Cameron-Liebler $k$-classes in \(\mathrm{PG}(2k+1,q)\) (Q1715065)

Let $A$ be the point line incidence matrix of \(\mathrm{PG}(n,q)\), then a set of lines \(L\) is a Cameron-Liebler line class if the characteristic function \(\chi_L\) belongs to \(\mathrm{row}(A)\). The authors generalize this definition to the following. A set of \(k\)-spaces in \(P(n,q)\) is a Cameron-Liebler \(k\)-class if the characteristic function \(\chi_L\) belongs to \(\mathrm{row}(A_k)\), where \(A_k\) is the points \(k\)-spaces incidence matrix of \(\mathrm{PG}(n,q)\). The authors give many results about Cameron-Liebler \(k\) classes analogous to results about Cameron-Liebler line classes. Further, they give a connection to \(k\)-spreads which parallels the situation for line classes in \(\mathrm{PG}(3,q)\). Additionally, they give connections to other geometric objects and Erdős-Ko-Rado sets, proving some results about the existence of these objects.