t-analogue of the q-characters of finite dimensional representations of quantum affine algebras

Frenkel-Reshetikhin introduced q-characters of finite dimensional representations of quantum affine algebras [6]. We give a combinatorial algorithm to compute them for all simple modules. Our tool is t-analogue of the q-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition of t-analogues and the proof. But it appear only in the last section. The rest of the paper is devoted to an explanation of the algorithm, which one can read without the knowledge about quiver varieties. A proof is given only in part. A full proof will appear elsewhere. 1. The quantum loop algebra Let g be a simple Lie algebra of type ADE over C, Lg = g⊗C[z, z] be its loop algebra, and Uq(Lg) be its quantum universal enveloping algebra, or the quantum loop algebra for short. It is a subquotient of the quantum affine algebra Uq(ĝ), i.e., without central extension and degree operator. Let I be the set of simple roots, P be the weight lattice, and P ∗ be its dual lattice (all for g). The algebra has the so-called Drinfeld’s new realization: It is a C(q)-algebra with generators q, ek,r, fk,r, hk,n (h ∈ P , k ∈ I, r ∈ Z, n ∈ Z \ {0}) with certain relations (see e.g., [1, 12.2]). The algebra Uq(Lg) is a Hopf algebra, where the coproduct is defined using the Drinfeld-Jimbo realization of Uq(Lg). So a tensor product M ⊗C(q) M ′ of Uq(Lg)-modules M , M ′ has a structure of a Uq(Lg)-module. Let Uε(Lg) be its specialization at q = ε ∈ C . For precise definition of the specialization, we first introduce an integral form UZq (Lg) of Uq(Lg) and set Uε(Lg) = U Z q (Lg)⊗Z[q,q−1] C, where Z[q, q ] → C is given by q 7→ ε. See [3] for detail. But we assume ε is not a root of unity in this paper. So we just replace q by ε in the definition of Uq(Lg). The quantum loop algebra Uq(Lg) contains the quantum enveloping algebra Uq(g) for the finite dimensional Lie algebra g as a subalgebra. The specialization Uε(Lg) contains the specialization Uε(g) of Uq(g). Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan. 1