Galois module structure of Deligne--Lusztig varieties for GL₃

Let Ȳ(w)̄(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 TwF on the coherent cohomology of the structure sheaf of Ȳ(w). The Lefschetz numbers for all elements of TwF determine the virtual TwF-representation given by the alternating sum of these cohomology groups. In the two cases of Deligne--Lusztig surfaces of type A₂ and ²A₂, 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 Ȳ(w) is not virtually free as a virtual module over the covering group.