Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables

Abstract

In this paper, we analyze differential operators of first order acting between vector bundles associated to G/P where G = Spin(n+2, 2) and P is a parabolic subgroup. The operators in question are invariant with respect to the group G and we identify them with operators on the flat space that are invariant with respect to its subgroup SL(2)× Spin(n). For n even, the list of all such invariant operators of first order is obtained using certain algebraic conditions on the highest weights of the representations in question. In some cases, an explicit realization of the operator is given in coordinates.