ABSTRACT Alan Roche Purdue University DMS-98 01131 Professor Roche will study the complex representation theory of a reductive p-adic group by restricting to compact open subgroups. Building on an idea due to Bushnell and Kutzko known as the method of types, the investigator expects to be able to connect precisely components of the Bernstein decomposition of a reductive p-adic group with data from the open compact parts of the group. Professor Roche also more fully understand associated Hecke algebras using theses new methods. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and consider the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples above. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, these examples have been systematized into a branch of number theory called the Langlands program. This project is an important part of the Langlands program.
