# Borel's Paris Lectures

I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or know how I could get it? Ensembles fondamentaux pour les groupes arithmétiques et formes automorphes, Lectures at Institut H. Poincaré, Paris 1966 (Unpublished, notes by H. Jacquet, J.J. Sansuc and B. Schiffmann) Thanks!

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Some fragments of these occur in Borel's little book on SL(2,R), and in his Park City/IAS note, although the latter also partly refers to the original unpublished notes. The "theory of the constant term" arguments in the little book seem to scale up well. The general reduction theory is probably that in the Borel-HarishChandra book on arithmetic groups. But, no, I've never seen these notes. – paul garrett Apr 29 '13 at 12:51
Thanks a lot for your comment. In Borel's Paris lecture notes, I was hoping to find a proof of Prop 5.6, page 23 of the Park City/IAS volume, as I don't quite follow Harish-Chandra's proof. Do you know where I can find an alternate proof of Prop 5.6? – admissiblecycle Apr 29 '13 at 13:49
I don't know an alternate reference, but I think 5.6 (that $z$-finite functions on a split component are exponential polynomials) should follow from the Harish-Chandra isomorphism (that the center of the enveloping algebra is identifiable with Weyl-group-symmetric elements of the enveloping algebra of $\mathfrak a$), and thus reducing to the rank-one case, which amounts to solving $P(y\cdot {d\over dy})u=0$ or $P({d\over dx})u=0$ after taking logarithms, where $P$ is a polynomial. Thus, this reduces to constant-coefficient ODEs on the real line. – paul garrett Apr 29 '13 at 15:12
@paul garrett Thanks a lot for the explanation, it makes a lot more sense now! :) – admissiblecycle Apr 29 '13 at 17:12

## 1 Answer

The Paris lectures, along with others he gave later, were spliced together into a publication: Introduction aux groupes arithmetiques (softcover, Hermann, Paris, 1969). As his nominal assistant at IAS in 1968-69, I tried to help with the splicing process but didn't manage to clean up all the inconsistent notation and typos (especially on pages 90-94) which I later jotted down for myself. There was some time pressure, not to mention his many other projects at the time, so the finished product is useful but imperfect. By now it's also probably hard to locate outside some libraries.

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Thanks a lot for the answer and the reference! – admissiblecycle Apr 29 '13 at 13:20
P.S. My notes are by now a bit hard for me to decipher, but at the time I went on proofreading even after the monograph was published. The trouble spots are basically notational, since he was trying to reconcile different notations used in his lectures. – Jim Humphreys Apr 29 '13 at 13:31
Here it is: dropbox.com/s/yaocircrp2ygaaw/… – Nick Gill Apr 29 '13 at 14:36
@Jim: "Hard to locate" is quite correct, for the book version: I was lucky to find my own copy at a "bouquiniste", along the Seine in Paris, a couple of years ago! – Alain Valette Apr 29 '13 at 20:07
If someone knows a deserving library, I am in possession of a copy- via the estate of Walter Baily*. I don't want money, but if the library was willing to make a nominal donation to the AMS in Walter's name, that would be great. It isn't signed, but on the inside cover there is a type written note saying "with the compliments of the author. Borel and Baily were good friends. Feel free to delete this comment if deemed too commercial. – aginensky Apr 30 '13 at 23:31