Memories of Sir Michael Atiyah

Abstract

I was very fortunate to have the opportunity to study with Michael for two months in Oxford in 1968 and for two years (1969–71) at the Institute for Advanced Study in Princeton, where he arranged forme to be invited. Michael Atiyah is famous for his work in algebraic geometry, algebraic topology, index theory, and, later, physics, but on the side he had important contributions to representation theory. I will try to explain some of them here. Michael’s 1967 paper with Bott contains a proof of the holomorphic Lefschetz fixed point formula that provides a wonderfully simple explanation for Weyl’s character formula for tr(g,V) (g is a regular semisimple element, and V is an irreducible rational representation of a complex semisimple groupG). The explanation was in terms of the (finite) fixed point set of the automorphism defined by g on the flag manifold of G. This was a model for me when I later worked with Deligne on representation theory of finite reductive groups, where the holomorphic Lefschetz fixed point formula was replaced by a Lefschetz fixed point formula in l-adic cohomology for certain automorphisms of finite order of a flag manifold and certain subvarieties