The Weil Representation

Abstract

There is a whole aspect of SL ₂(R) into which we shall not go, namely the various models which can be found in an infinitesimal equivalence class of representations, and the possibility of finding canonical models, e.g. the Whittaker model in such a class. We refer the reader to Jacquet-Langlands [Ja, La], Knapp-Stein [Kn, St], and Stein [St 2] for more information in this direction, and a discussion of intertwining operators among various models. Helgason [He 3] gives a particularly interesting model of representations in eigenspaces of the Laplacian. I include here just the special model of the Weil representation because of its particular interest in number theoretic applications, and the possibility of constructing automorphic forms with it, as in Shalika-Tanaka [Sh, Ta]. Besides, since Weil’s Acta paper [We] is written in an extremely general context, it may be useful to have a naive treatment of the special case as an introduction. Finally, the way the Weil representation is constructed provides an excuse for giving generators and relations for SL ₂, and for mentioning the Brahat decomposition. I did not want to get very much involved in the matters discussed here, and so the chapter is somewhat arbitrary.