# A Crash Course on Kleinian Groups by American Mathematical Society By American Mathematical Society

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4) with g ∈ L G((t)). 6). We denote this set by LocL G (D× ). Thus, we have LocL G (D× ) = {∂t + A(t), A(t) ∈ L g((t))}/L G((t)). 6) has regular singularity if and only if A(t) has a pole of order at most one at t = 0. If A−1 is the residue of A(t) at t = 0, then the corresponding monodromy of the connection around the origin is an element of L G equal to exp(2πiA−1 ). Two connections with regular singularity are (algebraically) gauge equivalent to each other if and only if their monodromies are conjugate to each other.

26 Local Langlands correspondence replaced this representation by something that makes sense from the point of view of the representation theory of the complex algebraic group G (rather than the corresponding discrete group); namely, the category C0 . 4 Back to loop groups In our quest for a complex analogue of the local Langlands correspondence we need to decide what will replace the notion of a smooth representation of the group G(F ), where F = Fq ((t)). As the previous discussion demonstrates, we should consider representations of the complex loop group G((t)) on various categories of D-modules on the ind-schemes G((t))/K, where K is a “compact” subgroup of G((t)), such as G[[t]] or the Iwahori subgroup (the preimage of a Borel subgroup B ⊂ G under the homomorphism G[[t]] → G), or the categories of representations of the Lie algebra g((t)).

10) 38 Vertex algebras Then we have [Sn , Am ] = − κ − κc nAn+m . 11) Since the invariant inner products on a simple Lie algebra g form a onedimensional vector space, the ratio appearing in this formula is well-defined (recall that κ0 = 0 by our assumption). 11) comes as a surprise. It shows that the Segal–Sugawara operators are indeed central for one specific value of κ, but this value is not κ = 0, as one might naively expect, but the critical one, κ = κc ! This may be thought of as a “quantum correction” due to our regularization scheme (the normal ordering).