARAJAN: “These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them.” The papers are arranged chronologically, Volume I collects Harish-Chandra's articles written between 1944 and 1954.

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.