Peng Zhou

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====== 2024-09-08 ======
Let's summarize what can be written. 
  -HMS for K-theoretic Coulomb branches

===== HMS for K-theoretic Coulomb branches =====
basically we want to prove half, or even less than half (no generation result). 
so, this can only be called as, symplectic realization of multiplicative KLRW algebra. 

So, what is cKLRW and why is it related to (additive resolved) Coulomb branch? We need to say a few sentences:
  * additive resolved Coulomb branch is a symplectic resolution of conic symplectic singularity, so one can use Bezrukavnikov-Kaledin to construct tilting bundles
  * When Webster did that procedure (char p quantization) for Coulomb branch. If we do it honestly, then we need to consider the convolution algebra of $BFN=map(bubble, N/G)$, but with $F_p$ coefficient. This way, the resulting algebra is over $F_p$. Then, we turn on deformation quantization for the Coulomb branch algebra, by that we mean we choose a $\C^*$-action on that BFN space (where $\C^*$ acts on the domain, but could also acts on the target $N$). So the equivariant BM homology (or K-theory) picked up another equivariance over $\C^*$, and the convolution algebra has one more parameter. //Then, somehow, it become $\mu_p$ fixed point of the BFN space.// 

ok, apparently I still don't know the Webster's story. 

But that's not my side. I think I will follow Ginzburg's note on Hecke, affine Hecke, degenerate affine Hecke, Nil affine Hecke algebra to introduce these different versions. 

It would be nice to just do a summary for the symplectic realization
  - finite Hecke algebra, Tian-Yuan-Honda
  - affine Hecke algebra (for $GL_n$?) the one with the $S^1 \times \R$,
  - double affine Hecke should be on $S^1 \times S^1$.
  - this paper: degenerate affine Hecke is Fukaya category of Horizontal Hilb for $\C^* \times \C \to \C$, no superpotential. 
  - Nil affine Hecke, our last paper, which has potential.