Formulae relating the Bernstein and Iwahori–Matsumoto presentations of an affine Hecke algebra

We consider the “anti-dominant” variants Θ−λ of the elements Θλ occurring in the Bernstein presentation of an affine Hecke algebra . We find explicit formulae for Θ−λ in terms of the Iwahori–Matsumoto generators Tw (w ranging over the extended affine Weyl group of the root system R), in the case (i) R is arbitrary and λ is a minuscule coweight, or (ii) R is attached to GLn and λ=mek, where ek is a standard basis vector and m 1. In the above cases, certain minimal expressions for Θ−λ play a crucial role. Such minimal expressions exist in fact for any coweight λ for GLn. We give a sheaf-theoretic interpretation of the existence of a minimal expression for Θ−λ: the corresponding perverse sheaf on the affine Schubert variety X(tλ) is the push-forward of an explicit perverse sheaf on the Demazure resolution . This approach yields, for a minuscule coweight λ of any R, or for an arbitrary coweight λ of GLn, a conceptual albeit less explicit expression for the coefficient Θ−λ(w) of the basis element Tw, in terms of the cohomology of a fiber of the Demazure resolution.