Loop groups and twisted K-theory I

This is the first in a series of papers investigating the relationship between the twisted equivariant K‐theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we set up the foundations of twisted equivariant K‐groups, and more generally twisted K‐theory of groupoids. We establish enough basic properties to make effective computations. We determine the twisted equivariant K‐groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.