Cohomological Disjointness for Parabolic Deligne–Lusztig Varieties

Definitions and mathematical statement

Write $G^F=\{g\in G:F(g)=g\}$ for the fixed-point group of the endomorphism $F$ appearing in the intended Deligne--Lusztig construction. Let $X_P$ denote the parabolic Deligne--Lusztig variety in the source, and put \[ V_i=H_c^i(X_P,\mathbb Q_\ell),\qquad V=\bigoplus_{i\geq 0}V_i, \] where $H_c^i$ denotes compactly supported $\ell$-adic étale cohomology, with its intended $G^F$-action. Such cohomology requires a prime $\ell$ different from the characteristic of the base field. The record does not specify the underlying group, field, endomorphism, parabolic data, or the construction of $X_P$.

For finite-dimensional representations of a finite group over $\mathbb Q_\ell$, call two representations disjoint if they have no common irreducible constituent. By semisimplicity, this is equivalent to the vanishing of the space of equivariant homomorphisms between them. For a finite group $W_S$, the notation $\mathbb Q_\ell W_S$ means its group algebra over $\mathbb Q_\ell$. The record does not define $W_S$ or $S$, so this interpretation remains conditional on identifying the intended group.

Requested assertion, subject to unresolved source hypotheses. For every datum in the intended, but unspecified, class of parabolic Deligne--Lusztig varieties, prove \[ \operatorname{Hom}_{\mathbb Q_\ell[G^F]}(V_i,V_j)=0 \qquad\text{for all }i,j\geq 0\text{ with }i\ne j, \] and prove an isomorphism of $\mathbb Q_\ell$-algebras \[ \operatorname{End}_{\mathbb Q_\ell[G^F]}\!\left(\bigoplus_{i\geq 0}H_c^i(X_P,\mathbb Q_\ell)\right) \cong \mathbb Q_\ell W_S. \] Here the endomorphisms are those of the total representation displayed in the supplied statement. This transcription does not assert the claim for arbitrary parabolics: the admissible data and the meaning of $W_S$ must first be recovered from the source. Consequently, the supplied record alone does not determine a fully specified conjecture.

Short English statement

The requested claim is that distinct compactly supported cohomology degrees of the intended parabolic Deligne--Lusztig variety share no irreducible $G^F$-representation, and that the endomorphism algebra of their direct sum is $\mathbb Q_\ell W_S$. The construction, admissible hypotheses, and group $W_S$ are unspecified. Supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002015$, source problem code catalogue entry, title Cohomological Disjointness for Parabolic Deligne--Lusztig Varieties, in the Oberwolfach Reports Open Problems collection. Project registry: data/unsolvedmath/problems.json; supplementary registry: catalog/problems.json.
• Representations of Finite Groups (2012), Oberwolfach Reports, pp. 963--1019, DOI 10.4171/owr/2012/16. Supplied report reference: https://doi.org/10.4171/owr/2012/16. The record attributes the problem to workshop open-problem discussions and supplies no individual proposer or precise location within the report.
• Supplied literature reference: Parabolic Deligne--Lusztig varieties, https://arxiv.org/abs/1110.4863.
• Supplied literature reference: Translation by the full twist, https://www.sciencedirect.com/science/article/pii/S0001870819305447.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the endomorphism-algebra conjecture is proved in important cases, including the split type-A disjointness consequence, while the full general pairwise-disjointness and endomorphism statement remains open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent checking.