• Action by correspondence, history of quiver Hecke algebra
• Lie superalgebra

Let $\Gamma$ be the quiver with a single dot. The $A_1$ quiver. Corresponding to the single root of $sl_2$.

We consider the moduli stack of quiver representation, it is classified by the dimension vector, and in this case it is just $d \in \Z_{\geq 0}$. Let $X_d = [pt / GL(d)]$. Map from a space $X$ to this stack (ok, I like it better than BGL(d)) is the same as having a principal $GL(d)$ bundle on $X$, or we can get an associated $\C^d$-bundle, from which we can always recover the $GL(d)$ bundle as the endomorphism bundle.

What is the constructible sheaf on the stack $X_d$? What is the stratification? Is it just local system?

Now, suppose we have $X_{d_1}, X_{d_2}$ and $X_{d}$ where $d=d_1 + d_2$. What's the relation between the three moduli spaces? Well, consider the parabolic subgroup $P \In GL(d)$, of block size $d_1, d_2$. We have Levi subgroup? Let me think: $P$ is defined as the stablizer of some flag, so $P$ acts on the associated graded, then $P$ maps to Levi $P \to L = GL(d_1) \times GL(d_2)$. OK, so we have $$ GL(d_1) \times GL(d_2) \gets P \to GL(d_1+d_2). $$

But, I don't get it, why not directly build some $GL(d_1) \times GL(d_2) \into GL(d_1+d_2)$? hmm, maybe this induction is too wild. So, we first use restriction, to take a representation of $GL(d_1) \times GL(d_2)$ to that of $P$. Then, we take a $P$ representation, and freely extend ('induction') to get a $G$ representation. So, basically, we extends trivally on one half of the generators (those in $P$), and extends freely the other half. But Coh on these stacks are like representations. How about constructible sheaves?

What's the moduli space map? If I have a map of groups $G \to H$, then given a G-bundle $P$, I can consider $P \times_G H$, to make it into an $H$ bundle. This defines a map from $G$ bundle to $H$-bundle. So we have a map of $[pt / G] \to [pt / H] $.

In general, if we have subgroup $G \In H$, then we can take $EG = EH$, and $[EH / G] \to [EH / H]$ the map of orbit is well-defined. But, if we have two arbitrary groups $G \to H$, we can first build $EG$, that is a principal $G$ bundle, then, extend it to an $H$-bundle, $EG \times_G H$, then that is no longer contractible. But it should be able to map to $EH$, $EG \times_G H \to EH$ in an $H$-equivariant way, so we get $[pt/G] \to [pt / H]$. This is not the simple fraction, but rather the universal induction. There is no map in the other direction, no universal restriction.

So, given this, what do we know?