Reading BFM

They want to understand what is the coherent Satake category, and the first step is to understand the K-theory, and the graded version.

Do you have any intuitive reason that, horizontal Hilbert scheme, or universal centralizer, or open Toda lattice would show up?

Notation convention, the centralizer, subscript one is regular one, where by regular we mean the one with the minimal centralizer.

Ginzburg computed cohomology. Kostant-Kumar, computed the general Kac-Moody group's B\G/B's convolution (K)-homology. But the answer is not presented using the dual group. (well, who needs dual group, I don't)

Hamiltonian reduction of $T^* G^\vee$? and internal fusion double of $D(G^\vee)$?

Why this easy to state, well-defined computation has these meanings?