The Relative Trace Formula and its Applications

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ABSTRACTTECHNICAL DESCRIPTIONThe relative trace formula (RTF) is used to study automorphic representations of a reductive group G that are distinguished by a subgroup H obtained as the fixed-point set of an involution. One hopes to characterize distinguished generic representations as functorial transfers from a third group G' (which can be specified conjecturally in terms of the involution) by comparing RTF with the Kuznetzov trace formula on G'. To this end, it is important to obtain a fine spectral expansion of the RTF and, in particular, to write the spectral expansion in terms of relative Bessel distributions. Progress towards this goal was made in prior joint work by the PI, Jacquet and Lapid for the case of the standard Galois involution on GL(n). They developed a procedure for defining regularized periods of Eisenstein series, which turns out to be key ingredients in the fine spectral expansion. In some cases, it has been possible to express the regularized periods in terms of certain integrals analagous to intertwining operators which have been called intertwining periods. The PI intends, in collaboration with E. Lapid, to extend this previous work to the general case. This will involve analytic problems related to Eisenstein series and combinatorial problems related to the structure of the orbits of the Borel subgroup on G/H. It will also be necessary to develop a suitable truncation operator. One would like to express the regularized periods of cuspidal Eisenstein series in terms of L-functions. In general, however, the regularized period will be equal to an infinite sum of intertwining periods. To deal with this problem, the PI and Lapid intend to develop a formalism for forming linear combinations of the intertwining periods, in analogy with the linear combinations of characters that occur in the endoscopic theory of Langlands-Shelstad. Closely related is the problem of establishing identities between relative Bessel distributions for the pair (G,H) and Bessel distributions on G'. In a related project to be carried out with D. Ramakrishnan, the PI will investigate certain limit formulas connected with relative trace formulas. This will lead to a new method of proof and more precise versions of previously known results of W. Duke and others on the distribution of certain special values of GL(2) L-functions. The distribution results will involve certain measures on the spherical dual. Higher rank cases will be investigated and a general context in which to place the results will be sought.NON-TECHNICAL DESCRIPTIONThe history of mathematics has shown that the simplest phenomena are sometimes the hardest to understand deeply. The correct explanation may emerge only after the right theoretical framework has been found. The reciprocity laws of number theory fall into this category of mathematical phenomena. The simplest law of this type, the so-called law of quadratic reciprocity, is a beautiful and mysterious fact about ordinary whole numbers. It can be explained to a curious high school student, but its true structural meaning can only be understood within the context of a sophisticated and advanced part of number theory called class field theory. One of the great challenges of modern number theory is to fully explore the most general reciprocity laws. A framework for formulating such laws was developed 30 years ago by R. Langlands, and as a result, we know that there must exist a vast web of interrelated reciprocity laws. As a totality, these conjectural laws are called the functoriality principle. The functoriality principle seeks to explain the reciprocity laws within the context of a theory that originated in theoretical physics, the so-called representation theory of semisimple groups. In addition to ties with advanced theoretical physics, the theory of functoriality has found applications in diverse areas of combinatorics, coding theory, and cryptography. Enormous progess in the theory of functoriality has been made during the last thirty years which in turn has motivated much outstanding research, including the solution of the famous Fermat's Last Theorem. Despite this, our understanding of functoriality remains rudimentary in many respects. When a fully developed theory of functoriality is eventually developed, we can expect it to have a profound influence on mathematics and some areas of its applications. The goal of the project supported by this grant is to advance our understanding of the Relative Trace Formula, which is one of a handful of valuable tools that we have for studying functoriality. The results of this study will make it possible to study the functoriality principle from the point of view of "period integrals". Hopefully, this will play a role in advancing our knowledge of the general functoriality principle.