RUI: Mathematical Sciences: Spherical Characters on P-adic Coset Spaces and the Relative Trace Formula

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Rader This award provides funding for a project in the Langlands program. The Relative Trace Formula (as initiated by Jacquet) is a new tool for studying automorphic L-functions and functoriality in the Langland's program. Rader and S. Rallis have been collaborating to prove some of the local results they think necessary to make the Relative Trace Formula a theorem instead of just a collection of examples. They have shown that many of the results known for usual character and central orbital integrals generalize to the case of spherical characters and double coset integrals on certain p-adic symmetric spaces. They plan to expand their research to include spherical subgroups such in the sense of M. Brion. Although it will take many years to complete such a project, it seems to be the natural domain for relative trace formulae, because (probably) one dimensional representations of H occur in irreducible smooth infinite dimensional representations of G with finite multiplicity. Rader also plans to complete a collaborative effort with Marie-France Vigneras investigating the geometric Zelevinskii involution for nilpotent orbits in certain Lie algebras. This should have applications in constructing modular representations of Hecke algebras, after Kazhdan-Lusztig. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.