Symplectic branching laws and Hermitian symmetric spaces (Q2855942)

Let \(K\) denote an arbitrary compact Lie group and assume that \(K\) acts holomorphically and in a Hamiltonian fashion on the connected compact Kähler manifold \((M,\omega)\) with moment map \(\tau: M \to \mathfrak k^\ast\). Assume that \((M,\omega)\) admits a prequantum line bundle \(\mathcal L\). The action of \(K\) then lifts to an action on \(\mathcal L\), and hence we have a representation of \(K\) on the space \(H^0(M,\mathcal L)\). One may then ask how \(H^0(M,\mathcal L)\) decomposes under \(K\). For this purpose it is useful to realize the irreducible \(K\)-representation of highest weight \(\xi\) as the space of holomorphic sections \(H^0(\mathcal O^K_\xi ,{\mathcal L}_{\,\xi})\), where \(\mathcal O^K_\xi \subseteq \mathfrak k^\ast\) is the coadjoint orbit through \(\xi\) and \({\mathcal L}_{\,\xi}\) is the prequantum line bundle attached to the Kostant-Kirillov symplectic form on \(\mathcal O^K_\xi\). The multiplicity of the representation \(H^0(\mathcal O^K_\xi ,{\mathcal L}_{\,\xi})\) in \(H^0(M,\mathcal L)\) is given by the dimension of the space, \(H^0(M\times \overline{\mathcal O^K_\xi} ,\mathcal L\boxtimes \overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^K\), of \(K\)-invariant holomorphic sections of the line bundle \(\mathcal L\boxtimes \overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} \to M \times\overline{\mathcal O^K_\xi}\). Here \(\mathcal O^K_\xi\) denotes the topological space \(\mathcal O^K_\xi\) equipped with the reverse complex structure and the symplectic form given by \(-1\) times the Kostant-Kirillov form. The fibre over \(x \in \mathcal O^K_\xi\) of the line bundle \(\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}}\) consists of the space of antilinear complex-valued functionals on \(({\mathcal L}_{\,\xi})_x\). The group \(K\) now acts holomorphically and in a Hamiltonian fashion on \(M \times\overline{\mathcal O^K_\xi}\) with moment map \(\tau^\xi : M \times\overline{\mathcal O^K_\xi} \to \mathfrak k^\ast\) given by \(\tau^\xi(m, f) := \tau (m)-f\).NEWLINENEWLINEAn obvious question is whether the space \(H^0(M \times \overline{\mathcal O^K_\xi} ,\mathcal L\boxtimes\overline{ {\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^K\) can be interpreted as the space of all holomorphic sections of some line bundle over some ``quotient'' of \(M \times\overline{\mathcal O^K_\xi}\) by \(K\). Indeed, this holds for the Mumford quotient NEWLINE\[NEWLINE(M \times\overline{\mathcal O^K_\xi} )_0 := (M \times\overline{\mathcal O^K_\xi} )_{ss}/\!/K^{\mathbb C},NEWLINE\]NEWLINE where \((M \times \overline{\mathcal O^K_\xi} )_{ss}\) is the open subset consisting of the semistable points of \((M \times\overline{\mathcal O^K_\xi} )\). An interesting feature, and one which links the Mumford quotient to symplectic geometry, is that \((M\times\overline{\mathcal O^K_\xi} )_0\) is homeomorphic to the topological quotient \((\tau^\xi)^{-1}(0)/K\), the symplectic reduction at 0. Moreover, the following result holds.NEWLINENEWLINE{Theorem (\textit{R.~Sjamaar}).} If \(\xi\) does not lie in the image \(\tau (M)\), then the irreducible representation of highest weight \(\xi\) does not occur in \(H^0(M,\mathcal L)\).NEWLINENEWLINEThe space \((M \times\overline{\mathcal O^K_\xi} )_0\) also carries the structure of a complex projective variety. In fact, it is isomorphic to Proj\((\oplus_{k =0}^\infty H^0(M \times\overline{\mathcal O^K_\xi} , (\mathcal L\boxtimes\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^k)^K)\). Moreover, for large enough \(q\), the line bundle \(\mathcal L^q\boxtimes (\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^q\) induces a line bundle \((\mathcal L^q\boxtimes (\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^q)_0\) over \((M \times\overline {\mathcal O^K_\xi })_{ss}/\!