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

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 ρIrr(G)\rho\in\operatorname{Irr}(G), there exists χIrr(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.

Sources & referencesView supporting material

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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