Quasi-projective Brauer characters

We study p-Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p-singular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character φ there exists a minimal p-power, say pa(φ), such that pa(φ)φ is quasi-projective. The exponent a(φ) only depends on the Cartan matrix of the block to which φ belongs. Moreover pa(φ) is bounded by the vertex of the module affording φ, and equality holds in case that G is p-solvable. We give some evidence for the conjecture that a(φ)=0 occurs if and only if φ belongs to a block of defect 0. Finally, we study indecomposable quasi-projective Brauer characters of a block B. This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B.