Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings

We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.