Today, I did

1. more study on the slices of affine Grassmannian
2. Return to CKL

I run into a note by Vasily Krylov's note on slices.

$\gdef\Gr{\mathcal{Gr}}$ There are two key new ideas,

• 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.

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 $\dim_\C gl(2) - 2 = 2$.

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 $(T^*Gr(1,n) )_{aff}$.

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 $G=GL(n)$.

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 $\overline{\Gr^\lambda}$.

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.

There are two stories, which I will tell separatedly, then relate in the end hopefully.

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 $sl_2$? Remember, this is before KLR, but after Khovanov and Stroppel. 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.
• There is a correspondence between $Y_{n-2}$ and $Y_n$

As a summary, here is what I understood

• Geometric Satake relates tensor product of representations $V_\lambda$ with convolution product. For example, $G=SL(2)$, consider the representation $\C^2$, it goes to IC of $Gr^{(1,0)}_{PGL_2}$.
• At this moment, there is no slices. They want to do $G(O)$ orbit closure and do coherent sheaves category.