Actions of symplectic groups on a product of quaternion projective spaces (Q802178)

The paper is one of a series of the author's work on classification theorems for transformation groups on spheres and projective spaces. In his previous paper [ibid. 20, 513-520 (1983; Zbl 0533.57017)] he considered smooth SU(n)-actions on orientable closed manifolds with the rational cohomology ring isomorphic to that of \(P_ a({\mathbb{C}})\times P_ b({\mathbb{C}})\), the product of complex projective spaces, and classified such actions under some restrictions on a,b, and n. The paper under review deals with smooth Sp(n)-actions on orientable closed manifolds with the rational cohomology ring isomorphic to that of \(P_ a({\mathbb{H}})\times P_ b({\mathbb{H}})\), the product of quaternionic projective spaces. Under restrictions \(2\leq a+b\leq 2n-2\) and \(n\geq 7\) he classifies such actions into four kinds of possible cases. The first step to the classification is to classify closed connected subgroups G of Sp(n) such that dim Sp(n)/G\(<8n\), and orthogonal representations of Sp(r) of degree \(k<8r\). In a final section he notices that the similar classification of SU(n)-actions is obtained under the same restrictions on a,b, and n as for Sp(n)-actions, which generalizes the previous result.