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--- blog:2023-06-11 [2023/06/12 00:34] – [KWWY, just Lie algebra] pzhou
+++ blog:2023-06-11 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 14: / Line 14: @@
   * $R^P(G, N)$ is the convolution space of $N(O)/I_P \times_{N(K)/G(K)} N(O)/I_P$. This lives over the Hecke modification space $pt/I_P \times_{pt/G(K)} pt/I_P$. This is like $I_P \ G(K) / I_P$. Since basically, we have a pair-of-flags.
+Try again. $T_{G,N}$ is the data of a section of $N \times D \to D$, and together with a meromorphic gauge transformation (modulo gauge transformation on the disk $D$).
+$R_{G,N}$ is the submoduli space of $T_{G, N}$, such that section in the trivial $N$-bundle, after doing that meromorphic section, extends from $N \times D^*$ to $N \times D$.
+The parabolic version $T^P_{G,N}$, is still a section of $N$ over $D$, together with a group element $G(K)$, modulo the equivalence relation that we can modify both the group action, and the section, by a gauge transformation. It is just, the gauge transformation group is restricted.
+The parabolic version $R^P_{G,N}$, is alsomostthe same as before, except we impose the regularity condition on the N-section, after the action.
+These are the BFN spaces. Then, we take equivariant $K$ theory or cohomology on the spaces. And then, we define convolutions. So, again, what is the parabolic induction?
+We first have a restriction, and