Drinfeld Functor and Finite-dimensional Representations of Yangian

We extend the results of Drinfeld on the Drinfeld functor to the case l ≥ n. We present the character of finite-dimensional representations of the Yangian Y (sln) in terms of the Kazhdan-Lusztig polynomials as a consequence. Introduction In this article we study the representations of the Yangian Y (sln). The Yangian is a quantum group introduced by V. G. Drinfeld ([D1]). The parameterization of the simple finite-dimensional representations of Y (sln) was obtained in [D3] by the sequences of monic polynomials Q(u) = (Q1(u), . . . , Qn−1(u)) called the Drinfeld polynomials. Furthermore, he has constructed in [D2] a functor from the category CHl of finite-dimensional representations of the degenerate affine Hecke algebra Hl to the category CY (sln) of finite-dimensional representations of Y (sln). This functor is called the Drinfeld functor. It was stated in [D2] that as well as the classical Frobenius-Schur duality, the Drinfeld functor gives the categorical equivalence between CHl and the certain subcategory of CY (sln) in the case l < n. Chari-Pressley generalized this duality to the case between the affine Hecke algebra and the quantum affine algebra. They proved that the categorical equivalence holds in this case as well provided that l < n ([CP2]). However, due to the restriction l < n, the above categorical equivalence does not describe all the finite-dimensional representations of the Yangian Y (sln). In particular, even the characters of finite-dimensional representations of Y (sln) have not been known, except for the case n = 2 ([CP3]) and the special class of the representations called tame ([NT1]). The main purpose of this article is to extend the Drinfeld’s results [D2] to the case l ≥ n. To be more precise, we first show the followings without restriction l < n: 1. The Drinfeld functor sends the standard modules of Hl to zero or the highest modules of Y (sln) (Theorem 8). 2. The Drinfeld functor sends the simple modules of Hl to zero or the simple modules of Y (sln) (Theorem 10). Here the standard modules are certain induced Hl-modules which have unique simple quotients (see subsection 1.4). We also determine the explicit images of the standard modules. It turns out that the highest weight modules obtained as the images of the standard modules are those tensor product modules of the evaluation representations studied in [AK]. We note that any simple Y (sln)-module is isomorphic to the image of a simple Hl-module for some l. 1