Formulae relating the Bernstein and Iwahori–Matsumoto presentations of an affine Hecke algebra
✍ Scribed by Thomas J. Haines; Alexandra Pettet
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 193 KB
- Volume
- 252
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
We consider the "anti-dominant" variants Θ - λ of the elements Θ λ occurring in the Bernstein presentation of an affine Hecke algebra H. We find explicit formulae for Θ - λ 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 λ is a minuscule coweight, or (ii) R is attached to GL n and λ = me k , where e k 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 GL n . 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(t λ ) is the push-forward of an explicit perverse sheaf on the Demazure resolution m : X(t λ ) → X(t λ ). This approach yields, for a minuscule coweight λ of any R, or for an arbitrary coweight λ of GL n , a conceptual albeit less explicit expression for the coefficient Θ - λ (w) of the basis element T w , in terms of the cohomology of a fiber of the Demazure resolution.