Cuspidal representations in the cohomology of Deligne-Lusztig varieties for \(\mathrm{GL}(2)\) over finite rings
Let \(K\) be a non-Archimedean local field, \(O\) its ring of integers, \(P\) its unique maximal ideal, \(k=O/P\) the residue field. Assume that \(K\) has the characteristic of \(k\). Let \(G\) be a reductive group over \(k\). \textit{G. Lusztig} generalized the Deligne-Lusztig theory for \(G(O/P)\) by constructing a variety over \(\bar{k}\) whose étale cohomology classes realize some irreducible representations of \(G(O/P^n)\) for general \(n\). These results appear in [Proc. Symp. Pure Math. 33, 171--175 (1979; Zbl 0421.22009); Represent. Theory 8, 1--14 (2004; Zbl 1085.20029)]. The authors refer to the theory in these papers as Deligne-Lusztig theory over finite rings. In the [2004, loc. cit.], the Deligne-Lusztig variety for \(\mathrm{SL}_2(O/P^2)\) has been explicitly studied. In the 1979 paper, Lusztig describes a construction for division algebras as well. As the cohomology of this variety for any division algebra contains many irreducible components with lower conductor, it is difficult to study them in general. In the present work, the authors study the étale cohomology of certain closed subvarieties of Deligne-Lusztig varieties of \(\mathrm{GL}_2\) over such finite rings and prove that cuspidal representations appear in the cohomology. This generalizes some of the results in [Lusztig, 2004, loc. cit.]. In addition, the authors also prove analogous results for the Deligne-Lusztig variety corresponding to the quaternion division algebra case. A product of such varieties may be regarded as an affine bundle over a curve where the base curve appears as an open subscheme of irreducible components of the stable reduction of the Lubin-Tate curve in a special case. The authors state a conjecture on a part of the stable reduction using the above varieties. This work can be thought of as trying to understand bad reduction of Lubin-Tate curves using Deligne-Lusztig varieties. The conjecture is described as follows. Let \(\pi\) be an irreducible, cuspidal representation of \(\mathrm{GL}_2(K)\). Let \(X(P^n)\) be the Lubin-Tate curve with Drinfeld level \(P^n\)-structures; it is a rigid analytic curve over the \(p\)-adic completion of the maximal unramified extension of \(K\) in an algebraic closure. A morphism of rigid analytic varieties \(f : X \rightarrow Y\) induces a morphism of affine schemes \(\bar{f} : \bar{X} \rightarrow \bar{Y}\); here \(\bar{X}\) etc. refer to the canonical reduction. Denote by \(X_C\), for any rigid analytic variety \(X\), the base change to the completion \(C\) of the algebraic closure of \(K\). With these notations, the authors conjecture: Conjecture. For \(i \leq n\), consider the projection \(p_{n,i} : X(P^n) \rightarrow X(P^i)\) and the group \(G_n = \mathrm{GL}_2(O/P^n) \times O_{2n-1}^{\ast}\). Let \(I_K\) denote the inertia subgroup of \(K\). Then, for \(i \leq n\), there exist \(G_n\)-stable affinoid submonoids \(Y_{n,i}\) in \(X(P^n)\) satisfying: {\parindent=8mm \begin{itemize}\item[(a)] \(Y_{n,i} \cap Y_{n,j} = \emptyset\) for \(i \neq j\); \item[(b)] \(\exists\) a \(G_n \times I_K\)-equivariant isomorphism \(\bar{Y}_{n,n,C} \cong Y_n\); \item[(c)] \(p_{n,i}(Y_{n,i}) = Y_{i,i}\); \item[(d)] the map \(\overline{p_{n,i}} : \bar{Y}_{n,i,C} \rightarrow \bar{Y}_{i,i,C}\) is purely inseparable and compatible with the map \(G_n \times I_K \rightarrow G_i \times I_K\). \end{itemize}}
scientific article; zbMATH DE number 6923341
scientific article