Representation Theory of p-adic Groups

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It is proposed here that support be provided for four projects as part of the investigator's continuing work in the representation theory of p-adic groups with applications to local number theory. The first of these projects would extend to classical groups much of the investigator's work with Bushnell on the groups GL(N). The ultimate goal is to construct inducing representations for the supercuspidal representations of these groups and then to produce covers for the resulting supercuspidal types. This would allow the use of earlier methods in studying the harmonic analysis of these groups and would hopefully have arithmetic applications, especially to Langlands functoriality. The second project involves working out what might be called a Plancherel theory for affine Hecke algebras with possibly unequal parameters. These algebras play a critical role in applying the techniques alluded to above; it is to be hoped that results already obtained in low rank may be extended to this general situation. The third project involves the K-theory and homology of p-adic groups and builds upon some recent results on the module theory of Hecke algebras. The fourth project is to apply methods involving compact, open subgroups to study the oscillator representation, especially in case p=2.A basic problem in the mathematics of almost every ancient culture was to determine solutions to equations where the solutions were constrained to be whole numbers. Such equations, called Diophantine equations after the late Greek mathematician Diophantos, have motivated much of the development of that part of mathematics referred to as the theory of numbers. As an example, one has the so-called Fermat equation, an equation that has been much in the news of late due to the recent determination of its lack of solutions except in case n=2. One way of attempting to solve a Diophantine equation is to replace the equation by a congruence: that is, one picks an integer N and one replaces the condition that the two sides of the equation be equal by the less stringent condition that the difference of the two sides of the equation be divisible by N. Solving these congruences for enough integers N often gives good information about the existence or non-existence of solutions to the original equation. Often it is useful to fix a prime number p and then to study the congruences that result when N runs through all powers of p. In this case, one says that one has localized the problem to the prime p. This process of localization has, over the last two hundred years, led to the development of a part of number theory called local number theory. This is the general area in which this project is to be carried out. One of the most powerful conceptualizations in local number theory is to be found in the conjectures of R.P. Langlands which, if verified, would go a long way to clarifying the nature of local Diophantine problems. The project proposed here would build on the investigator's earlier work in an attempt to provide some of the tools necessary to verify these conjectures.