Intertwining Operators, Plancherel Measures and Automorphic L-functions

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David GoldbergDMS-98 01340Professor Goldberg will work on four related questions in the local Langlands program. He continue his work on Langlands conjecture about Plancherd measures for certain non-generic representations of some classical groups. He will also attempt to determine the poles of standard intertwining operators attached to arbitrary parabolic subgroups of quasisplit unitary groups. In addition, Professor Goldberg will study semisimple types of the special linear group of a p-adic field. Finally he will test another conjecture of Langlands through the systematic study of R-groups.One of the most important mathematical ideas of the second half of this 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, the subject has been systematized. The Langlands program is a 25 year old project involving this systematization. This project is an important part of the Langlands program.