Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}_F)$-biinvariant $\mathbb{Z}$-valued functions on $G(F)$. Let $R(G^L)$ be the Grothendieck ring of finite dimensional representation of $G^L$. It is stated in Gross's survey about Satake isomorphism that $H \otimes \mathbb{Z}[q^{\pm1/2}]$ is isomorphic to $R(G^L)\otimes \mathbb{Z}[q^{\pm1/2}]$, where $q$ is the cardinality of residue field of $F$.
My question is whether $H$ is isomorphic to $R(G^L)$? I suspect that they are non isomorphic when $G$ is some exceptional group.
