  A bosonic Laplacian, which is a generalization of Laplacian, is constructed
as a second order conformally invariant differential operator acting on
functions taking values in irreducible representations of the special
orthogonal group, hence of the spin group. In this paper, we firstly introduce
some properties for homogeneous polynomial null solutions to bosonic
Laplacians, which give us some important results, such as an orthogonal
decomposition of the space of polynomials in terms of homogeneous polynomial
null solutions to bosonic Laplacians, etc. This work helps us to introduce
Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces,
in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the
unit ball and a description of bosonic Bergman projection are given as well. At
the end, we investigate bosonic Hardy spaces, which are considered as
generalizations of harmonic Hardy spaces. Analogs of some well known results
for harmonic Hardy spaces are provided here. For instance, connections to
certain complex Borel measure spaces, growth estimates for functions in the
bosonic Hardy spaces, etc.
