Gelfand Pairs and Automorphic L-functions

The goal of the proposal is to study certain integrals of automorphic

forms on a reductive group G. The integrals are over a subgroup H of G

which is large in G, in the sense that over the algebraic closure of the

ground-field, the group H has an open orbit in the flag variety of G.

The integrals may be thought of being periods attached to automorphic

forms and turn out to be, in all cases, interesting numbers. In

particular, in many cases, the square of the periods are conjectured to be

the central values of automorphic L-functions. Examples of such a

phenomenon are results of Waldspurger and others and conjectures of

Gross-Prasad. In other cases, the forms for which the integrals are

non-zero are interesting in their own right. The investigator studies

theses periods and their properties by using a variant of the trace

formula, called the relative trace formula.

The proposal focuses on one aspect of the so called Langlands program. The

goal of the program is to study the automorphic objects (called

automorphic representations); they can be described in terms of the

spectrum of certain operators (generalized Fourier analysis). The data

attached to those objects is encoded in a family of functions of one

complex variable, the automorphic L-functions. The conjectured properties

of these functions contain all possible information on the automorphic

representations. Moreover, other L-functions coming from geometry or

Diophantine problems should be automorphic L-functions. The value of the

automorphic L-functions at a particular point (central value) is a

especially interesting number which is difficult to compute. In many

cases, it is conjectured that this number is non-negative and yet this is

cannot be established directly. The investigator is using a new method

(the relative trace formula) to obtain information about some of these

numbers by representing them as square of integrals. This a novel way to

connect the automorphic L-functions and the automorphic representations.