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.**