  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.

Representation Theory