Gelfand Pairs and Automorphic Representations

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9619766 Jacquet The goal of the project is to study certain period integrals. Let G be a reductive group over a number field F; let H be a subgroup of G; we assume that over the algebraic closure of F the group H has an open orbit in the flag variety of G. If f is an automorphic form on G and t a suitable character of H the corresponding period integral is the integral of f times t taken over the adelic quotient of H. In many cases, it is conjectured that at every place of F the triple (G,H,t) satisfies Gelfand multiplicity one condition. Furthermore, it is conjectured that the period integral is related to the value of an automorphic L-function at its center of symmetry. For instance, the value of the L-function may be essentially the square of a period integral. In other cases, automorphic representations that admit a vector f for which the integral is non-zero are especially interesting from the point of view of the Langlands program. In this proposal, the tool for studying the period integrals is a variant of the trace formula, the relative trace formula. There will be applications to the value of the L-functions at their center of symmetry. In some cases, it might be possible to prove that this value is non-negative. Other results will be more directly related to the Langlands program, and will provide information on the principle of functoriality or the structure of the L-packets. In general, this information cannot be obtained from the standard trace formula or the theory of dual pairs. The questions in local harmonic analysis which arise may be of independent interest. A notable example is a conjectural identity between certain Kloosterman integrals (or sums). This proposal deals with the Langlands program, and the Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diver se parts of the discipline. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.