I still want to understand what Cautis-Kamnitzer-Licata did.

One motivation is the geometric of geometric Satake, which says $Rep(G)^{fd}$ and $Perv_{G^\vee(O)}(Gr_{G^\vee})$ are related, simple representation $V(\lambda)$ goes to IC sheaf $IC_\lambda$. The guess is, if $\lambda$ is miniscule, then $Gr_\lambda$ is smooth projective, and the graded dg category $Coh_{\C^*}(Gr_\lambda)$ can be used to do categorification of $V(\lambda)$. Recall $\C^*$ acts on $G^\vee(K)$ and $G^\vee(O)$ by rotating the domain disk.

There is something called convolution Grassmannian, just sequences of nested lattices. It also have $\C^*$-action, and it is a (Bott-Samuelson) resolution of singular space $Gr_{\sum_i \lambda_i}$. Conjecturally, this convolution Grassmannian corresponds to categorified tensor product.

How to discuss the 'relative position' of two elements in Grassmannian? I mean $G\RM (G/P \times G/P) = P \RM G / P$. If $G = GL(n), P=P_{n_1,n_2}$, then I know $G = \sqcup BwB$, but many $wB$ can be absorbed into $P$, so we have classification by double coset $W_P \RM W / W_P$. This is like 'shuffle'. Now similarly, $W_{aff}$ contains a copy of $W$ and a copy of $\Lambda$, but $W_{G(O)}=W$, so after quotienting, we are left with just $\Lambda/W = \Lambda_+$.

A state is a sequence of miniscule $\lambda_i$, which corresponds to a convolution Grassmannian's Coh category. A braid corresponds to an invertible functor from one to another convolution Grassmannian.

• Is the convolution Grassmannian a moduli stack of a type of objects inside some category? I tried to use Legendrian sheaves to get this, but it is not natural at all. Pause this thought.

Here is a puzzle:

• later they uses Nakajima quiver variety, and took the B-model there, what's the relation?
• Does it matter whether we do compactification or not? Which one is more natural? So far in the paper sl2 and slm, they do the compactified version.

They posted three papers in a row, 0902.179x. In these papers, their main examples is $T^*Gr(k,N)$. This is the Higgs branch of the $[N]-(k)$ quiver. What's the relation to previous work?

The spherical object and the $\P^n$ object. https://arxiv.org/pdf/math/0507040, I have no idea what is the Atiyah-class and the Kodaira-Spencer class. Is there a categorical notion for these classes?

In this paper, https://arxiv.org/pdf/0902.1795, they consider the (derived) equivalence $$ Coh( Gr_{\lambda} \wt \times Gr_{\mu}) \to Coh( Gr_{\mu} \wt \times Gr_{\lambda}). $$

How does this go about? It is not geometrical, doing a fiber product or stuff. One can express it as FM kernel, but it is not useful unless the kernel is geometrical.

Let $G = GL_m$. The affine Grassmannian for $Gr_G$ is $G(K)/G(O)$. Given an element $M$ in $G(O)$, it is an $m \times m$ matrix with entries in $O$, such that its determinant is invertible element in $O$, in particular one can plug in $z=0$ to get $G$. For an element in $G(K)$, if we do determinant, we would get $z^n$, $n \in \Z$. If we do $SL_m$'s affine Grassmannian, we just get the 'boring' piece where $n=0$; if we do $PGL_m$, then we get the quotient up version, $\Z/m\Z$ many components. Hmm, it seems to be related to $\pi_1(G)$.

$$\gdef\ncal{\mathcal{N}}$$ Now, they consider something really weird (Is that already in MVy paper? https://arxiv.org/pdf/math/0206084) Here we have some basic story for Nilpotent orbit and slices for $GL_m$. It is always healthy to learn some basic rep theory. Here we go:

1. Let $D$ be an $N$ dimensional vector space. Let $\ncal$ be the nilpotent cone in $gl(N)$. If $x \in \ncal$, we can ask for its Jordan block type, which is an un-ordered partition of $N$. Denote this partition by $\mu=(\mu_1 \geq \mu_2 \cdots \geq \mu_m > 0)$. Then, we can do the dual partition $\mu^\vee=(m=\mu^\vee_1 \geq \mu^\vee_2 \geq \mu^\vee_n > 0).$, let $\vec a$ be a permutation of $\mu^\vee$. Now, we are ready to consider a particular of n-flag variety. $$ F_{\vec a} = \{0 =F_0 \In F_1 \cdots F_n \mid \dim(F_i/F_{i-1}) = a_i \}, \quad T^*F_{\vec a} = \{(u,F) \mid u (F_i) \In F_{i-1} \}. $$ Well, I don't understand why such an endomorphism $u$ provides the cotangent direction. I can tell this is true for Grassmannian, how about 2-step flag? Well, we can first say that, the space of flags is transitive under the global $GL(D)$ action, so any infinitesimal action is generated by $End(D)$. There are certain parabolic sub-algebra $\frak p$ preserving the partial flag, so the tangent space is $\frak g / \frak p$, and its dual is $\frak p^\perp$, namely those dual $(\frak g)^\vee$ element that vanishes on $\frak p$. How does that translate to $u(F_i) \In F_{i-1}$? The things in $\frak p$ are those $x \in \frak g$, where $x(F_i) \In F_i$. The way dual $\frak g$ is identified with $\frak g$ is via taking trace. So, if we want to have an element $u \in \frak p^\perp$, the necessary-sufficient condition is that $Tr(u x) = 0$ for all $x \in \frak p$. Then, by an explicit calculation of trace with basis, we can see the cotangent fiber is parametrized by this, $\frak p^\perp \cong \frak n_p$.
2. Now I am very confused. Given a Jordan block type, meaning a partition, we can have many ordered partition corresponding to it. What's the meaning of the ordered partition? OK, just like $sl_3$ Weyl group acting on the weight lattice. There are different 'singular block' I would say, where we can have highest weight be $(a,a,b)$ or $(a,b,b)$, for $a>b$. Do they corresponds to different representation of $sl_3$? I think so. (weight lattice of $sl_3$ is the diagonal quotient of that for $gl_3$).