Differences

This shows you the differences between two versions of the page.

--- blog:2024-06-10 [2024/06/11 06:23] – pzhou
+++ blog:2024-06-10 [2024/06/11 07:18] (current) – [Ben Webster's story (following BLPW, KWWY)] pzhou
@@ Line 36: / Line 36: @@
 BGG category O is a special case of Cat O of quantized additive (resolved? deformed?) Coulomb branch.
+(really? Cat O as reg. hol. D-mod on flag variety G/B, so it is the quantization of $T^*(G/B)$. I GUESS, all $T^*(G/B)$ are Coulomb branches. )
+Are all Coulomb branches have mirror Higgs branches? In other words, is 3d mirror symmetry 'always holds', or like 2d mirror symmetry, sometimes holds? What is the statement?
+Webster's story is that, quantized Coulomb branch's module, pick a full subcategory of things supported somewhere, then you get DQ module on Higgs branch, supported elsewhere. So, you get two algebra (from quantized Coulomb branch and Higgs branches) are Koszul dual.
+Webster's proof is like this: he cook up some hom space, that is simultaneously interpreted as endo of simple on Higgs side; or endo of projectives on the quantized Coulomb side. The endo of simple is roughly easy to see, we have a pushforward for $V'/G'$ to $V/G$ (not sure), is $G'$ Levi or parabolic, and what is $V'$?
+Quantized Coulomb branch.