Let $G=GL_n$ over $\mathbb{Q}_p$ and $H$ be the Hecke algebra, ie. bi-$K$-invariant functions on $G$, where $K$ is $GL_n(\mathfrak{o})$. It seems that the Stake isomorphism makes it isomorphic to a polynomial algebra in indeterminate $X_1, X_1^{-1}, \ldots, X_n, X_n^{-1}$ that are invariant by permutations.
Where can I find a statement of the Stake isomorphism in this formalism?