Witten Non Abelian Localization for Equivariant K-Theory, and the [𝑄,𝑅]=0 Theorem

Equivariant Dirac operators and differentiable geometric invariant theory

“…‚ In Section 4, we define our K-theoretical analogue of the Witten deformation and recall some of its properties (proved in [28,32]). It allows us to reduce the computation of Q K pM, Sq to indices q Z of simpler transversally elliptic operators defined in neighborhoods of connected components of Z S " tκ S " 0u.…”

Kirillov’s orbit method: The case of discrete series representations

“…The set Z M , which is not necessarily smooth, admits the following description. Choose a Weyl chamber t˚Ă t˚in the dual of the Lie algebra of a maximal torus T of K. We see that For any K-invariant open subset U Ă M such that U X Z M is compact in M , we see that the restriction σpM, S, Φ K q| U is a transversally elliptic symbol on U , and so its equivariant index is a well defined element in p RpKq (see [1,31]).…”

“…The quantity Q spin K pMˆP´, Z 0 q P p RpKq is computed as an index of a K-transversally elliptic operator D 0 acting on the sections of S b S P´. The argument used in the compact setting still work (see Lemma 1.3 in [31]): if rS b S P´s γ " 0 then rkerpD 0 qs K and rcokerpD 0 qs K are reduced to 0. l Another important property of the formal geometric quantization procedure is the functoriality relatively to restriction to subgroup. Let H Ă K be a closed connected subgroup.…”

Formal Geometric Quantization III: Functoriality in the Spinc Setting

“…Proof. The proof is done in Section 3.4.2 of [18] in the Hamiltonian setting. The same proof works here.…”

“…where ν P T˚M »ν P TM is an identification associated to an invariant Riemannian metric on M . For any K-invariant open subset U Ă M such that U X Z S is compact in M , we see that the restriction σpM, S, Φ S q| U is a transversally elliptic symbol on U , and so its equivariant index is a well defined element inRpKq (see [1,18]).…”