Spinors over a cone, Dirac operator, and representations of \(Spin\)(4,4) (Q2367691)

The Dirac operator on the pseudo-Riemannian product \(S^ 3\times S^ 3\) considered as projective cone in \((R^ 8,Q(x) = x^ 2_ 1 + x^ 2_ 2 + x^ 2_ 3 + x^ 2_ 4-x^ 2_ 5-x^ 2_ 6-x^ 2_ 7-x^ 2_ 8)\) is studied. The author describes the space \(F\) of positive harmonic spinors and proves that it is an irreducible pseudo-unitary representation of \(Spin(4,4)_ e\). The weight spaces of \(SO(2)\) in \(F\) are irreducible representations of \(Spin(4,2)_ e\). Analogous facts for the Laplacian were proved by Kostant.