Harmonic spinors on semisimple symmetric spaces (Q1868693)

A geometric construction of the discrete series representations of semisimple Lie groups treated as the solutions of a certain twisted Dirac operator \(D\) on Riemannian symmetric spaces (1972, Parthasarathy; 1974, Wolf; 1977, Atiyah, Schmid) is extended to the case of non-Riemannian semisimple symplectic spaces \(G/H\). Here a set \(H\) defines the identity component of the fixed point group of an involution of a connected linear semisimple Lie group \(G\). It is assumed that the complex rank of \(G\) is equal to the complex rank of \(H\). An explicit integral formula representing the solutions of the equation \(D=0\) is given in the form of a \(G\)-intertwining operator from a principal series representation into the sections of twisted spin bundles on \(G/H\). This formula is very similar to the classical Poisson integral representations of harmonic functions. It is proved that the image of the intertwining operator lies in the kernel of the Dirac operator.