Smooth \(SU(p,q)\)-actions on the \((2p+2q-1)\)-sphere and on the complex projective \((p+q-1)\)-space (Q2772933)

In this paper the orthogonal \(S(U(p)\times U(q))\)-action on the standard \((2p+2q-1)\)-sphere and the induced \(S(U(p)\times U(q))\)-action on the standard complex projective \((p+q-1)\)-space are called standard. The author studies smooth \(SU(p, q)\)-actions on the spaces for \(p, q\geq 3\) whose restricted action to the maximal compact subgroup \(S(U(p)\times U(q))\) is standard. The characterization of such non-compact Lie group actions on a standard sphere was first introduced by \textit{T. Asoh} [Osaka J. Math. 24, 271-298 (1987; Zbl 0706.57021)]. Actions on each space are characterized by pairs \((\varphi, f)\) where \(\varphi\) is an action of some subgroup of \(SU(p, q)\) on the fixed point set \(F\) of the principal isotropy subgroup of the standard action, and \(f: F\rightarrow P_1(\mathbb{C})\) is a smooth map. In the case of the projective space, the pair \((\varphi, f)\) is also characterized by a triple constructed in [\textit{K. Mukōyama}, Tôhoku Math. J., II. Ser. 48, No. 4, 543-560 (1996; Zbl 0890.57050)]. This implies that there are infinitely many smooth \(SU(p, 1)\)-actions on the projective space.