Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences
Contemporary Mathematics Volume 614, 2014 http://dx.doi.org/10.1090/conm/614/12253
Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences
Dihua Jiang
Abstract. A general framework for constructions of endoscopy correspon- dences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms, which is called the Principle of Endoscopy Correspon- dences, extending the well-known Howe Principle of Theta Correspondences. This suggests another principle, called the (τ,b)-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski- Shapiro, Rallis, Kudla and others on standard L-functions of classical groups and theta correspondence.
1. Introduction Automorphic forms are transcendental functions with abundant symmetries, and fundamental objects to arithmetic and geometry. In the theory of automorphic forms, it is an important and difficult problem to construct explicitly automorphic forms with specified properties, in particular, to construct cuspidal automorphic forms of various types. Ilya Piatetski-Shapiro is a pioneer to use representation theory to study au- tomorphic forms. The main idea is to construct certain models of representation- theoretic nature to obtain cuspidal automorphic forms, that is, to explicitly realize cuspidal automorphic forms in the space of functions over certain geometric spaces. The celebrated theorem of Gelfand and Piatetski-Shapiro shows that all cuspidal automorphic forms are rapidly decreasing on a fundamental domain D when ap- proaching the cusps, and hence can be realized discretely in the space L2(D)ofall square integrable functions on D. Therefore, it is fundamental to understand the space L2(D) or more precisely the discrete spectrum of L2(D). In the modern theory of automorphic forms, this problem is formulated as follows. Let G be a reductive algebraic group defined over a number field (or a global field) F and consider the quotient space
XG = ZG(A)G(F )\G(A)
where A is the ring of adeles of F and ZG is the center of G. As a consequence of the reduction theory in the arithmetic theory of algebraic groups, this space XG has
2010 Mathematics Subject Classification. Primary 11F70, 22E50; Secondary 11F85, 22E55. Key words and phrases. Fourier coefficients, automorphic integral transforms, endoscopy correspondence, discrete spectrum, automorphic forms.