Let $G$ be your favorite complex reductive algebraic group, and consider its (algebraic) loop group $G((t))$. A role very similar to the Borel $B\subset G$ is played in the loop group by the Iwahori $$I=\{g\in G[[t]] : g\in B \pmod{t}\}.$$
In particular, $$G((t))=\bigsqcup_{w\in \widehat{W}} IwI$$ for $\widehat W$ the affine Weyl group of $G$. Now, $G((t))$ is an affine ind-scheme, so for any $w\in \widehat W$, it makes sense to talk about the ideal defined by $\overline{IwI}$.
In the comparable case of $\overline{BwB}$, this ideal is relatively easily described: the generalized minors of the fundamental representations which vanish on $\overline{BwB}$ generate the ideal.
Is there any comparable set of representations for $G((t))$?
You would make me a very happy man if you were to say the answer was just the fundamental representations of $G$ with $((t))$ adjoined. I should emphasize that I know this is true as sets; I am asking about ideals, which is notably stronger.
I should note that I would be OK with replacing $I$ with the parahoric $G[[t]]$, which is strictly weaker.