/K^\mathbb C\), the total space of which is \(((\mathcal L\boxtimes\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^q|_{(M\times\overline{\mathcal O^K_\xi})_{ss}})/K^{\mathbb C}\). For such \(q\), there is an isomorphism NEWLINE\[NEWLINEH^0(M \times\overline{\mathcal O^K_\xi}, (\mathcal L\boxtimes\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^q)^K\cong H^0((M \times\overline{\mathcal O^K_\xi} )_0, (\mathcal L^q\boxtimes(\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^q)_0).NEWLINE\]NEWLINE Under favourable conditions, e.g. the vanishing of all cohomology groups \(H^i((M \times\overline{\mathcal O^K_\xi} )_0, (\mathcal L^{kq}\boxtimes (\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^{kq})_0)\), for \(k > 0\) and \(i > 0\), the asymptotics of dim\(H^0((M \times\overline{\mathcal O^K_\xi} )_0, (\mathcal L^{kq}\boxtimes(\overline{{\mathcal L}^{\,\,\,\,\ast}_{\,\xi}} )^{kq})_0)\) as \(k \to\infty\) is given by the Riemann-Roch theorem for (singular) complete schemes.NEWLINENEWLINEAlthough this machinery works well in principle, a major obstruction to applying it in particular cases in order to obtain explicit asymptotic expressions for multiplicities is that the moment maps \(\tau^\xi\), and hence their fibres, are in general notoriously hard to compute.NEWLINENEWLINEIn this paper, the authors are able to compute the moment map \(\mu_{\mathfrak k} : X \to \mathfrak k^\ast\) for the \(K\)-action by using an explicit Jordan-theoretic description of \(X\). The authors also prove that for any \(\nu \in\mu_{\mathfrak k}(X)\), the stabilizer, \(K_\nu\), of \(\nu\) acts transitively on the fibre \(\mu^{-1}_{\mathfrak k} (\nu)\). As a consequence, the decomposition of \(H^0(X,{\mathcal L}^{\,\,\,\,k})\) under \(K\) is multiplicity free for every \(k \in\mathbb N\). The authors also explicitly describe the moment polytope for \(k\mu_{\mathfrak k}\) for any \(k \in\mathbb N\), i.e., the intersection of \(k\mu_{\mathfrak k}(X)\) with a closed Weyl chamber, as well as the integral points in the moment polytope. By Theorem, these are the only weights that can occur as highest weights of irreducible \(K\)-representations in \(H^0(X,{\mathcal L}^{\,\,\,\,k})\). The authors prove that all these integral points in fact do occur. In fact, from the particular form of the integral weights in the moment polytopes for the \(k\mu_{\mathfrak k}\) it turns out that it suffices to prove this for \(k = 1\), i.e., that all the integral points in the moment polytope for \(\mu_{\mathfrak k}\) parameterize irreducible \(K\)-representations in \(H^0(X,{\mathcal L}\,\,\,)\).NEWLINENEWLINEIn the special case \(k = 1\), the authors prove that the integral points stand in a one-to-one correspondence with the closed \(K^{\mathbb C}\)-orbits, \(X_0, \dots, X_r\), in \(X\). The number \(r\) is the rank of \(X\) as a symmetric space. The authors also give a Jordan-theoretic characterization of these orbits. Using this characterization, the authors give a geometric decomposition of \(H^0(X,{\mathcal L}\,\,\,)\) under \(K\). The \(K\)-equivariant embedding \(W_i \to H^0(X,{\mathcal L}\,\,\,)\) of the irreducible representation \(W_i\) corresponding to the orbit \(X_i\) is a section for the restriction map \(H^0(X,{\mathcal L}\,\,\,) \to H^0(X_i,{\mathcal L}\,\,\,|_{X_i})\).NEWLINENEWLINEThe authors also define an Okounkov body for the line bundle \({\mathcal L}\,\,\,\) and the \(K\)-action by using a canonical local trivialization of sections. The semigroup defining the Okounkov body describes the initial monomial terms of the polynomials that are local trivializations of highest weight vectors for irreducible \(K\)-subrepresentations. Using the decompositions for the spaces \(H^0(X,{\mathcal L}^{\,\,\,\,k})\) under \(K\) the authors are able to prove that this semigroup is finitely generated.