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