Refinements of the orthogonality relations for blocks

For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a certain cyclotomic field. After that, we use the refinements in order to give upper bounds for the number of irreducible characters (of height 0) in B. In this way we generalize results from [H'ethelyi-K"ulshammer-Sambale, 2014]. These ideas are applied to blocks with abelian defect groups of rank 2. Finally, we address a recent conjecture by Navarro.