By Edwin Weiss

ISBN-10: 0486401898

ISBN-13: 9780486401898

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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.

### Algebraic Number Theory by Edwin Weiss

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