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Admissible Invariant Distributions on Reductive $$p$$-adic Groups
Harish-Chandra, and Notes by Stephen DeBacker and Paul J. Sally, Jr., University of Chicago, IL
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University Lecture Series
1999; 97 pp; softcover
Volume: 16
ISBN-10: 0-8218-2025-7
ISBN-13: 978-0-8218-2025-4
List Price: US$25 Member Price: US$20
Order Code: ULECT/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.

Graduate students and research mathematicians interested in representations of Lie groups.

Reviews

"This branch of representation theory is particularly hard going. In addition, Harish-Chandra's notes were extremely terse, and were tucked away in an obscure source ... the authors have done us all a favour by writing a complete modern treatment which should prove more accessible (in both senses) to modern PhD students."

-- Bulletin of the London Mathematical Society

"DeBacker and Sally are to be commended for their excellent work."

-- Mathematical Reviews