Dirac Cohomology of Tempered Modules

Definitions and mathematical statement

Source question. Does every tempered module with a solvable central character have nonzero Dirac cohomology?

Conditional formulation. Fix the ambient algebra or representation-theoretic setting intended by the source. Let $\mathcal T$ denote its class of tempered modules, and let $\chi_X$ denote the central character of $X$, whenever defined. Write $S(\chi_X)$ for the source's condition that this central character is "solvable." If the relevant Dirac operator is $D_X:V_X\to V_X$, its Dirac cohomology is \[ H_D(X):=\frac{\ker D_X}{\ker D_X\cap\operatorname{im}D_X}. \] The proposed assertion has the form \[ \forall X\in\mathcal T,\qquad S(\chi_X)\Longrightarrow H_D(X)\ne 0. \] Here $V_X$ denotes the space on which the intended Dirac operator acts; its construction is not specified in the supplied record.

Unresolved qualifications. The record does not identify the ambient algebra, its parameters, the relevant reflection group, the definition of temperedness, or whether the modules must be irreducible. It also does not define "solvable central character." The background mentions parameters with solvable centralizer, but supplies neither the object being centralized nor its relationship to a central character. These phrases therefore cannot be identified without additional source evidence. The displayed assertion is a formulation schema, not a fully specified conjecture.

The supplied background describes a conjecture for arbitrary parameters and reflection groups and reports results for parameters with solvable centralizer and related special cases. It does not provide the hypotheses or constructions needed to turn that description into additional precise quantifiers.

Short English statement

The question asks whether temperedness together with the source's "solvable central character" condition forces nonzero Dirac cohomology. A precise conjecture requires the intended module category, Dirac operator, and solvability condition. The supplied record labels the problem open; that label and its literature assessment are not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $30002484$, source problem code catalogue entry, title Dirac Cohomology of Tempered Modules; supplied collection: Oberwolfach Reports Open Problems. The supplied record lists proposed year 2014 and names no proposer.
• Supplied report citation: Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras (2014), Oberwolfach Reports, pp. 145--212, DOI 10.4171/owr/2014/03. https://doi.org/10.4171/owr/2014/03. The record describes these reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or passage is supplied.
• Supplied literature source, labeled "Oberwolfach representation theory report": https://ems.press/content/serial-article-files/46494?nt=1.
• Supplied literature source, labeled "Chan, twisted Euler--Poincare pairing": https://arxiv.org/abs/1407.0956.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports Dirac-cohomology results for parameters with solvable centralizer and related special cases, and describes the conjecture for arbitrary parameters and reflection groups as unproved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.