Peng Zhou

stream of notes

User Tools

Site Tools

You are here: Hi, welcome! » blog » 2024-09-08
blog:2024-09-08

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
blog:2024-09-08 [2024/09/09 06:22] – pzhoublog:2024-09-08 [2024/09/10 16:38] (current) – pzhou
Line 10: Line 10:
   * additive resolved Coulomb branch is a symplectic resolution of conic symplectic singularity, so one can use Bezrukavnikov-Kaledin to construct tilting bundles   * 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.//    * 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. 
 +
 +
  
  
blog/2024-09-08.1725862933.txt.gz · Last modified: 2024/09/09 06:22 by pzhou