Formally self adjointness for the Dirac operator on homogeneous spaces

Akira Ikeda
1975
Introduction. In [5] , Wolf proved that the Dirac operator is essentially self adojoint over a Riemannian spin manifold M and he used it to give explicit realization of unitary representations of Lie groups. Let K be a Lie group and a a Lie group homomorphism of K into SO(n) which factors through Spin (ή). He defined the Dirac operator on spinors with values in a certain vector bundle under the assumption that the Riemannian connection on the oriented orthonormal frame bundle P over M can be
more » ... P over M can be reduced to some principal A>bundle over M by the homomorphism a. The purpose of this paper is to give the Dirac operator on a homogeneous space in a more general situation using an invariant connection, and to determine connections that define the formally self adjoint Dirac operator. Let G be a unimodular Lie group and K a compact subgroup of G. We assume GjK has an invariant spin structure. First, we define the Dirac operator D on spinors using an invariant connection on the oriented orthonormal frame bundle P over G/K. Next, we introduce an invariant connection V to a homogeneous vector bundle C{? associated to a unitary representation of K, then we define the Dirac operator D ® 1 on spinors with values in GJ according
doi:10.18910/7675 fatcat:kucv4kkvcna3hjre2ukedjgn3u