Abstract: Deligne and Lusztig defined a finite \'etale cover \(Y(w) \to X(w)\) of varieties which is attached to any connected reductive group \(\mathbf{G}\) defined over a finite field \( \mathbb{F}_q \), with an isogeny \(F\) whose some positive power is a Frobenius endomorphism, and a Weyl group element \(w \in W\). We consider the ramified Galois covers \( \overline{Y}(w)\rightarrow \overline{X}(w) \) of the natural compactifications of these Deligne-Lusztig varieties. Our goal is to calculate the Lefschetz numbers for the action of the covering group, which is given by the points \(\mathbf{T}^{wF}\) of a torus with values in \(\mathbb{F}_p\), on the coherent cohomology of the structure sheaf of \( \overline{Y}(w)\).
The full collection of Lefschetz numbers
determine the virtual representation of \(\mathbf T^{wF}\) on the
alternating sum of these cohomology groups. In the two cases of Deligne--Lusztig surfaces of type \(A_2\) and \(^{2}A_2\) for \( \mathbf{G} = \mathrm{GL}_3 \), we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component of \(\overline{Y}(w)\) .
It follows that the alternating coherent cohomology class of the full compactification \(\overline{Y}(w)\) is not virtually free
over the full covering group.