Equivariant Dirac operators and differentiable geometric invariant theory (Q1691138)

Let \(M\) be an oriented, compact connected manifold of even dimension. We consider a \(\operatorname{spin}^{c}\) structure on \(M,\) and denote by \({\mathcal S}\) the corresponding spinor bundle. Let \(K\) be a compact connected Lie group acting on \(M\) and \({\mathcal S}\). We denote by \(D:\Gamma(M,{\mathcal S}^{+})\to\Gamma(M,{\mathcal S}^{-})\) the corresponding \(K\)-equivariant \(\operatorname{spin}^{c}\) Dirac operator. Let us denote by \({\mathcal Q}_{K}(M,{\mathcal S})\) the equivariant index of \(D.\) It belongs to the Grothendieck group of representations of \(K:\) \({\mathcal Q}_{K}(M,{\mathcal S}) = \sum\limits_{\pi\in\hat{K}}m(\pi)\pi,\) where \(\hat{K}\) denotes the set of classes of unitary irreducible representations of \(K.\) Consider the determinant line bundle det(\({\mathcal S}\)) of the \(\operatorname{spin}^{c}\) structure. This is a \(K\)-equivariant complex line bundle on \(M.\) The choice of a \(K\)-invariant Hermitian metric and a \(K\)-invariant Hermitian connection \(\nabla\) on det(\({\mathcal S}\)) determines a moment map \(\Phi_{{\mathcal S}}:M\to\mathfrak{k}^{\ast}.\) Here \(\mathfrak{k}\) is the Lie algebra of \(K.\) If \(M\) is spin and \({\mathcal S} = {\mathcal S}_{\operatorname{spin}}\otimes L,\) then det(\({\mathcal S}) = L^{\otimes^{2}}\) and \(\Phi_{{\mathcal S}}\) is equal to the moment map \(\Phi_{L}\) associated with a connection on \(L.\) Let \(\mathfrak{h}\) be a subalgebra of \(\mathfrak{k}\) and \((\mathfrak{h})\) its the conjugacy class. If \(\xi\in\mathfrak{k}^{\ast},\) we shall denote by \(\mathfrak{k}_{\xi}\) its infinitesimal stabilizer. The set \(\mathcal{H}_{\mathfrak{k}}\) of conjugacy classes of the algebras \(\mathfrak{k}_{\xi},\hskip 2mm \xi\) running in \(\mathfrak{k}^{\ast},\) is a finite set. Indeed the complexified Lie algebras of \(\mathfrak{k}_{\xi}\) varies over the Levi subalgebras of \(\mathfrak{k}_{\mathbb C}\). For \((\mathfrak{h})\in\mathcal{H},\) we say that a coadjoint orbit \(K\xi\) is of type \((\mathfrak{h})\) if \(\mathfrak{k}_{\xi}\) belongs to the conjugacy class \((\mathfrak{h}).\) The semi-simple part of \(\mathfrak{k}_{\xi}\) is \([\mathfrak{k}_{\xi},\mathfrak{k}_{\xi}].\) Let \((\mathfrak{k}_{M})\) be the generic infinitesimal stabilizer of the \(K\)-action on \(M.\) In the paper authors prove the following. Theorem 1. If \(([\mathfrak{k}_{M},\mathfrak{k}_{M}])\) is not equal to some \(([\mathfrak{h},\mathfrak{h}]),\) for \(\mathfrak{h}\in\mathcal{H}_{\mathfrak{k}},\) then for any \(K\)-equivariant line bundle \(\mathcal{L},\) \(\mathcal{Q}_{K}(M,\mathcal{S})=0.\) \vskip 1mm \noindent Hence the authors may assume that \(([\mathfrak{k}_{M},\mathfrak{k}_{M}])=([\mathfrak{h}_{M},\mathfrak{h}_{M}])\) for \((\mathfrak{h}_{M})\in\mathcal{H}_{\mathfrak{k}}\). We say that a coadjoint orbit \(\mathcal{P}\subset\mathfrak{k}^{\ast}\) is admissible if \(\mathcal{P}\) carries a \(\operatorname{spin}^{c}\)-bundle \(\mathcal{S}_{\mathcal{P}}\) such that the corresponding moment map is the inclusion \(\mathcal{P}\hookrightarrow\mathfrak{k}^{\ast}.\) We denote simply by \(\mathcal{Q}^{\operatorname{spin}}_{K}(\mathcal{P})\) the element \({\mathcal Q}_{K}(\mathcal{P},{\mathcal S}_{\mathcal{P}})\in R(K).\) It is either zero or an irreducible representation of \(K,\) and the map \(\mathcal{O}\longmapsto\pi_{\mathcal{O}}:=\mathcal{Q}^{\operatorname{spin}}_{K}(\mathcal{O})\) defines a bijection between the regular admissible orbits and the dual \(\hat{K}.\) When \(\mathcal{O}\) is a regular admissible orbit, an admissible coadjoint orbit \(\mathcal{P}\) is called an ancestor of \(\mathcal{O}\) (or a \(K\)-ancestor of \(\pi_{\mathcal{O}}\)) if \(\mathcal{Q}^{\operatorname{spin}}_{K}(\mathcal{P})=\pi_{\mathcal{O}}\). Denote by \(\mathcal{A}((\mathfrak{h}_{M}))\) the set of admissible orbits of type \((\mathfrak{h}_{M}).\) If \(\mathcal{P}\in\mathcal{A}((\mathfrak{h}_{M})),\) we can define the \(\operatorname{spin}^{c}\) index \(\mathcal{Q}^{\operatorname{spin}}(M_{\mathcal{P}})\in\mathbb Z\) of the reduced space \(M_{\mathcal{P}}=\Phi^{-1}_{\mathcal{S}}(\mathcal{P})/K\) (by a deformation procedure if \(M_{\mathcal{P}}\) is not smooth). The main result of the paper under review is the following theorem. Theorem 2. Assume that \(([\mathfrak{k}_{M},\mathfrak{k}_{M}]=[\mathfrak{h}_{M},\mathfrak{h}_{M}])\) for \((\mathfrak{h}_{M})\in\mathcal{H}_{\mathfrak{k}}.\) The multiplicity of the representation \(\pi_{\mathcal{O}}\) in \(\mathcal{Q}_{K}(M,\mathcal{S})\) is equal to \(\sum\limits_{\mathcal{P}}{\mathcal{Q}^{\operatorname{spin}}}(M_{\mathcal{P}}),\) where the sum runs over the ancestors of \(\mathcal{O}\) of type \((\mathfrak{h}_{M}).\) In other words, \({\mathcal Q}_{K}(M,{\mathcal S})=\sum\limits_{\mathcal{P}\in\mathcal{A}((\mathfrak{h}_{M}))}\mathcal{Q}^{\operatorname{spin}}(M_{\mathcal{P}})\mathcal{Q}_{K}^{\operatorname{spin}}(\mathcal{P}).\) Moreover, the authors present previous results on this subject. One of them is the article by \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 515--538 (1982; Zbl 0503.58018)]. There are also many examples, and considerations of results when \(K\) is the torus.