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Symmetric spaces and strongly isotropy irreducible spaces. (English) Zbl 0804.53075
The authors establish a correspondence between the compact simply connected irreducible symmetric spaces and the compact strongly isotropy irreducible quotients of the classical groups (that is the quotients with irreducible connected component of the isotropy group). It is based on the following observation by Wall, who compared the classification lists.
Let \(G/K\) be a compact simply connected irreducible symmetric space. Then either the isotropy group \(K\) is maximal in \(\text{SO}(n)\) and then \(\text{SO}(n)/K\) is strongly isotropy irreducible, or, up to small number of exceptions, there exists a maximal subgroup \(L\) of \(\text{SO}(n)\) such that \(K\subset L\) and \(L/K\) is strongly isotropy irreducible. Here, \(L\) can be either U\((n/2)\) if \(G/K\) is Hermitian symmetric or \(\text{Sp}(1)\cdot \text{Sp}(n/4)\) if \(G/K\) is quaternionic symmetric. The authors show that all exceptions may be also derived from some general principles. As a consequence, the classification of the non- symmetric strongly isotropy irreducible quotients of the classical groups is read off from that of the irreducible symmetric spaces.
An other principal result is a characterization of the isotropy representations of the irreducible symmetric spaces in terms of the highest weights. It gives a new elegant method for classification of symmetric spaces. It may be regarded as a modern version of the Cartan’s first method of classifying symmetric spaces, based on the holonomy representation.

MSC:
53C35 Differential geometry of symmetric spaces
53C30 Differential geometry of homogeneous manifolds
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