By Armand Borel

This ebook presents an advent to a few facets of the analytic conception of automorphic varieties on G=SL2(R) or the upper-half aircraft X, with admire to a discrete subgroup ^D*G of G of finite covolume. the perspective is galvanized via the idea of endless dimensional unitary representations of G; this is often brought within the final sections, making this connection specific. the themes handled comprise the development of basic domain names, the concept of automorphic shape on ^D*G\G and its courting with the classical automorphic varieties on X, Poincaré sequence, consistent phrases, cusp kinds, finite dimensionality of the gap of automorphic types of a given style, compactness of convinced convolution operators, Eisenstein sequence, unitary representations of G, and the spectral decomposition of L2(^D*G/G). the most necessities are a few leads to sensible research (reviewed, with references) and a few familiarity with the ordinary thought of Lie teams and Lie algebras.