SYMPLECTIC DIRAC OPERATORS FOR LIE ALGEBRAS AND GRADED HECKE ALGEBRAS

The aim of this paper is to define a pair of symplectic Dirac operators ( D + , D – ) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of Z/2-graded quadratic Lie algebras g = k + p and of graded affine Hecke algebras H. In these contexts, we show analogues of the Parthasarathy’s formula for [ D + , D – ] and certain generalisations of the Casimir inequality.