  This paper aims to focus on Richardson varieties on symplectic groups,
especially their combinatorial characterization and defining equations.
Schubert varieties and opposite Schubert varieties have profound significance
in the study of generalized flag varieties which are not only research objects
in algebraic geometry but also ones in representation theory. A more general
research object is Richardson variety, which is obtained by the intersection of
a Schubert variety and an opposite Schubert variety. The structure of
Richardson variety on Grassmannian and its combinatorial characterization are
well known, and there are also similar method on quotients of symplectic
groups. In the first part of this paper, we calculate the orbit of the
symplectic group action, and then rigorously give a method to describe the
corresponding quotient by using the nesting subspace sequence of the linear
space, i.e. flags. At the same time, the flag is used to describe the Schubert
variety and Richardson variety on quotient of symplectic group. The flag
varieties of Sp_{2n}(k)/P_d can be viewed as closed subvarieties of
Grassmannian. Using the standard monomial theory, we obtain the generators of
its ideal, i.e. its defining equations, in homogeneous coordinate ring of
Grassmannian. Furthermore, we prove several properties of the type C standard
monomial on the symplectic group flag variety. Defining equations of Richardson
varieties on Sp_{2n}(k)/P_d are given as well.
