Symplectic Dirac operators for Lie algebras and graded Hecke algebras - Ciubotaru, D; De Martino, M; Meyer, P (Springer, 2022)
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.