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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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[1] Beilinson, A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100 (1982) · Zbl 0536.14011
[2] Chriss, N.; Ginzburg, V., Representation Theory and Complex Geometry (1997), Birkhäuser: Birkhäuser Basel · Zbl 0879.22001
[3] Deligne, P., La conjecture de Weil II, Publ. Math. IHÉS,, 52, 313-428 (1980)
[4] Haines, T., The combinatorics of Bernstein functions, Trans. Amer. Math. Soc., 353, 1251-1278 (2001) · Zbl 0962.14018
[5] Haines, T., Test functions for Shimura varieties: The Drinfeld case, Duke Math. J., 106, 19-40 (2001) · Zbl 1014.20002
[6] T. Haines, B.C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math., to appear; T. Haines, B.C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math., to appear
[7] Iwahori, N.; Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math., 25, 5-48 (1965) · Zbl 0228.20015
[8] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184 (1979) · Zbl 0499.20035
[9] Kazhdan, D.; Lusztig, G., Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 87, 153-215 (1987) · Zbl 0613.22004
[10] Kottwitz, R.; Rapoport, M., Minuscule Alcoves for \(Gl_n\) and \(GSp_{2n} \), Manuscripta Math., 102, 403-428 (2000) · Zbl 0981.17003
[11] Lusztig, G., Representations of finite Chevalley groups, (Regional Conf. Ser. in Math., 39 (1978)), (expository lectures from the CBMS Regional Conf. held at Madison, WI, August 8-12, 1977) · Zbl 0372.20033
[12] Lusztig, G., Singularities, character formulas, and a \(q\)-analog of weight multiplicities, Astérisque, 101-102, 208-229 (1983) · Zbl 0561.22013
[13] Lusztig, G., Some examples of square integrable representations of semisimple \(p\)-adic groups, Trans. Amer. Math. Soc., 277, 2, 623-653 (1983) · Zbl 0526.22015
[14] Lusztig, G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc., 2, 3, 599-635 (1989) · Zbl 0715.22020
[15] Lusztig, G., Cells in affine Weyl groups and tensor categories, Adv. Math., 129, 85-98 (1997) · Zbl 0884.20026
[16] Schiffmann, O., On the center of affine Hecke algebras of type A · Zbl 1068.20002
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