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Let us consider the so-called Ran Grassmannian $Gr_{Ran}$, i.e. the geometric object defined e.g. in [Zhu, An Introduction to the affine Grassmannian and the Geometric Satake equivalence, Definition 3....

Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$. What is $\text{Pic}(\text{Gr}_{T,X^n})$? Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...