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--- blog:2025-01-05 [2025/01/05 20:44] – pzhou
+++ blog:2025-01-05 [2025/01/06 10:12] (current) – pzhou
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 Today, I did
   - more study on the slices of affine Grassmannian
+  - Return to CKL
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 So, in the affine Gr slice presentation, we have $\lambda = (n,0,0,\cdots, 0)$ and $\mu=(1,\cdots,1)$ for $G=GL(n)$.
+==== any partial flag ====
+For any partial flag, we get an ordered partition $\vec a$ of $n$, forget the ordering get the partition $\lambda^\vee$ of $n$, $T^*Fl_{\vec a}$, passing to affinization, and get closure of nilorbit $N_{\lambda^\vee}$. This will be $\overline{W_0^\lambda}$, an open cell in $\overline{\Gr^\lambda}$.
+==== other Higgs branch of type $A$ ====
+How about other Nakajima quiver variety? Like $[1] - (1) - (1) - [1]$, whose $M_H$ is like $\C^2 / \mu_3$.
+Here is a guess, get the dual quiver, which is $[3] - (1)$, then we get the top dominant weight $\lambda = 3 \omega_1$(partition is $3=3$), and the bottom dominant weight $\mu = \omega_1 + \omega_2$ (partition is $3 = 2+1$). From the pair of partitions, we get nilpotent slices; from the pair of dominant weights, we get affine Gr slices.
+But, how do we get the dual quiver in general? Apart from the Hanany-Witten move, TBD.
+===== CKL's work =====
+There are two stories, which I will tell separatedly, then relate in the end hopefully.
+==== Knot Homology for $sl_2$ ====
+The first story is just Cautis-Kamnitzer. They want to categorify tangle-invariant of $sl_2$ (or $sl_n$), they want to turn linear spaces to categories, and linear maps to functors. This is about categorifying morphisms between representations of $sl_2$ to functors between categories.
+How did they do it, even in $sl_2$?  They constructed the cup, cap, and braiding kernel.
+  * The space $Y_n$, lives in a (truncated version of) affine Gr for $GL(2)$, or rather the convolution space of it, a resolution of $\Gr^{(n,0)}$. One can either view it as living in an ambient big space with a given Nilpotent endomorphism (Jordan block size $(N,N)$) and with some condition of the flags. Or, take certain open subset of $Y_n$, call it $U_n$, then we can view it as a full-flag in $\C^n$, but with a varying nilpotent endormorphism, as resolution of a slice.
+  * The cup/cap correspondence between $Y_{n-2}$ and $Y_n$. We can view it as a sub-lattice of $Y_n$, setting $L_i = z^{-1} L_{i+2}$. This guy then forget to $Y_{n-2}$ after deleting $L_{i+1}$.
+  * The braiding correspondence is something natural as well.
+  * 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?