Heard a talk by Tim Logvinenko, about generalized braid group action. There are many interesting ideas, very concrete diagramatic. Of course, Kapranov can say, it is all in his old work with Schechtmann on perverse schober on $Sym^n \C$, but I still like this work.

Here are the series of papers by Kapranov-Schechtmann,

• Parabolic induction and perverse sheaves on h/W,
• PROBs and perverse sheaves I. Symmetric products
• PROBs and perverse sheaves II. Ran spaces and 0-cycles with coefficients

I doubt a bit, about Kapranov's comment that extending the braid group action to GBr action, is just direct image functor. I need to really learn how this works, for Ed Segal's construction.

Vadim's talk note is useful.