Modules for Relative Yangians (Family Algebras) and Kazhdan-Lusztig Polynomials

Let g be a complex simple Lie algebra and let V be a finite dimensional U(g) module. A relative Yangian is defined with respect to this pair. According to recent work of Khoroshkin and Nazarov 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 for some V using the appropriate simple Lie algebra 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. 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.