Weighted intriguing sets of finite generalised quadrangles

We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1-17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne's tight sets, we give a new proof of Thas' result that an m-ovoid of a generalised quadrangle of order (s,s (2)) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.