I still want to understand what Cautis-Kamnitzer-Licata did.

One motivation is the geometric of geometric Satake, which says $Rep(G)^{fd}$ and $Perv_{G^\vee(O)}(Gr_{G^\vee})$ are related, simple representation $V(\lambda)$ goes to IC sheaf $IC_\lambda$. The guess is, if $\lambda$ is miniscule, then $Gr_\lambda$ is smooth projective, and the graded dg category $Coh_{\C^*}(Gr_\lambda)$ can be used to do categorification of $V(\lambda)$. Recall $\C^*$ acts on $G^\vee(K)$ and $G^\vee(O)$ by rotating the domain disk.

There is something called convolution Grassmannian, just sequences of nested lattices. It also have $\C^*$-action, and it is a (Bott-Samuelson) resolution of singular space $Gr_{\sum_i \lambda_i}$. Conjecturally, this convolution Grassmannian corresponds to categorified tensor product.

How to discuss the 'relative position' of two elements in Grassmannian? I mean $G\RM (G/P \times G/P) = P \RM G / P$. If $G = GL(n), P=P_{n_1,n_2}$, then I know $G = \sqcup BwB$, but many $wB$ can be absorbed into $P$, so we have classification by double coset $W_P \RM W / W_P$. This is like 'shuffle'. Now similarly, $W_{aff}$ contains a copy of $W$ and a copy of $\Lambda$, but $W_{G(O)}=W$, so after quotienting, we are left with just $\Lambda/W = \Lambda_+$.

A state is a sequence of miniscule $\lambda_i$, which for them, corresponds to a convolution grassmannian's Coh category. A braid corresponds to an invertible functor from one to another convolution Grassmannian. We also have merge and split these webs.

Is the convolution Grassmannian a moduli stack of a type of objects inside some category? If so, which category?