SIMPLE SPINORS AND REAL STRUCTURES

A concept of a real index associated with any maximal totally null subspace in a complexified vector space endowed with a scalar product, and also with any complex simple (pure) spinor, is introduced and elaborated. It is shown that the real Pin group acts transitively on the projective space of simple spinors with any given real index. The connection between the real index and multivectors associated with a simple spinor and its charge conjugate is established. The simple spinors with extreme values of the real index deserve a special attention: The generic simple spinors for which the real index is minimal and simple-r spinors for which it is maximal.