Softcover ISBN:  9780821820254 
Product Code:  ULECT/16 
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Electronic ISBN:  9781470421656 
Product Code:  ULECT/16.E 
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Book DetailsUniversity Lecture SeriesVolume: 16; 1999; 97 ppMSC: Primary 22;
HarishChandra 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 HarishChandra's original lecture notes.
The main purpose of HarishChandra'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, HarishChandra 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 \(\mathfrak g\).
HarishChandra'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 HarishChandra's original lectures on this subject, including his extension and proof of Howe's Theorem.
In addition to the original HarishChandra 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.ReadershipGraduate students and research mathematicians interested in representations of Lie groups.

Table of Contents

Chapters

Introduction

Part I. Fourier transforms on the Lie algebra

Part II. An extension and proof of Howe’s Theorem

Part III. Theory on the group


Additional Material

Reviews

This branch of representation theory is particularly hard going. In addition, HarishChandra'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


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HarishChandra 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 HarishChandra's original lecture notes.
The main purpose of HarishChandra'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, HarishChandra 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 \(\mathfrak g\).
HarishChandra'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 HarishChandra's original lectures on this subject, including his extension and proof of Howe's Theorem.
In addition to the original HarishChandra 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.

Chapters

Introduction

Part I. Fourier transforms on the Lie algebra

Part II. An extension and proof of Howe’s Theorem

Part III. Theory on the group

This branch of representation theory is particularly hard going. In addition, HarishChandra'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