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--- blog:2025-01-05 [2025/01/05 19:11] – created pzhou
+++ blog:2025-01-05 [2025/01/06 10:12] (current) – pzhou
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 ====== 2025-01-05 ======
+Today, I did
+  - more study on the slices of affine Grassmannian
+  - Return to CKL
+===== Vasily Krylov's note =====
 I run into a note by Vasily Krylov's [[https://people.math.harvard.edu/~vkrylov/On%20isomorphisms%20between%20quiver%20varieties%20of%20type%20A%20and%20slices%20in%20the%20affine%20Grassmannian.pdf | note on slices]].
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   * one is that $\Gr$ is $Bun_G(\P^1)$ with a trivialization on $\P^1 \RM 0$
   * the other is that, there is a $\C^*_t$-action on $\Gr$, either viewed as rotation $z$, as $G((z) )/G[[z] ]$, or viewed as acting on the domain $\P^1$. The limit that $t \to 0$ is like zooming in at $z=0$, and $t \to \infty$ is zooming in at $z=\infty$. We have $$(\Gr)^{\C^*_t} = \sqcup_{\mu \in \Lambda_+} G z^\mu$$ where $\Lambda_+$ is the dominant cocharacters in $T \In G$.
+Given $\lambda \in \Lambda_+$, we have $$\Gr^\lambda = \{ x \in \Gr \mid \lim_{t \to 0} t \cdot x \in G z^\lambda\}. $$ $$\Gr_\lambda = \{ x \in \Gr \mid \lim_{t \to \infty} t \cdot x \in G z^\lambda\}. $$
+$$  W_\lambda = \{ x \in \Gr \mid \lim_{t \to \infty} t \cdot x = z^\lambda\} $$
+It probably is a good idea to twist $\C^*_t$ with some $\C^*$ subgroup in $T\In G$, compatible with the choice of $B \In G$. (is there some $\C^*$-action on $GL_n$ by conjugation, where the attracting manifold is $B$?) This hopefully will break the symmetry of all the $T$-fixed points on $\Gr$. The goal for the additional twist is to cut down the fixed loci, so that they become points, maybe the intersection of these stable and unstable manifold will be the open loci $W_\mu^\lambda$ on the nose.
+You see, the $G$-action and $\C^*_t$ action on $\Gr$ does not generate $G(O)$-action at all, the $\C^*_t$-action is really powerful. Its effect is hard to see on $Bun_G(\P^1)$, but is visible on the 'trivialization' data.
+==== Example of $T^*\P^1$ ====
+In the example of $T^*\P^1$, let $G=GL(2)$. We have $\mu = (1,1)$ and $\lambda = (2,0)$.
+Claim 1: We have $G \cdot z^{\lambda} \cong \P^1$ (view $z^\lambda$ as $z^\lambda G(O) / G(O)$). So that
+$$ W_\mu^\lambda = (T^*\P^1)_{aff}. $$
+Confusion, what's the dimension of the nilpotent cone for $gl(2)$? Well, it is 2 complex dimension, because $\dim_\C gl(2) - 2 = 2$.
+==== Example of $T^*\P^{n-1}$ ====
+I think this is $Gr(1,n)$, $T^*Gr(1,n)$ is a resolution of some strata of $\mathcal{N}_{gl(n)}$. Which strata?
+First, we need to say what is the transverse slice in the nilcone of $gl(n)$. The top strata is given by partition $\lambda$ of $n$ as $n=2+1+1+\cdots+1$, the bottom strata is given by partition $n=1+\cdots+1$. They only point in the bottom strata is $0$. So, the slice is just the top strata's closure itself.
+Second, I would like to embed this to affine Grassmannian. We turn partition to dominant weights $\lambda = (2,1,\cdots,1,0)$ and $\mu=(1,\cdots,1)$. So $W_\mu \cap \overline{Gr^\lambda}$ (which contains the point $z^\mu$) is isomorphic to $(T^*Gr(1,n) )_{aff}$.
+==== Example of $T^*Fl_n$ ====
+In this case, the comparison to the nilpotent slice is easy, the top strata is given by partition $n=n$, and the bottom is given by  $n=1+\cdots+1$.
+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?