A Fourier–Mukai approach to the K-theory of compact Lie groups

Abstract

Let G be a compact, connected, simply-connected Lie group. We use the Fourier-Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.

The Weyl map and bundle gerbes

“…, p + 1. Notice that this means that the two maps Y [2] → Y are (perhaps confusingly) π 1 ((y 1 , y 2 )) = y 2 and π 2 ((y 1 , y 2 )) = y 1 . If g : (M ).…”

“…If (P, Y ) is a bundle gerbe over M and (Q, X) is a bundle gerbe over N , then a morphism of bundle gerbes (P, Y ) → (Q, X) is a triple of maps f : M → N , g : Y → X and h : P → Q. These have to satisfy: g covers f and thus induces a map g [2] : Y [2] → X [2] and h : P → Q is a bundle morphism covering g [2] .…”

“…If (P, Y ) is a bundle gerbe over M and f : N → M , then f −1 (Y ) → N is a surjective submersion and we have f [2] : f −1 (Y ) [2] → Y [2] . The pullback of P by f [2] , more conveniently called f −1 (P ), then inherits a natural bundle gerbe multiplication. The bundle gerbe (f −1 (P ), f −1 (Y )) over N is called the pullback of (P, Y ) by f .…”

“…Any bundle gerbe (P, Y, π) admits a bundle gerbe connection ∇ which is a Hermitian connection on P preserving the bundle gerbe multiplication. As a result of this condition, the curvature F ∇ ∈ Ω 2 (Y [2] ) of a bundle gerbe connection satisfies δ(F ∇ ) = 0. It follows from exactness of the fundamental complex (2.1) that there exist imaginary 2-forms f ∈ Ω 2 (Y ) satisfying δ(f ) = F ∇ .…”

“…There is a long history of exploiting the attractive features of the Weyl map going back to Weyl's proof of the Integral Formula [18] and the K-theory of compact Lie groups [1]. Cup product constructions similar to ours have been used by Brylinski [5] to construct projective unitary group bundles, in index theory [10] and in twisted K-theory [6,2]. We note also that [9,Section 8.2] gives a general construction of the cup product bundle gerbe for a decomposable Dixmier-Douady class.…”