By Armand Borel

ISBN-10: 0521580498

ISBN-13: 9780521580496

This booklet offers an creation to a couple points of the analytic idea of automorphic types on G=SL2(R) or the upper-half airplane X, with appreciate to a discrete subgroup ^D*G of G of finite covolume. the viewpoint is galvanized through the idea of endless dimensional unitary representations of G; this can be brought within the final sections, making this connection particular. the subjects taken care of comprise the development of primary domain names, the idea of automorphic shape on ^D*G\G and its dating with the classical automorphic varieties on X, Poincaré sequence, consistent phrases, cusp kinds, finite dimensionality of the distance of automorphic sorts of a given kind, compactness of convinced convolution operators, Eisenstein sequence, unitary representations of G, and the spectral decomposition of L2(^D*G/G). the most must haves are a few leads to useful research (reviewed, with references) and a few familiarity with the straightforward conception of Lie teams and Lie algebras.

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**Example text**

V ^ oo, so y(V) is also a horodisc tangent to the real axis. Therefore y(V) intersects the horizontal line y = t. Since every point on y = t is the image of an element in C by some y G Foo, it follows that there exists y' such that y'(V) Pi C ^ 0, but this contradicts (2). This proves (3) and the proposition. 11 Some geometric definitions. Our last goal in this section is to construct and establish some properties of a fundamental domain for F when F\G has finite volume or, equivalently, when F\X has finite hyperbolic area.

A Siegel set has finite invariant measure. It suffices to see this when & is associated to 00. y~2 = r2. f dx < 00. J(o Similarly, a Siegel set in G has finite Haar measure. (b) Assume that u is cuspidal for F and that co contains an interval including a fundamental domain for FM on N. & = HUJ is a horodisc tangent to 3X at u. 8). Therefore, the projection 7r:Xp —> F\Xp is a finite-to-one mapping of {«} U & onto a relatively compact neighborhood of 7T(M), which is equal to ru\Hu,t. The sets {u} U & form a fundamental set of neighborhoods of u in F \X£.

Then Fz is isomorphic to a discrete subgroup of K and hence is finite cyclic, generated by an elliptic element. The point z is said to be elliptic for F. Let now z e 3X. Its isotropy group in G is a parabolic subgroup P. Let N = NP. (b) The point z is said to be cuspidal for F, or F-cuspidal, if FN = F D N ^ {1}. Then F# is infinite cyclic, generated by a parabolic element. We claim that F^ has index ^ 2 in FP := F D P. TN or FP is infinite cyclic, generated by an element in — N. In both cases, Fz' is equal to F^ and is infinite cyclic.

### Automorphic Forms on SL2 by Armand Borel

by Joseph

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