Macdonald’s polynomials and representations of quantum groups.

*(English)*Zbl 0833.17007From the introduction: In [A new class of symmetric functions, Sémin. Lothar. Comb. 20, B20a, 41 p. (1988; Zbl 0962.05507)] I. G. Macdonald defined a family of systems of orthogonal symmetric polynomials depending on two parameters \(q\), \(k\) which interpolate between Schur’s symmetric functions and certain spherical functions on \(\mathrm{SL} (n)\) over the real and \(p\)-adic fields. These polynomials are labeled by dominant integral weights of \(\mathrm{SL} (n)\), and are uniquely defined by two conditions: 1) they are orthogonal with respect to a certain weight function, and 2) the matrix transforming them to Schur’s symmetric functions is strictly upper triangular with respect to the standard partial ordering on weights (“strictly” means that the diagonal entries of this matrix are equal to 1).

In this paper the authors present a formula for Macdonald’s polynomials which arises from the representation theory of the quantum group \(U_q(\mathfrak{gl}_n)\). This formula expresses Macdonald’s polynomials as vector-valued characters – (weighted) traces of intertwining operators between certain modules over \(U_q(\mathfrak{gl}_n)\). This result was announced in their paper [Funct. Anal. Appl. 28, No. 1, 73–76 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 91–94 (1994; Zbl 0868.33010)]. One should also relate this construction to a recent paper of M. Noumi [Adv. Math 123, No. 1, 16-77 (1996; Zbl 0874.33011)] which gives an interpretation of Macdonald’s polynomials for special values of \(k\) as zonal spherical functions on a homogeneous space for a quantum group.

Finally they construct Macdonald’s operators from the generators of the center of \(U_q(\mathfrak{gl}_n)\), and derive an explicit formula for generic (non-symmetric) eigenfunctions of Macdonald’s operators using this construction.

In this paper the authors present a formula for Macdonald’s polynomials which arises from the representation theory of the quantum group \(U_q(\mathfrak{gl}_n)\). This formula expresses Macdonald’s polynomials as vector-valued characters – (weighted) traces of intertwining operators between certain modules over \(U_q(\mathfrak{gl}_n)\). This result was announced in their paper [Funct. Anal. Appl. 28, No. 1, 73–76 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 91–94 (1994; Zbl 0868.33010)]. One should also relate this construction to a recent paper of M. Noumi [Adv. Math 123, No. 1, 16-77 (1996; Zbl 0874.33011)] which gives an interpretation of Macdonald’s polynomials for special values of \(k\) as zonal spherical functions on a homogeneous space for a quantum group.

Finally they construct Macdonald’s operators from the generators of the center of \(U_q(\mathfrak{gl}_n)\), and derive an explicit formula for generic (non-symmetric) eigenfunctions of Macdonald’s operators using this construction.

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

05E05 | Symmetric functions and generalizations |