## Admissible Invariant Distributions on Reductive P-adic Groups (Google eBook)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 $\frak g$, the Lie algebra of $G$. In particular, Harish-Chandra shows that if the support of a $G$-invariant distribution on $\frak g$ is compactly generated, then its Fourier transform has an asymptotic expansion about any semisimple point of $\frak g$. 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. |

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admissible distribution assume bijection bounded on 9 Cartan subalgebra Cartan subgroup Cg(X character expansion characteristic function compact open subgroup compact set compact subset complex numbers constant on 9 Corollary 2.2 deﬁne denote the set distribution on 9 everywhere submersive f G D ﬁx Fix a compact Fourier transform function on 9 G D0 G M0 G-invariant distribution G-orbit H G I Haar measure Harish—Chandra Harmonic analysis Hence Howe’s Theorem implies invariant measure irreducible lattice in 9 Lemma Let f Lie algebra locally bounded locally constant locally constant function locally summable function Math Moreover nilpotent orbits Note orbital integrals proof of Lemma proof of Theorem reductive p-adic groups regular orbit remains bounded semisimple element Shalika germs subgroup of G subset of 9 supercuspidal representation Supp Supp(f Suppose surjective Theorem 3.1 X G 9 zero

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Page ix - Let fl be a p-adic field of characteristic zero with ring of integers R. Let G be the group of fi-rational points of a connected, reductive fi-group.

Page x - C"(G), the space of locally constant, compactly supported, complex-valued functions on G, then (0.2) JO is an operator of finite rank.

Page x - Motivated by the case of real reductive groups, we may ask if there exists a locally summable function F^ on G which is locally constant on G' such that ©,(/) = ff(x)-F-(x)dx JG for all / € C^(G).