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On the equivariant \(K\)-theory of the nilpotent cone. (English) Zbl 0986.20045

Let \(G\) be a simple simply connected algebraic group over the complex numbers. The author constructs a “Kazhdan-Lusztig” type canonical basis for the equivariant \(K\)-theory of the nilpotent cone on \(G\). Then the author conjectures the following regarding this basis: (1) this basis coincides with that of G. Lusztig [J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 2, 297-328 (1989; Zbl 0688.20020)], (2) certain elements related with irreducible local systems on nilpotent orbits belong to this basis, and (3) this basis is a generalization of that obtained by J. Humphreys [in AMS/IP Stud. Adv. Math. 4, 69-80 (1997; Zbl 0919.17013)]. A number of examples that support the conjectures are provided such as the groups of rank two.

MSC:

20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
19A49 \(K_0\) of other rings
20G10 Cohomology theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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