Cohomology of special unitary groups and congruence subgroups

We prove a homotopy invariance result for the first cohomology group of the special unitary group ${\mathrm{S}\mathrm{U}}_3(F[t])$ with coefficients in irreducible representations of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(F)$. The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(F)$.