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--- blog:2023-06-06 [2023/06/07 06:42] – created pzhou
+++ blog:2023-06-06 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 10: / Line 10: @@
 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 $d_1, d_2$
+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?