Motive Representation Theory

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A new type of integration, called motivic integration, has been proposedby M. Konsevitch and developed by Denef and Loeser. They show that manyof the classical properties of p-adic integration can be extended to tothe motivic context. The research of this proposal will adapt motivicintegration to the representation theory and the harmonic analysis ofreductive groups over fields of formal Laurent series in characteristiczero. This new theory will be developed from first principles, startingwith the existence of motivic Haar measures. The starting point of much of modern mathematics is the theory ofintegration, as developed by Isaac Newton, and generations ofmathematicians that have followed him. An unexpected development came in1995, when the mathematician M. Kontsevich developed an entirely new wayto integrate. This new tool will allow mathematicians to significantlyenlarge the scope of mathematics. The research of this grant willaccomplish part of this project, by using this new tool to enlarge thescope of representation theory, a branch of modern algebra.