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Đề tài " Quiver varieties and tanalogs of q-characters of quantum affine algebras "

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. (The reason why we use “ ” will be explained at the end. | Annals of Mathematics Quiver varieties and t-analogs of q-characters of quantum affine algebras By Hiraku Nakajima Annals of Mathematics 160 2004 1057 1097 Quiver varieties and t-analogs of ợ-characters of quantum affine algebras By Hiraku Nakajima Abstract 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 -characters for all simple modules. The result is based on computations of Betti numbers of graded cyclic quiver varieties. The reason why we use will be explained at the end of the introduction. Contents Introduction 1. Quantum loop algebras 2. A modified multiplication on Yt 3. A t-analog of the -character Axioms 4. Graded and cyclic quiver varieties 5. Proof of Axiom 2 Analog of the Weyl group invariance 6. Proof of Axiom 3 Multiplicative property 7. Proof of Axiom 4 Roots of unity 8. Perverse sheaves on graded cyclic quiver varieties 9. Specialization at e 1 10. Conjecture References Introduction Let g be a simple Lie algebra of type ADE over C Lg g G C z z-1 be its loop algebra and Ug Lg be its quantum universal enveloping algebra or the quantum loop algebra for short. It is a subquotient of the quantum Supported by the Grant-in-aid for Scientific Research No.11740011 the Ministry of Education Japan. 1058 HIRAKU NAKAJIMA affine algebra Ug fl i.e. without central extension and degree operator. Let Ue Lg be its specialization at q E a nonzero complex number. See 1 for definition. It is known that Ue Lg is a Hopf algebra. Therefore the category RepUe Lfl of finite dimensional representations of Ue Lg is a monoidal or tensor abelian category. Let Rep Ue Lfl be its Grothendieck ring. It is known .