2 1 A pr 1 99 8 LIE ALGEBRAS AND DEGENERATE AFFINE HECKE ALGEBRAS OF TYPE

We construct a family of exact functors from the BernsteinGelfand-Gelfand category O of sln-modules to the category of finite-dimensional representations of the degenerate affine Hecke algebra Hl of GLl. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple Hl-module can be thus obtained. Introduction The classical Frobenius-Schur-Weil duality gives a remarkable correspondence between the category of finite-dimensional representations of the symmetric group Sl and the category of finite-dimensional representations of the special (or general) linear group SLn. Its generalizations have been studied in e.g. [5, 6, 12, 14, 20] where Sl is replaced by other algebras, e.g. the Hecke algebras, the (degenerate) affine Hecke algebras or the double affine Hecke algebras, and SLn is replaced by the corresponding quantum groups. In this paper, we present a new direction in generalizing the classical duality. Let O(sln) denote the BGG category of representations of the complex Lie algebra sln, and let R(Hl) denote the category of finitedimensional representations of the degenerate (or graded) affine Hecke algebra Hl of GLl. To each weight λ of sln such that λ+ρ is dominant integral (where ρ is the half sum of the positive roots), we associate a functor Fλ from O(sln) to R(Hl). When we take λ = 0 and restrict the functor F0 to the category of finite-dimensional representations of sln, we obtain the classical duality. To be more precise, let Vn = C n be the vector representation of sln and M(λ) the highest weight Verma module with highest weight λ. † Supported by JSPS the Research Fellowships for Young Scientists.