Complexified Hermitian symmetric spaces, hyperkähler structures, and real group actions (Q6594713)

Let \(G/K\) be a complexified Hermitian symmetric space. It is well-known that it has a hyper-Kähler structure, see [\textit{O. Biquard} and \textit{P. Gauduchon}, C. R. Acad. Sci., Paris, Sér. I 323, No. 12, 1259--1264 (1996; Zbl 0866.58007); Lect. Notes Pure Appl. Math. 184, 287--298 (1997; Zbl 0879.53051); in: Séminaire de théorie spectrale et géométrie. Année 1997--1998. St. Martin D'Hères: Université de Grenoble I, Institut Fourier. 127--173 (1998; Zbl 0943.53029)]. This hyper-Kähler structure relies on two realizations of \(G/K\). One such realization is as the adjoint orbit \(\mathcal{O}\) of an element \(\Upsilon\in\mathfrak{g}\) by a complex semisimple Lie group \(G\). On this space there is a complex structure \(J_1\) and the Kirillov-Kostant-Souriau holomorphic symplectic form \(-\omega_3+i\omega_2\). The second realization of the complexified Hermitian symmetric space \(G/K\) is as the cotangent bundle of the flag manifold \(G/Q\), where \(Q\subseteq G\) is a parabolic subgroup. This gives a complex structure \(J_3\) as well as the usual symplectic form on a cotangent bundle given by \(\omega_1+i\omega_2\). The flag manifold \(G/Q\) can be viewed as the adjoint orbit of \(\Upsilon\) by a compact real form \(G_u\) of \(G\).\medskip\N\NOn every hyper-Kähler manifold \((M,g,J_1,J_2,J_3)\) there is a complex structure for each unit linear combination \[J_{(\lambda_1,\lambda_2,\lambda_3)}=\lambda_1J_1+\lambda_2J_2+\lambda_3J_3,\] which has an associated symplectic form \[\omega_{(\lambda_1,\lambda_2,\lambda_3)}=\lambda_1\omega_1+\lambda_2\omega_2+\lambda_3\omega_3.\]\N\NWe can identify the triple \((\lambda_1,\lambda_2,\lambda_3)\in\mathbb{S}^2\subset\mathbb{R}^3\) with a point \(\lambda\) in the extended plane \(\mathbb{C}\cup\{\infty\}\). The complex number \(\lambda\) plays a role in an explicit \(G_u\)-equivariant bijection \(T_\lambda:G/K\to G/K\) (see Definition 4.16), which provides a new understanding of how the complex structure \(J_{(\lambda_1,\lambda_2,\lambda_3)}\) and the symplectic form \(\omega_{(\lambda_1,\lambda_2,\lambda_3)}\) vary with \((\lambda_1,\lambda_2,\lambda_3)\). In particular, the author proves the following:\N\begin{itemize}\N\item Via the map \(T_\lambda\), each \(J_{(\lambda_1,\lambda_2,\lambda_3)}\), except \(\pm J_3\), is equivalent to \(J_1\) (see Theorem 10.1).\N\item Via a map related to \(T_\lambda\), for any \((\lambda_1,\lambda_2,\lambda_3)\) with \(\lambda_3\neq0\), the symplectic form \(\omega_{(\lambda_1,\lambda_2,\lambda_3)}\) is equivalent to a scalar multiple of \(\omega_3\) (see Theorem 10.4).\N\end{itemize}\N\NThe author also studies the action of \(G_0\) (the real form of \(G\) corresponding to the Hermitian symmetric space of non-compact type) on the complexified Hermitian symmetric space \(G/K\). There is a family of actions of \(G_0\) parametrized by a 2-sphere, each action corresponding to a choice of \((\lambda_1,\lambda_2,\lambda_3)\), with its complex structure, symplectic form and moment map. Then most actions are equivalent to the usual one on the \(G\)-orbit of \(\Upsilon\) and most moment-critical subsets are equivalent to the ones for the action on the cotangent bundle of \(G/Q\). Finally, the author presents a detail account for \(G=\mathrm{SL}(2,\mathbb{C})\).