# 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fondamentaux pour les groupes arithmétiques et formes automorphes, Lectures at Institut H. Poincaré, Paris 1966 (Unpublished, notes by H. Jacquet, J.J. Sansuc and B. Schiffmann) Thanks!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Borel's little book on SL(2,R), and in his Park City/IAS note, although the latter also partly refers to the original unpublished notes.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