By Armand Borel

ISBN-10: 0521580498

ISBN-13: 9780521580496

This ebook offers an advent to a few elements of the analytic thought of automorphic varieties on G=SL2(R) or the upper-half airplane X, with appreciate to a discrete subgroup ^D*G of G of finite covolume. the perspective is electrified through the speculation of limitless 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 thought of automorphic shape on ^D*G\G and its courting with the classical automorphic varieties on X, Poincaré sequence, consistent phrases, cusp varieties, finite dimensionality of the distance of automorphic varieties of a given sort, compactness of yes convolution operators, Eisenstein sequence, unitary representations of G, and the spectral decomposition of L2(^D*G/G). the most necessities are a few ends up in useful research (reviewed, with references) and a few familiarity with the ordinary idea of Lie teams and Lie algebras.

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W. g~l. 1(1), (1) [iW, Y] = 2F, [iW, Z] = - 2 Z , [Y, Z] = iW. Note that since U(g) is an algebra over C, it may be identified with ZY(gc). (iW)c (2) (a,b,ceN). 5(1), we have (3) C = -\W2 + YZ + ZY = -\W2 + iW + 2ZF = -\W2 - iW + 2YZ. Now let M be a complex vector space that is a U(g)-module. v = kv. The set of such elements is the weight space Mk. v is of weight X + 2(a — b). Zv = (X - 2)Zu, hence (5) so that our assertion follows immediately by induction on a and b. v for some v e M. 19 Theorem.

In particular G acts properly on X (but not on X, though, since the isotropy groups on 3X are not compact). 5 An element g e G (g ^ ±1) is said to be elliptic (resp. hyperbolic, parabolic) if it is conjugate to an element of K (resp. A, N). The following equivalences are elementary: (a) g is elliptic <=> the eigenvalues of g are complex of modulus 1 and distinct <=>> tr g < 2 <=> g has exactly one fixed point on X and none on 3X; (b) g is hyperbolic <=> the eigenvalues of g are real distinct 4 = ^ tr g > 2 <=> g has two fixed points on 3X and none on X; (c) g is parabolic <<=>> the eigenvalues of g are equal to 1 <<=>• tr g = 2 <<=>- g has one fixed point on 9X and none on X.

Downloaded from University Publishing Online. 250 on Tue Jan 24 03:47:54 GMT 2012. 19 on ^ ( 0 ) / - We first make some preparation for its proof. Let Gc = SL2(C) be the group of complex 2 x 2 matrices of determinant 1. Its Lie algebra QC is the Lie algebra of 2 x 2 complex matrices of trace zero. W. g~l. 1(1), (1) [iW, Y] = 2F, [iW, Z] = - 2 Z , [Y, Z] = iW. Note that since U(g) is an algebra over C, it may be identified with ZY(gc). (iW)c (2) (a,b,ceN). 5(1), we have (3) C = -\W2 + YZ + ZY = -\W2 + iW + 2ZF = -\W2 - iW + 2YZ.

### Automorphic Forms on SL2 by Armand Borel

by Charles

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