v 1 2 8 M ay 1 99 6 SHIFTED SCHUR FUNCTIONS

The classical algebra Λ of symmetric functions has a remarkable deformation Λ, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions sμ, where μ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in Z(gl(n)), the center of the universal enveloping algebra U(gl(n)), n = 1, 2, . . . . The functions sμ are closely related to the factorial Schur functions introduced by Biedenharn and Louck and further studied by Macdonald and other authors. A part of our results about the functions sμ has natural classical analogues (combinatorial presentation, generating series, Jacobi–Trudi identity, Pieri formula). Other results are of different nature (connection with the binomial formula for characters of GL(n), an explicit expression for the dimension of skew shapes λ/μ, Capelli–type identities, a characterization of the functions sμ by their vanishing properties, ‘coherence property’, special symmetrization map S(gl(n)) → U(gl(n)). The main application that we have in mind is the asymptotic character theory for the unitary groups U(n) and symmetric groups S(n) as n → ∞. The authors were supported by the International Science Foundation (under grants MBI300 and MQV000, respectively) and by Russian Foundation for Basic Research (grant 95-01-00814) current address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, e-mail: [email protected]