Symplectic Dirac operators on Hermitian symmetric spaces

Abstract

Abstract. We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

On Mpc-structures and Symplectic Dirac Operators

“…) . We now proceed as in the pseudo-Hermitean case to construct a lift using a smooth square root of the last determinant which exists since the term 1 2 (I F + (A T g A g ) −1 − iS g ) has strictly positive real part in the complex general linear group. We set…”

“…with the right action of MU c (V, Ω, j) on the right-hand side given via σ × λ by the right action of the group U(V, Ω, j) on U(M, ω, J) and of U(1) on L (1) . Define…”

“…The action of U(1)-principal bundles on Mp c -structures is made explicit and canonical (not depending on the choice of an almost complex structure) in the following two lemmas: Lemma 3.8 Given an Mp c -structure on M, (P π → M, φ : P → Sp(M, ω)), and given a principal U(1)-bundle over M, L (1) π → M, one can define a new Mp c -structure on M, (P ′ π ′ → M, φ ′ : P ′ → Sp(M, ω)) denoted L (1) • P as follows. One first considers the fibrewise product of L (1) and P over M L (1)…”

“…At the level of isomorphism classes this Lemma gives an action of H 2 (M, Z) on the set of isomorphism classes of Mp c -structures for a fixed symplectic structure. Lemma 3.9 Given two Mp c -structures (P, φ) and (P ′ , φ ′ ) over the same symplectic manifold (M, ω), there is a canonical principal U(1)-bundle L (1) over M constructed in the following way. One first defines the fibrewise product of P and P ′ viewed as principal U(1) bundles over Sp(M, ω):…”

“…This Lemma shows in particular that the action is simply transitive. Lemma 3.12 Given a symplectic action ρ M of the Lie group G on the symplectic manifold (M, ω) and given two G-invariant Mp c -structures P, φ, ρ P and P ′ , φ ′ , ρ P ′ over (M, ω, ρ M ), the canonical principal U(1)-bundle L (1) over M constructed in Lemma 3.9 is G-invariant in a canonical way.…”