Quantum affine algebras and affine Hecke algebras

Abstract

One of the most beautiful results from the classical period of the representation theory of Lie groups is the correspondence, due to Frobenius and Schur, between the representations of symmetric groups and those of general or special linear groups. If V0 is the natural irreducible (n+ 1)–dimensional representation of SLn+1(CI), the symmetric group Sl acts on V ⊗l 0 by permuting the factors. This action obviously commutes with the action of SLn+1(CI). It follows that one may associate to any right Sl–module M a representation of SLn+1(CI), namely FS(M) = M⊗SlV ⊗l 0 , the action of SLn+1(CI) on FS(M) being induced by its natural action on V ⊗l 0 . The main result of the Frobenius–Schur theory is that, if l ≤ n, the assignment M → FS(M) defines an equivalence from the category of finite–dimensional representations of Sl to the category of finite–dimensional representations of SLn+1(CI), all of whose irreducible components occur in V ⊗l 0 . Around 1985, Drinfeld and Jimbo independently introduced a family of Hopf algebras Uq(g), depending on a parameter q ∈ CI , associated to any symmetrizable Kac–Moody algebra g. Assuming that q is not a root of unity, Jimbo [7] proved an analogue of the Frobenius–Schur correspondence in which SLn+1(CI) is replaced by Uq(sln+1), V0 by the natural (n+1)–dimensional irreducible representation V of Uq(sln+1), and Sl by its Hecke algebra Hl(q 2). In [5], Drinfeld announced an analogue of the Frobenius–Schur theory for the Yangian Y (sln+1), which is a “deformation” of the universal enveloping algebra of the Lie algebra of polynomial maps CI → sln+1. The role of Sl in this theory is played by the degenerate affine Hecke algebra Λl, an algebra whose defining relations are obtained from those of the affine Hecke algebra Ĥl(q 2) by letting q → 1 in a certain non–trivial fashion. 1 Both authors were partially supported by the NSF, DMS-9207701.