MODULES FOR RELATIVE YANGIANS (FAMILY ALGEBRAS) AND KAZHDAN-LUSZTIG POLYNOMIALS

Abstract

Let g be a complex simple Lie algebra and let mod U (g) be the category of finite dimensional U (g)-modules. The relative Yangian Y V (g) with respect to the pair g, V: V ∈ mod U (g) is defined to be the g invariant subalgebra of End V ⊗ U (g) with respect to the natural “diagonal” action. According to recent work (see [12, Sect. 5] and references therein), the finite dimensional simple modules of the Yangians for g = sl(n) or the twisted Yangians for g = sp(2n), so(n) are described by the simple modules of relative Yangians Y V (g) : V ∈ mod U (g). Here a classification of the simple modules of a relative Yangian is obtained simply and briefly as an advanced exercise in Frobenius reciprocity inspired by a Bernstein–Gelfand equivalence of categories [2]. An unexpected fact is that the dimension of these modules are determined by the Kazhdan-Lusztig polynomials, and conversely the latter are described in terms of dimensions of certain extension groups associated to finite dimensional modules of relative Yangians.