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.

Author notes

2010 Mathematics Subject Classification 19L50 (primary), 22E67.
During the course of this work the first author was supported by the National Science Foundation under the grants DMS-0072675 and DMS- 0305505, the second by grants DMS-9803428 and DMS-0306519, and the third by DMS-0072675. Collaboration between the authors was greatly facilitated by the KITP of Santa Barbara (NSF Grant PHY99-07949) and the Aspen Center for Physics.