Quiver varieties and \(t\)-analogs of \(q\)-characters of quantum affine algebras. (English) Zbl 1140.17015

Nakajima, Hiraku

Ann. Math. (2) 160, No. 3, 1057-1097 (2005).

Summary: We consider a specialization of an untwisted quantum affine algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of “computable” polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we “compute” \(q\)-characters for all simple modules. The result is based on ”computations” of Betti numbers of graded/cyclic quiver varieties.

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
33D80	Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics