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--- blog:2024-09-08 [2024/09/09 05:26] – created pzhou
+++ blog:2024-09-08 [2024/09/10 16:38] (current) – pzhou
@@ Line 1: / Line 1: @@
 ====== 2024-09-08 ======
 Let's summarize what can be written.
+  -HMS for K-theoretic Coulomb branches
-  - 'HMS' where both sides are 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.
-===== 'HMS' where both sides are K-theoretic Coulomb branches =====