By Soufi A. E., Sandier E.

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The Tannakian formalism discussed above then implies that there is a “universal” element of g whose action on V is given by this formula. 5 Regular vs. irregular singularities In the same way as in the case of GLn , we obtain that a holomorphic principal G-bundle on a complex variety X with a holomorphic flat connection gives rise to an equivalence class of homomorphisms from the fundamental group of X to G. But does this set up a bijection of the corresponding equivalence classes? If X is compact, this is indeed the case, but, if X is not compact, then there are more flat bundles than there are representations of π1 (X).

5) n+m=N ;n<0 n+m=N ;n≥0 The first of them belongs to Uκ (sl2 ), but the second one does not: the order of the two factors is wrong! This means that this element does not give rise to a well-defined operator on a module from the category sl2,κ -mod. Indeed, we can write em fn = fn em + [em , fn ] = fn em + hm+n . Thus, the price to pay for switching the order is the commutator between the two factors, which is non-zero. Therefore, while the sum fn em n+m=N ;n<0 belongs to Uκ (sl2 ) and its action is well-defined on any module from sl2,κ -mod, the sum em fn n+m=N ;n<0 that we are given differs from it by hm+n added up infinitely many times, which is meaningless.

Both categories are equipped with natural actions of the group GC . Let us pause for a moment and spell out what exactly we mean when we say that the group GC acts on the category C0 . † This means the following: each element g ∈ G gives rise to a functor Fg on C0 such that F1 is the identity functor, and the functor Fg−1 is quasi-inverse to Fg . Moreover, for any pair g, h ∈ G we have a fixed isomorphism of functors ig,h : Fgh → Fg ◦ Fh so that for any triple g, h, k ∈ G we have the equality ih,k ig,hk = ig,h igh,k of isomorphisms Fghk → Fg ◦ Fh ◦ Fk .