An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by ℚ. This proves a weak form of a conjecture of Ginzburg proposed in 1987.
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Nanhua Xi, Kazhdan–Lusztig basis and a geometric filtration of an affine Hecke Algebra, II. J. Eur. Math. Soc. 13 (2011), no. 1, pp. 207–217DOI 10.4171/JEMS/249