Galois module structure of Deligne–Lusztig varieties for $\mathrm{GL}_3$

Let $\overline{\mathbf{Y}}(w)\to\overline{\mathbf{X}}(w)$ be the ramified Galois cover of compactified Deligne–Lusztig varieties. Our objects of study are the Lefschetz numbers for the action of the finite covering group $T^{wF}$ on the coherent cohomology of the structure sheaf of $\overline{\mathbf{Y}}(w)$. The Lefschetz numbers for all elements of $T^{wF}$ determine the virtual $T^{wF}$-representation given by the alternating sum of these cohomology groups. In the two cases of Deligne–Lusztig surfaces of type $A_2$ and ${}^{2}A_2$, we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component. It follows that the equivariant Euler characteristic of the compactification $\overline{\mathbf{Y}}(w)$ is not virtually free as a virtual module over the covering group.