Polarization algebras and the geometry of commuting varieties

We prove a reduced version of the Chevalley restriction conjecture on the commuting scheme posed by T.H. Chen and B.C. Ng\^o, extending the results of Hunziker for classical groups. In particular, we prove that for any connected reductive group, the ring of $G$-invariant functions on the commuting variety restricts to an isomorphism with the invariants of the d-fold product of a Cartan subalgebra under the Weyl group $k[\mathfrak{t}^d]^W$. The full conjecture implies that this isomorphism extends to the ring of $G$-invariants on the non-reduced commuting scheme, $k[\mathfrak{C}_{\mathfrak{g}}^d]^G$ (hence the invariant ring is nilpotent free). We then prove an analogous restriction theorem for general polar representations which we apply to resolve an important case of a conjecture posed by Bulois, C Lehn, M Lehn and Terpereau about symplectic reductions of $\theta$-representations. Throughout this work, we focus on the connection between the invariant subring generated by polarizations and commutativity.