• write up the notes that are useful for myself.
• why quiver gauge theory has anything to do with Kac-Moody algebra?

For $G=GL_n$, you give me an orbit $G(O) t^\lambda G(O)/G(O)$ in the affine Grassmannian $Gr_G = G(K)/G(O)$. Then you want to do some convolution product.

Why not directly work with the $G(O)$ orbit, but rather with its closure, and objects extended from the cell to the closure?

Why you don't do IC homology, but the usual BM homology?

Is (BM) homology well-defined for singular space? i.e. for $xy = z^2$ in $\C^3$.

What do I need smoothness of the space for? Do I need Poincare duality? No, I will just work with the dualizing complex $\omega$ on $Gr_G$.

On one hand, BFM tells me that the pure gauge theory is super-easy.

• First, one unfolds $H^G(Gr_G)$ to $H^T(Gr_G)$ (enlarge the ring)
• Then, we study $H^T(Gr_G)$ using localization.

meaning, we look at fixed points of $Gr_G$ under various element $t \in T$, or $X \in Lie(T)$. We get that $H^T(Gr_G)_X = H^T(Gr_{G'})_X$. Then, we learned that $H^T(Gr_{G'})$ is easy. What does $G'$ look like? $G'$ is a Levi subgroup of $G$, simplest non-trivial block-diagonal matries.

Then, how does equivariant BM homology class restricts to fixed points? We need to do rescaled restriction $\wt {res}$, since that latter is the inverse of the $\iota_*$.