Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.
Starting from the 2001 Thomas Friedrich’s work on Spin(9), we review some interactions
between Spin(9) and geometries related to octonions. Several topics are discussed in this respect:
explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry
as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the
geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9)
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