Gelfand Pairs and Automorphic L-functions

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The goal of the proposal is to study certain integrals of automorphicforms on a reductive group G. The integrals are over a subgroup H of Gwhich is large in G, in the sense that over the algebraic closure of theground-field, the group H has an open orbit in the flag variety of G. The integrals may be thought of being periods attached to automorphicforms and turn out to be, in all cases, interesting numbers. Inparticular, in many cases, the square of the periods are conjectured to bethe central values of automorphic L-functions. Examples of such aphenomenon are results of Waldspurger and others and conjectures ofGross-Prasad. In other cases, the forms for which the integrals arenon-zero are interesting in their own right. The investigator studiestheses periods and their properties by using a variant of the traceformula, called the relative trace formula.The proposal focuses on one aspect of the so called Langlands program. Thegoal of the program is to study the automorphic objects (calledautomorphic representations); they can be described in terms of thespectrum of certain operators (generalized Fourier analysis). The dataattached to those objects is encoded in a family of functions of onecomplex variable, the automorphic L-functions. The conjectured propertiesof these functions contain all possible information on the automorphicrepresentations. Moreover, other L-functions coming from geometry orDiophantine problems should be automorphic L-functions. The value of theautomorphic L-functions at a particular point (central value) is aespecially interesting number which is difficult to compute. In manycases, it is conjectured that this number is non-negative and yet this iscannot be established directly. The investigator is using a new method(the relative trace formula) to obtain information about some of thesenumbers by representing them as square of integrals. This a novel way toconnect the automorphic L-functions and the automorphic representations.