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Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert variety $X_w$ are all linear. In this case, this is proved in Ramanathan, "Equations defining Schubert varieties and Frobenius splitting of diagonals", Pub. IHES 65 (1987) 61-90.

Does anyone have a reference for this fact in the Kac-Moody setting? I only need this for affine Grassmannians, but certainly a more general reference would be preferred.

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