Peng Zhou

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====== 2023-07-07 ======

yesterday, I was stuck on abelianization. 

One question is, given $G$ acting on $V$, and $T$ a maximal torus in $G$, what can we say about the Coulomb branch spaces $M(T,V)$ and $M(G,V)$? 

First, consider the case where $V=0$. We know $M(T,0) = T^\vee \times Lie(T)$, or 
$$ A(T, 0) = H_*(T(O) \RM T(K) / T(O) ) =  \C[T^\vee] \times H_T^*(pt). $$
What does a $T(O)-T(O)$ orbit look like in $T(K)$? well, it is indexed by the pole order $T(K) \to \Z$. And for each pole order, the action factors through $T(O) \times T(O) \to T(O)$, so one copy-worth of $T(O)$ acts freely, and there are some non-free action that give rises to $H^*(pt/T)$. 

If you want to add two vectors, for which you only know the length, then the result can be wild, you only get some bound from triangle inequality. 

If you want to multiply two matrices, for which you only know them up to conjugation, i.e., you only know there eigenvalues, but don't know their eigenvectors, then you don't know about the ambiguity of their product. 

Here, for this group $G(K)$, we are not looking at its conjugacy classes. The set of isomorphism class of a $G(K)$ local system on $S^1$ corresponds to the set of conjugacy classes.

What are we looking at, when we say $B \RM G / B$? We are looking at a $G$-local system on an interval with a decoration of flag on each endpoint. This the same as $G \ (G/B \times G/B)$. So, it doesn't have to be an interval, can be any contractible space. 

Can one do something like this: $A_3 = H_*(G \RM (G/B \times G/B \times G/B))$, where $G = GL_n(F( ( t ) ) )$, and $B=  GL_n(F[ [t ] ] )$, and we can multiply $A_3$ with $A_2$, by taking fiber product over $G\RM G/B$. So have, given a (left) action of $G$ on $A,B,C$, we can do
$$ A/G \times_{B/G} C/G = (A \times_B C)/G $$

What we have is that, we have $n_1$ marked points in the first component, $n_2$ marked points in the 2nd component, and we choose a gluing point, we glue and get $n_1+n_2-1$ marked points.

OK, this was some thinking in convolution algebra. This is basically matrix multiplication. For actual matrices, you can do multiplication and additions. But, for homology cycles, how to deal? 

Consider $(G/B)_1 \times (G/B)_2$ and $(G/B)_2 \times (G/B)_3$, on the level of set.