Siegel slash operator (awaiting review)

For a positive integer $g$, let $\GSp(2g,\Q)^+=\{\gamma\in\GL(2g,\Q)\, \mid \, \exists\, r(g)\in\Q_{>0}: \gamma^t J\gamma=r(g)J\}$ and $f \colon \mathcal{H}_g \to V$, the slash operator (of weight $\rho$) is defined by \[ (f\vert_{\rho}\alpha)(\tau)=r(\alpha)^{\lambda_1+\ldots+\lambda_g-\frac{g(g+1)}{2}}\rho(c\tau+d)^{-1}f((a\tau+b)(c\tau+d)^{-1}) \] for $\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\GSp(2g,\Q)^+$ and $\rho\colon \GL(g,\C) \to \GL(V)$ an irreducible finite-dimensional representations of $\GL(g,\C)$ of highest weight $\lambda_1\geqslant \lambda_2\geqslant\ldots\geqslant \lambda_g$.