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We study Cameron-Liebler $k$-sets in the affine geometry, so sets of$k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler$k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraiccombinatorics, Cameron-Liebler $k$-sets of $\text{AG}(n, q)$ correspond tocertain equitable bipartitions of the Association scheme of $k$-spaces in$\text{AG}(n, q)$, while in the analysis of Boolean functions, they correspondto Boolean degree $1$ functions of $\text{AG}(n, q)$. We define Cameron-Liebler$k$-sets in $\text{AG}(n, q)$ by intersection properties with $k$-spreads andshow the equivalence of several definitions. In particular, we investigate therelationship between Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ and$\text{PG}(n, q)$. As a by-product, we calculate the character table of theassociation scheme of affine lines. Furthermore, we characterize the smallestexamples of Cameron-Liebler $k$-sets. This paper focuses on $\text{AG}(n, q)$for $n > 3$, while the case for Cameron-Liebler line classes in $\text{AG}(3,q)$ was already treated separately.

Cameron-Liebler $k$-sets in $\text{AG}(n,q)$