Shifted Schur Functions Ii. Binomial Formula for Characters of Classical Groups

Let G be any of the complex classical groups GL(n) , SO(2n + 1) , Sp(2n) , O(2n) , let g denote the Lie algebra of G , and let Z(g) denote the subalgebra of G -invariants in the universal enveloping algebra U(g) . We derive a Taylortype expansion for finite-dimensional characters of G (the binomial formula) and use it to specify a distinguished linear basis in Z(g) . The eigenvalues of the basis elements in highest weight g-modules are certain shifted (or factorial) analogs of Schur functions. We also study an associated homogeneous basis in I(g) , the subalgebra of G -invariants in the symmetric algebra S(g) . Finally, we show that the both bases are related by a G -equivariant linear isomorphism σ : I(g) → Z(g) , called the special symmetrization.