Covering groups and projective representations

Abstract

In order to deduce Theorem 1.18 from the special case established in the last chapter, we will need to exploit a relationship between characters of tori in G and representations of G. This relationship is most natural when it is formulated in terms of certain coverings of the tori related to “ρ-shifts” for G (see for example Theorem 1.37 or Theorem 6.8 in [58]). We will therefore need for tori a version of Theorem 1.18 that describes representations of such coverings. It is just as easy to treat coverings of general groups; in any case this will be necessary when we discuss endoscopy.