Double Affine Hecke Algebras

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Double Affine Hecke AlgebrasIvan CherednikThe aim of the project is to study double affine Heckealgebras (DAHA) and their representations, especially thosewith applications in harmonic analysis. The focus is thestructure of the polynomial representation of DAHA, itsdecomposition and special quotients for generic q, at rootsof unity and for unimodular q. The main examples are finitedimensional quotients of the polynomial representationgeneralizing the Verlinde algebras from conformal fieldtheory; infinite dimensional and non-semisimple quotientsare a natural next step. Deep connections to the classicaltheory of affine Hecke algebras, the theory of Kac-Moodyalgebras, and modern mathematical physics are expected.Lie groups and Lie algebras formalize the concept of symmetryin the theory of special functions, physics, geometry, andcombinatorics. In a similar way, DAHA are candidates for aformalization of the notion of the Fourier transform. In thesimplest one-dimensional case, they are directly related tothe celebrated sl(2). There are indications that they servethe multidimensional theory of Fourier transform better thanLie and Kac-Moody algebras. DAHA are also a source of new unitaryinfinite dimensional theories, that, presumably, will findfundamental applications in mathematics and physics, thetheory of special functions and combinatorics. There are otherimportant directions of the DAHA theory, mainly of algebraicnature, but unitary representaions and Fourier analysis remainof high priority.