Huang's conjecture on parabolic induction from representations with nonzero Dirac cohomology

At least 5 years old · documented by 2020

Let $G$ be a complex classical Lie group, and let a unitary representation of $G$ be given. A unitary representation has nonzero Dirac cohomology if its Dirac cohomology does not vanish. Huang's conjecture. A unitary representation either has nonzero Dirac cohomology or is induced from a unitary representation with nonzero Dirac cohomology by parabolic induction. The paper states that this conjecture, raised by Huang in 2015, is disproved for the groups studied; hence the proposed description of the unitary dual by parabolic induction from representations with nonzero Dirac cohomology is false.

References

Primary source

Chao-ping Dong and Kayue Daniel Wong, “Scattered representations of complex classical Lie groups”, arXiv:2006.07806 (2020).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2024 paper reports counterexamples to Huang’s conjecture, but the available evidence does not confirm that they refute this exact parabolic-induction statement.

Huang raised the conjecture in 2015: every unitary representation without nonzero Dirac cohomology should arise by parabolic induction from representations with nonzero Dirac cohomology. The conjecture concerns the proposed description of the unitary dual for complex classical groups.

2024 counterexamples

A paper dated 2024 reports “some counterexamples” to a conjecture of Huang proposed in 2015. The retrieved description discusses a related conjecture about Dirac indices and does not explicitly identify the exact parabolic-induction formulation above; the claimed disproof is therefore unverified for this problem.

Current status (as of September 2026): A 2024 paper claims counterexamples to a Huang conjecture, but the exact parabolic-induction statement has not been verified as disproved by the retrieved evidence.

Sources

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