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Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra. (English) Zbl 1056.20003
Summary: We consider the “anti-dominant” variants \(\Theta^-_\lambda\) of the elements \(\Theta_\lambda\) occurring in the Bernstein presentation of an affine Hecke algebra \(\mathcal H\). We find explicit formulae for \(\Theta^-_\lambda\) in terms of the Iwahori-Matsumoto generators \(T_w\) (\(w\) ranging over the extended affine Weyl group of the root system \(R\)), in the case (i) \(R\) is arbitrary and \(\lambda\) is a ‘minuscule’ coweight, or (ii) \(R\) is attached to \(\text{GL}_n\) and \(\lambda=me_k\), where \(e_k\) is a standard basis vector and \(m\geq 1\). In the above cases, certain ‘minimal expressions’ for \(\Theta^-_\lambda\) play a crucial role. Such minimal expressions exist in fact for any coweight \(\lambda\) for \(\text{GL}_n\). We give a sheaf-theoretic interpretation of the existence of a minimal expression for \(\Theta^-_\lambda\): the corresponding perverse sheaf on the affine Schubert variety \(X(t_\lambda)\) is the push-forward of an explicit perverse sheaf on the Demazure resolution \(m\colon\widetilde X(t_\lambda)\to X(t_\lambda)\). This approach yields, for a minuscule coweight \(\lambda\) of any \(R\), or for an arbitrary coweight \(\lambda\) of \(\text{GL}_n\), a conceptual albeit less explicit expression for the coefficient \(\Theta^-_\lambda(w)\) of the basis element \(T_w\) in terms of the cohomology of a fiber of the Demazure resolution.

MSC:
20C08 Hecke algebras and their representations
14M15 Grassmannians, Schubert varieties, flag manifolds
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