Malle's degree-matching conjecture for cuspidal characters of finite Chevalley groups

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Let q=pfq=p^f be a power of the prime pp, let GG be a finite Chevalley group defined over Fq\mathbb F_q with pp a bad prime for GG, and let UU be a Sylow pp-subgroup of GG. For a character degree ρ(1)\rho(1), write ρ(1)p\rho(1)_p for its pp-part. Malle's conjecture. For every cuspidal character ρ∈Irr⁡(G)\rho\in\operatorname{Irr}(G), there exists χ∈Irr⁡(U)\chi\in\operatorname{Irr}(U) such that

χ(1)=ρ(1)p.\chi(1)=\rho(1)_p.

The conjecture predicts a strong connection between cuspidal character degrees of finite groups of Lie type and the degrees of irreducible characters of their Sylow pp-subgroups, supporting the use of induced characters from UU in studying modular representations and decomposition matrices. Its resolution status is not specified in the source.

References

Primary source

Tung Le, Kay Magaard and Alessandro Paolini, “On the characters of Sylow p-subgroups of finite Chevalley groups G(p^f) for arbitrary primes”, arXiv:1904.00638 (2019).

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