By Armand Borel

This e-book offers an creation to a few facets of the analytic idea of automorphic varieties on G=SL2(R) or the upper-half airplane X, with recognize to a discrete subgroup ^D*G of G of finite covolume. the viewpoint is electrified via the speculation of endless dimensional unitary representations of G; this is often brought within the final sections, making this connection particular. the themes taken care of comprise the development of basic domain names, the idea of automorphic shape on ^D*G\G and its dating with the classical automorphic types on X, Poincaré sequence, consistent phrases, cusp types, 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 ordinary thought of Lie teams and Lie algebras.

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**Sample text**

This material is classical. It is collected here for the sake of completeness and to fix notation. 1 We recall that PG = P SL2(M) = SL2(M)/{±1} is the group Aut X of conformal transformations of X. The subgroup #7{±1} is the isotropy subgroup of i:gi = i <<=>> g e K. i yields a diffeomorphism TC : NA = X = G/K, which maps 'l 0 x\ l)'{ (yW 0 0 onto x + yi (x, y e R, y > 0). 2 The action of G on X extends continuously t o X U R U o o . We let X = XUEUoo, which is diffeomorphic to the closed unit disc.

Then pm_x = Yn = 0 and we have (7) £ > , + Ar(Dj)) = n + m—\ m Assume now that the chain is infinite - say, that n has no maximum - and let n -> oo. The left-hand side of (6) is bounded by In + Ar(£2); hence the series on the right-hand side converges and so Yj also has a limit, say Yoo- We claim that Yoo ^ n/2- The sequence of distances d(zo, dj) (j = 0 , 1 , . . ) diverges, so there exist arbitrarily large j such that d(zo,aj+i) >d(zo,aj). This implies that fij > Yj in the triangle Dj and, since fij + Yj ^ TT, yields Yj ^ TT/2, whence our assertion.

Assume first / to be of type n. Denote by V the Downloaded from University Publishing Online. 250 on Tue Jan 24 03:47:54 GMT 2012. 16 SL2(M), differential operators, and convolution 23 smallest G-invariant closed subspace of C°°(G) containing / . Then it can be shown that its elements are annihilated by P(C) and that Vx = { v e V | v is of type x } is finite dimensional for every x £ K. This is the result we need. 21. 12). We see therefore that there exists afinitedimensional subspace L of C°°(G) containing / , stable under the convolutions *a (a e 7C°°(G)) and containing therefore all elements / * a.