respectively; the local rings are generated by $D_p{\operatorname{diag}}[\tilde{q}, q]D_p$ and $(D_0)_p{\operatorname{diag}}[\tilde{q}, q](D_0)_p$ respectively, for $q\in{\text{{{\fontfamily{pag}\selectfont}X}}}_p$. Let $W_n$ be the Weyl group of transformations generated by $x_i\mapsto x_i^{-1}; x_j\mapsto x_j$ for $j\neq i$, and let $\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}$ denote the ring of Weyl-invariant complex polynomials. The Satake map $\omega_p:\mathcal{R}_{0p}\to\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}$ is defined through the composition of two maps as follows $$\begin{aligned}
    \omega_p:\mathcal{R}_{0p}\xrightarrow{\Phi_p} \mathcal{R}({\text{{{\fontfamily{pag}\selectfont}O}}}_p, GL_n(\mathbb{Q}_p))\xrightarrow{\omega_{0p}}\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}.\end{aligned}$$

#### The map $\Phi_p$.

If $\sigma = {\operatorname{diag}}[\tilde{q}, q]$ with $q\in{\text{{{\fontfamily{pag}\selectfont}X}}}_p$ then by Lemma 2.1 of [@Shimuraint] we have the decomposition $$\begin{aligned}
    (D_{0})_p\sigma (D_{0})_{p} =\bigsqcup_{x\in X}\bigsqcup_{s\in Y_x}\bigsqcup_{d\in R_x}D_p\alpha_{d, s}, \hspace{20pt}\alpha_{d, s} = \begin{pmatrix} \tilde{d} & sd \\ 0 & d\end{pmatrix}, \label{satakecoset}\end{aligned}$$ where $X\subseteq GL_n(\mathbb{Q}_p)$, $R_x\subset x{\text{{{\fontfamily{pag}\selectfont}O}}}_p$ represents ${\text{{{\fontfamily{pag}\selectfont}O}}}_p{\backslash}{\text{{{\fontfamily{pag}\selectfont}O}}}_p x{\text{{{\fontfamily{pag}\selectfont}O}}}_p$, and $Y_x\subseteq S_p$. To define $\Phi_p$ we extend, by $\mathbb{C}$-linearity, the map $$\Phi_p((D_0)_p\sigma(D_0)_p) := \sum_{d, s}J(r_P(\alpha_{d, s}))^{-1}{\text{{{\fontfamily{pag}\selectfont}O}}}_pd,$$ where $J(\alpha) := J^{\frac{1}{2}}(\alpha, i)$ for $\alpha\in {\mathop{\rm pr}\nolimits}^{-1}(D[2, 2]Z_0D[2, 2])$ (see [@Shimurahalf (2.7) and Lemma 2.4] for the precise definition and characterisation of $J(\alpha)$).

#### The map $\omega_{0p}$.

Note that any coset ${\text{{{\fontfamily{pag}\selectfont}O}}}_pd$ with $d\in GL_n(\mathbb{Q}_p)$ contains an upper triangular matrix of the form $$\begin{aligned}
    \begin{pmatrix} 
    p^{a_{d_1}} & \star & \cdots & \star \\ 
    0 & p^{a_{d_2}} & \cdots & \star \\ \vdots & \vdots & \ddots 