write $e_{\infty}(x) = e(x_{\infty})$.

Write any $\alpha\in GL_{2n}(\mathbb{Q})$ as $$\alpha = \begin{pmatrix} a_{\alpha} & b_{\alpha} \\ c_{\alpha} & d_{\alpha}\end{pmatrix}$$ where $x_{\alpha}\in M_n(\mathbb{Q})$ for $x\in\{a, b, c, d\}$. Define an algebraic group $G$, subgroup $P\leq G$, and the Siegel upper half-space $\mathbb{H}_n$ by $$\begin{aligned}
    G:&= Sp_n(\mathbb{Q}) = \{\alpha\in GL_{2n}(\mathbb{Q})\mid \alpha^T\iota\alpha = \iota\},\ \hspace{20pt} \iota := \begin{pmatrix} 0 & -I_n \\ I_n & 0\end{pmatrix}, \\
    P:&= \{\alpha\in G\mid c_{\alpha} = 0\}, \\
    \mathbb{H}_n :&= \{z = x+iy\in M_n(\mathbb{C})\mid x, y\in M_n(\mathbb{R}), z^T = z, y>0\},\end{aligned}$$ and let $G_{\mathbb{A}}$ denote the adelisation of $G$.

A half-integral weight is an element $k\in\mathbb{Q}$ such that $k-\frac{1}{2}\in\mathbb{Z}$; an integral weight is an element $\ell\in\mathbb{Z}$. The factor of automorphy of half-integral weight involves taking a square root; to guarantee consistency of the choice of root one uses the double metaplectic cover $Mp_n$ of $Sp_n$. The localizations $M_p := Mp_n(\mathbb{Q}_p)$ and the adelisation $M_{\mathbb{A}}$ of $Mp_n(\mathbb{Q})$ can be described as groups of unitary transformations, respectively, on $L^2(\mathbb{Q}_p^n)$ and $L^2({\operatorname{\mathbb{A}_{\mathbb{Q}}}}^n)$ with the exact sequences $$\begin{aligned}
    1\to\mathbb{T}\to M_x\to G_x\to 1,\end{aligned}$$ where $x\in\{p, \mathbb{A}\}$. There are natural projections ${\mathop{\rm pr}\nolimits}_{\mathbb{A}}:M_{\mathbb{A}}\to G_{\mathbb{A}}$ and ${\mathop{\rm pr}\nolimits}_p:M_p\to G_p$, either of which will usually be denoted ${\mathop{\rm pr}\nolimits}$ as the context is clear. On the 