Symplectic Dirac operators for Lie algebras and graded Hecke algebras

Abstract

We 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 $\mathbb{Z}/2$-graded quadratic Lie algebras $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ and of graded affine Hecke algebras $\mathbb{H}$.

26 pages