& \vdots \\ 
    0 & 0 & \cdots & p^{a_{d_n}}
    \end{pmatrix},  \label{dform}\end{aligned}$$ with $a_{d_i}\in\mathbb{Z}$, and then define $$\omega_{0p}({\text{{{\fontfamily{pag}\selectfont}O}}}_pd) := \prod_{i=1}^n (p^{-i}x_i)^{a_{d_i}}.$$ Through the decomposition ${\text{{{\fontfamily{pag}\selectfont}O}}}_p x{\text{{{\fontfamily{pag}\selectfont}O}}}_p = \bigsqcup_d {\text{{{\fontfamily{pag}\selectfont}O}}}_pd$ and $\mathbb{C}$-linearity, we extend this to obtain $\omega_{0p}$.

By Lemma 1.1.3 of [@Andrianov] there exists an embedding ${\varepsilon}_p:\mathcal{R}_p\to\mathcal{R}_{0p}$; the restriction of $\omega_p$ to ${\varepsilon}_p(\mathcal{R}_p)$ defines the usual Satake isomorphism – see Section 4 of [@Shimurahalf]. The full map $\omega_p$ is not necessarily injective, however. Assume also that $p\nmid{\mathfrak}{b}^{-1}$. There exists an element $U_p\in\mathcal{R}_{0p}$ – called the *Frobenius* element – defined by $$U_p = \Gamma_0\begin{pmatrix} p^{-1}I_n & 0 \\ 0 & pI_n\end{pmatrix} \Gamma_0 =  \bigsqcup_{u\in S({\mathfrak}{b}^{-1}/p^2{\mathfrak}{b}^{-1})}\Gamma_0\begin{pmatrix} p^{-1}I_n & p^{-1}u \\ 0 & pI_n\end{pmatrix}.$$ If $n=1$, it is well-known that $U_p$ corresponds to the $p$th Hecke operator when $p\mid{\mathfrak}{c}$; for general $n>1$, this is no longer true. Denote the centraliser of $U_p$ in $\mathcal{R}_{0p}$ by $\mathcal{C} := \{ A\in \mathcal{R}_{0p}\mid U_pA = AU_p\}$; the map $\Phi_p$ restricted to $\mathcal{C}$ is injective – use Proposition 2.1.1, Lemma 2.1.1 of [@Andrianov p. 86] along with the argument of Lemma 4.3 of [@Shimurahalf] to see why. Note that $\omega_p(U_p) = p^{\frac{n(n+1)}{2}}x_1\cdots x_n$.

The definition of the $p$-stabilisation is now achieved through factorisations of a certain Hecke 