  This paper is devoted to the theory of vector-valued modular forms for
orthogonal groups of signature (2,n). Our purpose is multi-layered: (1) to lay
a foundation of the theory of vector-valued orthogonal modular forms; (2) to
develop some aspects of the theory in more depth such as geometry of the Siegel
operators, filtrations associated to 1-dimensional cusps, decomposition of
vector-valued Jacobi forms, square integrability etc; and (3) as applications
derive several types of vanishing theorems for vector-valued modular forms of
small weight. Our vanishing theorems imply in particular vanishing of
holomorphic tensors of degree <n/2-1 on orthogonal modular varieties, which is
optimal as a general bound.
  The fundamental ingredients of the theory are the two Hodge bundles. The
first is the Hodge line bundle which already appears in the theory of
scalar-valued modular forms. The second Hodge bundle emerges in the
vector-valued theory and plays a central role. It corresponds to the
non-abelian part O(n,R) of the maximal compact subgroup of O(2,n). The main
focus of this paper is centered around the properties and the role of the
second Hodge bundle in the theory of vector-valued orthogonal modular forms.
