The cohomology of semi-infinite Deligne-Lusztig varieties (Q2209458)

For an algebraic reductive group \(\mathbb{G}\) over a non-Archimedean local field \(K\) with residue field \(\mathbb{F}_{q}\) and an elliptic unramified maximal torus \(\mathbb{T}\) over \(K\), Lusztig constructs \(\widetilde{X}\) (named semi-infinite Deligne-Lusztig variety by the author) and suggests the construction of supercuspidal representations of \(\mathbb{G}(K)\) through the homology of \(\widetilde{X}\) as in the Deligne-Lusztig theory for representations of finite reductive groups [\textit{G. Lusztig}, Proc. Symp. Pure Math. 33, Part 1 (1979; Zbl 0403.00002)]. The paper under review realizes this goal in the case when \(\mathbb{G}(K) = D^{\times}\) for an \(n^{2}\)-dimensional division algebra over \(K\) and \(\mathbb{T}(K) = L^{\times}\) for a degree-\(n\) unramified extension \(L\) of \(K\). More precisely, for any smooth character \(\theta: L^{\times} \rightarrow \overline{\mathbb{Q}}^{\times}_l\), the author shows that the \(\theta\)-isotypic component of the homology group \(H_{i}(\widetilde{X}, \overline{\mathbb{Q}}_l)[\theta]\) is nonzero in exactly one degree \(r_{\theta}\), which can be determined by the Howe factorization of \(\theta\). The author also shows that the representation of \(D^{\times}\) on \(H_{r_{\theta}}(\widetilde{X}, \overline{\mathbb{Q}}_l)[\theta]\) is irreducible when \(\theta\) has trivial \(\mathrm{Gal}(L/K)\)-stabilizer, and this construction gives a bijection between \(\mathrm{Gal}(L/K)\)-orbits of smooth characters of \(L^{\times}\) with trivial \(\mathrm{Gal}(L/K)\)-stabilizer and the isomorphism classes of irreducible \(D^{\times}\)-representations invariant under the twist by any character \(\varepsilon\) of \(K^{\times}\) satisfying \(\ker(\varepsilon) = \mathrm{Nm}_{L/K}(L^{\times})\). The author checks the compatibility with the local Langlands correspondence and Jacquet-Langlands correspondence. The author's proof follows the work of Boyarchenko, which reduces the problem to the study of a family of finite type \(\mathbb{F}_{q^n}\)-scheme \(X_{h}\) for \(h \in \mathbb{Z}_{>0}\) and the representations of some finite unipotent group on \(H^{i}_{c}(X_{h}, \overline{\mathbb{Q}}_{l})\). Besides the construction of representations, the author also obtains interesting results on the geometry of \(X_{h}\), including purity and the Hasse-Weil zeta function.