This type of question is considered in the paper Lowest weight vectors and excellent filtrations by Wilberd van der Kallen (Math. Z. 201, 19-31 (1989)). van der Kallen works over an arbitrary algebraically closed field $k$, and states his results for the Borel subgroup $B$ of a connected simply-connected semisimple algebraic group $G$ defined over $k$ (e.g., $G = SL_n$). In this context, he shows that the coordinate algebra $k[B]$ admits a filtration as a $B \times B$-module with sections of the form $P(-\lambda) \otimes Q(\lambda)$ for $\lambda$ an integral weight, where $P(-\lambda)$ is a "dual Joseph module," and $Q(\lambda)$ is "minimal relative Schubert module."

I am not well-versed on all of the terminology and conventions in van der Kallen's paper, so will leave it to the reader to consult van der Kallen's paper for the precise definitions of the relevant modules appearing in the filtration. I'm also not sure how this description fits with Ben's description when $k = \mathbb{C}$.

This type of question is considered in the paper Longest weight vectors and excellent filtrations by Wilberd van der Kallen (Math. Z. 201, 19-31 (1989); MR). van der Kallen works over an arbitrary algebraically closed field $k$, and states his results for the Borel subgroup $B$ of a connected simply-connected semisimple algebraic group $G$ defined over $k$ (e.g., $G = SL_n$). In this context, he shows that the coordinate algebra $k[B]$ admits a filtration as a $B \times B$-module with sections of the form $P(-\lambda) \otimes Q(\lambda)$ for $\lambda$ an integral weight, where $P(-\lambda)$ is a "dual Joseph module," and $Q(\lambda)$ is "minimal relative Schubert module."

I am not well-versed on all of the terminology and conventions in van der Kallen's paper, so will leave it to the reader to consult van der Kallen's paper for the precise definitions of the relevant modules appearing in the filtration. I'm also not sure how this description fits with Ben's description when $k = \mathbb{C}$.