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.

中文翻译：

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有限域上辛形对称空间的特征图

对称群的特征映射是将对称群的表示理论与对称函数联系起来的同构。为有限域上辛形式的对称空间构造了一个类似的同构，球函数被发送到参数为 $(q,q^2)$ 的 Macdonald 多项式。抛物线归纳的类似物被解释为对称函数的某种乘法。给出了参数为 $(q,q^2)$ 的斜麦克唐纳多项式的 Schur 正性以及球函数值的组合公式的应用。