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--- blog:2025-01-05 [2025/01/06 06:30] – [Knot Homology for $sl_2$] pzhou
+++ blog:2025-01-05 [2025/01/06 10:12] (current) – pzhou
@@ Line 65: / Line 65: @@
   * What's subtle and mysterious is the twisting line-bundle in addition to the structure sheaf. really mysterious, and one really needs coherent sheaves here.
   * The 2nd grading comes from $\C^*$-action inherited from affine Gr, which comes from the domain curve $\P^1$, or punctured disk.
+==== geometric / categorical action by $sl_2$ ====
+Oh well, Ben Webster came along. Or rather Bezrukavnikov came along in 2008, saying, one should work in DQ-module on the Higgs branch. Then Ben's paper https://arxiv.org/pdf/1208.5957 explained the sl2-action that Cautis-Kamnitzer build, $Coh(T^*X)$ is actually the classical limit for the $DQ(T^*X)=D-mod(X)$. Cautis-Dodd-Kamnitzer wrote a careful paper explaining the process of taking associated graded.
+The stacky Higgs branch (2-stop version) is more fundamental, than the stable open loci version (one stop).
+The $sl_2$-action is given by Hecke correspondence, or parabolic induction in the quiver representation. We pullback from the sub and quotient to the correspondence, then push-forward to the bigger guy.
+So, it is not quite right to say CK is not related to us, or KLR. Somehow, the FM kernel does not care?