Peng Zhou

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====== 2023-07-22 ======

Reading BFM

===== 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? 


===== ring objects =====
I was diverted to watch Nakajima's talk, and also reading the paper on non-cotangent type Coulomb branch.

It was nice to get some pictures. The general setup should be like, $G$ acts Hamiltonianly on some variety $M$ (affine?), then there should be some mirror dual variety $M^\vee$ with $G^\vee$ action. 

The Whittaker reduction of $M^\vee$ by $G^\vee$ should be the desired Coulomb branch. 

How do we get $M^\vee$? Well, that's not important. The important thing is to construct a ring object. I don't know the general philosophy. 

Anyway, that's not relevant for this paper.