On the equivariant K-theory of the nilpotent cone

Details On the equivariant K-theory of the nilpotent cone On the equivariant K-theory of the nilpotent cone

1999-11-10

arxiv.org/abs/math/9911068v1

Mathematics

Algebraic Geometry

9 pages

Scientific paper

Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In this note we construct a Kazhdan-Lusztig type canonical basis of K_{G x C^*}(N) over representation ring of C^*. This basis is parametrized by the set of dominant weights for G. On the other hand we conjecture that this basis is close to the basis consisting of irreducible G-equivariant bundles on nilpotent orbits. This would give us a natural construction of Lusztig's bijection between two sets: \{dominant weights for G\} and \{pairs consisting of a nilpotent orbit O and irreducible G-equivariant bundle on O.

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