I'm not sure I have a lot more to say than the title. Let $G$ be your favorite simple algebraic group over $\mathbb{C}$, and let $$\overline {\mathrm{Gr}}_\lambda= \overline{G(\mathbb{C}[[t]])\cdot t^\lambda \cdot G(\mathbb{C}[[t]])}/ G(\mathbb{C}[[t]]).$$ It's a commonly cited theorem that $\overline {\mathrm{Gr}}_\lambda$ is a projective variety for every $\lambda$, but the usual tricks for finding the Picard group of a Schubert variety in the finite dimensional case don't work (the group $G(\mathbb{C}[[t]])$ is perfect if $G$ is semisimple). Is this Picard group computed anywhere in the literature?
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[From the comments] The proof of Proposition 13.2.19 in Kumar's "KacMoody groups, their flag varieties and representation theory" appears to provide the requested information. 

