  The characteristic map for the symmetric group is an isomorphism relating the
representation theory of the symmetric group to symmetric functions. An
analogous isomorphism is constructed for the symmetric space of symplectic
forms over a finite field, with the spherical functions being sent to Macdonald
polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is
interpreted as a certain multiplication of symmetric functions. Applications
are given to Schur-positivity of skew Macdonald polynomials with parameters
$(q,q^2)$ as well as combinatorial formulas for spherical function values.
