Reference number : GC-2017-Q1-01493, amendment number : , agreement type : G, recipient legal name : Casselman, William (The University of British Columbia), recipient country : CA, recipient province : nan, recipient city : British Columbia|Colombie-Britannique, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : 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., program purpose french : 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., agreement title english : The Satake transform and the trace formula, agreement title french : The Satake transform and the trace formula, agreement number : RGPIN, agreement value : 70000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada