The Satake transform and the trace formula

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The Satake transform and L-functions Classically, L-functions with Euler products are an indispensable tool for investigating properties of prime numbers, and more generally properties of complicated structures occurring in number theory. They were introduced by Dirichlet in the early nineteenth century, following observations of the Swiss mathematicain Leonhard Euler. Since then various similar functions have been introduced by many mathematicians, often with very appealing applications, but it was only in the winter of 1966/1967 that the Canadian mathematician Robert Langlands defined for the first time the most general known form of L-functions with Euler products, associated simultaneously to automorphic forms and representations of Galois groups. Since then, a number of cases of his conjectures regarding these functions have been verified, but until very recently essentially all paths to further cases have come to dead ends. Langlands himself suggested around 2000 a possible way to deal with the problem, but for a long time how to follow his suggestions was not very clear. In recent years, however, the Fields medallist Bao Chau Ngo and others have taken up this idea in the form of utilization of local functions not necessarily of compact support in James Arthur's extension of the Selberg Trace Formula. The functions concerned are called `basic functions'. They are parametrized by irreducible representations r of Langlands' L-groups, and they contain, among others, the functions on a p-adic group bi-invariant on left and right by a maximal compact subgroup whose Satake transform is the L-function associated by Langlands to r. There are by now several characterizations of basic functions, but all of them are rather abstract. My contribution so far has been to describe all of them in completely explicit terms in the simplest case of GL(2), and to find by computation conjectural and non-trivial examples for a few other groups of low rank. I hope to find a pattern to these examples so as to make a conjecture for all cases. Early stages of this work are reported on in a paper that will to appear soon in an issue of the Bulletin of the Iranian Mathematical Society honouring the work of the Iranian-American Freydoon Shahidi. One of the surprising by-products of this has been a number of intriguing examples of how the symmetric powers of irreducible representations of complex groups decompose, in which hitherto unseen phenomena appear. This is a classical question, first raised in some form over a hundred years ago. It is not at all clear to what extent these will become a real theory, but results so far are very promising. I expect these results to be applied soon by James Arthur and Salim Ali Altug in applications to the Trace Formula.