By Armand Borel

ISBN-10: 0521580498

ISBN-13: 9780521580496

This publication offers an creation to a couple elements of the analytic thought of automorphic types on G=SL2(R) or the upper-half aircraft X, with appreciate to a discrete subgroup ^D*G of G of finite covolume. the perspective is electrified by way of the idea of endless dimensional unitary representations of G; this is often brought within the final sections, making this connection particular. the subjects taken care of contain the development of primary domain names, the idea of automorphic shape on ^D*G\G and its dating with the classical automorphic kinds on X, Poincaré sequence, consistent phrases, cusp kinds, finite dimensionality of the distance of automorphic types of a given style, compactness of yes convolution operators, Eisenstein sequence, unitary representations of G, and the spectral decomposition of L2(^D*G/G). the most must haves are a few ends up in practical research (reviewed, with references) and a few familiarity with the easy conception of Lie teams and Lie algebras.

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Then, using (3), we obtain (5) hm{x) = f h(x)Xn(k)Xm(k-l)dk = h(x)Smtn, Downloaded from University Publishing Online. 250 on Tue Jan 24 03:47:54 GMT 2012. 20 hence (6) hn = 8m,nh (h of type m). This shows that *Xn is a projector of C°°(G) onto the space of elements of K-type n (on the right). By standard Fourier analysis, the Fourier series J2 nn converges absolutely to h, but we shall not need this fact. 20(1). Since Mim is finite dimensional, it is closed in V. It suffices therefore to show that any v e Vm is a limit of elements in M;m.

Xforalla e N. The proof is by induction on a. JC. Y. Assume our assertion proved up to a. Z. x by the induction assumption. Thus Ya+lZa+lx is a multiple of FZJC and hence of x. This proves (1). It follows that if a ^ b then r \ Z * . v € Thus M is spanned by the vectors Ymv and Zmi> of respective weights X + 2m and A — 2m (m e N). Therefore the weights belong to A + 2Z and have multiplicity 1. This establishes the theorem when P(C) = C — c. To prove the finite dimensionality of the M^ in the general case, we argue by induction.

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.

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