Cameron-Liebler k-sets in AG(n, q)

We study Cameron-Liebler k-sets in the affine geometry, so sets of k-spaces in AG(n, q). This generalizes research on Cameron-Liebler k-sets in the projective geometry PG(n, q). Note that in algebraic combinatorics, Cameron-Liebler k-sets of AG(n, q) correspond to certain equitable bipartitions of the association scheme of k-spaces in AG(n, q), while in the analysis of Boolean functions, they correspond to Boolean degree 1 functions of AG(n, q). We define Cameron-Liebler k-sets in AG(n, q) in a similar way as Cameron- Liebler k-sets in PG(n, q), such that its characteristic vector is a linear combination of point-pencils. In particular, we investigate the relationship between Cameron- Liebler k-sets in AG(n, q) and PG(n, q). As a by-pro duct, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler k-sets. This paper focuses on AG(n, q) for n > 3, while the case for Cameron-Liebler line classes in AG(3, q) was already treated separately.