Central character of a block (${\omega }_b:Z(kG)\to Z(kGb)/J(Z(kGb))$)

The center of an indecomposable finite-dimensional block of a group algebra is a commutative Artinian local ring. Its quotient by its Jacobson radical is a field, giving the block's central character. Over a splitting field for finite group representations the residue field is the coefficient field; without this assumption one retains the residue field rather than pretending every central character is scalar-valued over $k$.