We study the space of vector-valued conjugation-invariant functions on a semisimple group. Let $G$ be a simply-connected semisimple algebraic group over $\mathbb C$, and let $T$ be a maximal torus. Restricting a conjugation-invariant function on $G$ to $T$ gives rise to a Weyl group-invariant function on $T$. It turns out that this induces an isomorphism $\mathbb C[G]^G \cong \mathbb C[T]^W$. In this talk, we instead consider the space of functions $f$ on $G$ with values in a representation $V$ of $G$ such that $f(ghg^{-1}) = g(f(h))$ for $g,h \in G$. The analogous Chevalley restriction map is no longer an isomorphism. In this talk, we study certain invariants related to the failure of the Chevalley restriction map being an isomorphism. This is a joint work with Xinwen Zhu.