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--- blog:2024-12-25 [2024/12/26 06:49] – created pzhou
+++ blog:2024-12-25 [2024/12/26 07:05] (current) – pzhou
@@ Line 8: / Line 8: @@
 It is about going from $BGL(k_1)$ and $BGL(k_2)$ to $BGL(k)$. I just know $T^{k_1}$ and $T^{k_2}$ merges to $T^{k_1+k_2}$. Let's try parabolic induction: we have
-$$ GL(k_1) \times GL(k_2) \gets P_{k_1, k_2} \to GL(k_1+k_2) $$
+$$ G_1\times G_2 = GL(k_1) \times GL(k_2) \gets P=P_{k_1, k_2} \to GL(k_1+k_2)=G $$
+Assume everything is fine, then we have
+$$ BP \to B(G_1 \times G_2), \quad BP \to BG $$
+The first one is saying, if you have a principle $P$ bundle $E$ over something $M$, then you can build the associated bundle of $E \otimes_P L$.
+I guess the corresponding construction on sheaf is just doing pull-push along the parabolic induction.
+-----
+Is there classifying space for two step flags? well, one needs to say the ranks, so it would be the $BP_{k_1, k_2}$.