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The Eichler-Shimura isomorphism describes a certain cohomology group withcoefficients in a space of polynomials by using holomorphic modular/cusp forms.It determines a canonical decomposition of the corresponding de Rham cohomologygroup associated to a specific oper on a Riemann surface. One purpose of thepresent paper is to establish its analogue for opers in positivecharacteristic. We first discuss some basic properties on the (parabolic) deRham cohomology groups and deformation spaces of $G$-opers (where $G$ is asemisimple algebraic group of adjoint type) in a general formulation. Inparticular, it is shown that the deformation space of a $G$-oper induced froman $\mathrm{SL}_2$-oper decomposes into a direct sum of the (parabolic) de Rhamcohomology groups of its symmetric products. As a consequence, we obtain anEichler-Shimura-type decomposition for dormant opers on general pointed stablecurves by considering a transversal intersection of related spaces in the deRham moduli space.

Infinitesimal deformations of opers in positive characteristic and the de Rham cohomology of symmetric products