Fourier transforms on the basic affine space of a quasi-split group

For a quasi-split group $G$ over a local field $F$, with Borel subgroup $B=TU$ and Weyl group $W$, there is a natural geometric action of $G\times T$ on $L^2(X),$ where $X=G/U$ is the basic affine space of $G$. For split groups, Gelfand and Graev have extended this action to an action of $G\times (T\rtimes W)$ by generalized Fourier transforms $\Phi_w$. We define an analog of these operators for quasi-split groups. We also extend the construction of the Schwartz space $\mathcal S (X)$ by Braverman and Kazhdan to the case of quasi-split groups.