Integral representation of Langlands L functions

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The PI plans to continue his work (joint with David Ginzburg) on integral representation of Langlands L functions with the study of several new examples which appear in exceptional groups. These include an integral representation of the adjoint L function of GL(5), and an extension of the so-called doubling method to the exceptional group G2. Langlands L functions are certain functions of a single complex variable which appear in the subject of automorphic forms, at the intersection of number theory and representation theory. The simplest example is the Riemann zeta function, which plays a crucial role in estimating the number of prime integers less than a given number. The next simplest examples are the Dirichlet L functions, which play a similar role in estimating the distribution of primes in an arithmetic progression. Other examples were defined using modular forms and elliptic curves, resulting in functions with properties nearly identical to these classical examples. In the last century it emerged that it was possible to study modular forms and elliptic curves from the point of view of representation theory, which is essentially the study of collections of linear operators acting on a vector space. At the same time, certain complex functions with nearly the same properties as the classical number theoretic functions discussed above arose naturally in representation theory. These are Langlands L functions. The main strategy for studying these functions is via integral representations. This means, simply, that one tries to take a function of two variables, one of which is complex, and integrate it over the other one. The result is a function of one complex variable, and one hopes that by setting everything up right one can arrange for it to be a Langlands L function. To date, no one really knows exactly what it takes for this to work out, although a number of useful heuristics have been suggested. The PI wishes to continue his study of this intriguing phenomenon.