Phantom minimal series and the Peter--Weyl theorem for loop groups

Connection link: https://hu-berlin.zoom.us/j/61686623112
Contact: Gaetan Borot (gaetan.borot@hu-berlin.de)

Let G be a complex reductive group. The celebrated Peter--Weyl theorem decomposes the algebra of functions on G as a G x G module with respect to left and right translations. In this talk we introduce a natural analogue for the loop group G((z)). A key role is played by a family of G((z)) representations at negative level, the phantom minimal series. These are dual, in a precise but somewhat subtle homological sense, to the more familiar positive energy representations at positive level. Time permitting, we will discuss the existence of phantom minimal series for many related vertex algebras, and some interesting analytic properties of their characters.