Quaternionic representations of exceptional Lie groups. (Q1880086)

The simply connected complex exceptional Lie groups \(F_4(\mathbf C),\, E_6(\mathbf C),\, E_7(\mathbf C),\, E_8(\mathbf C)\) contain, respectively, the connected quaternionic real forms \(F_{4,4},\, E_{6,4},\, E_{7,4},\, E_{8,4}\) with real root systems of type \(F_4\). \textit{B. H. Gross} and \textit{N. R. Wallach} [J. Reine Angew. Math. 481, 73--123 (1996; Zbl 0857.22012)] constructed three unitary representations \(\sigma\) of the double cover \(\tilde G\) of each group \(G = F_{4,4},\, E_{6,4}\mathbf Z_2,\, E_{7,4},\, E_{8,4}\). As in his previous paper [J. Funct. Anal. 172, No. 2, 377--403 (2000; Zbl 0953.22018)], the author considers the subgroups \(G' = \text{Spin}(5,4)\mathbf Z_2^2,\, \text{Spin}(5,4)\mathbf Z_2^3\subset F_{4,4},\; F_{4,4}\mathbf Z_2\subset E_{6,4}\mathbf Z_2,\; E_{6,4}U_1 \subset E_{7,4},\; E_{7,4}SU_2 \subset E_{8,4}\) and investigates the restrictions of \(\sigma\) onto their double covers \(\tilde G'\). In the present paper, the irreducible components of these restrictions are determined and certain dual pair correspondences associated with the dual pairs \(\tilde G'\) of representations of \(\tilde G\) are described.