### Summary and Info

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

### More About the Author

Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.

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