Geometric Methods in the Theory of Automorphic Forms

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Geometric methods in the theory of automorphic forms. Abstract.This proposal consists of 2 parts. In the first (and main) partwe introduce certain functions on the set ofisomorphism classes of irreducible representations of a reductivegroup over a local field. We call them gamma-functions. The existenceof such functions is a corollary of the local Langlands conjecture.We formulate some conjectures about explicit construction of thesegamma-functions and explain how they can be applied to study theproblem of Langlands lifting itself (both locally and globally).In the second part we suggest how to generalize the results ofauthor's joint paper with D.Gaitsgory entitled "GeometricEisenstein series". In particular we plan to study intersectioncohomology sheaves on parabolic Drinfelds' spaces and also generalizeour previous results to the case of tamely ramified Eisenstein series. This proposal is in the part of mathematics known as the Langlands program.The Langlands program is part of number theory. Number theory is the studyof the properties of the whole numbers and is the oldest branch ofmathematics. From the beginning problems in number theory have furnished adriving force in creating new mathematics in other diverse parts of thediscipline. The Langlands program is a general philosophy that connectsnumber theory with calculus; it embodies the modern approach to the studyof whole numbers. One aspect of this proposal is to explore theapplications of geometric techniques within the Langlands program.