Geometry of the Jantzen region in Lusztig's conjecture

The Lusztig Conjecture expresses the character of a finite-dimensional irreducible representation of a reductive algebraic group $G$ in prime characteristic as a linear combination of characters of Weyl modules for $G$. The representations described by the conjecture are in one-to-one correspondence with the (finitely many) alcoves in the intersection of the dominant cone and the so-called Jantzen region. Each alcove has a length, defined to be the number of alcove walls (hyperplanes) separating it from the fundamental alcove (the unique alcove in the dominant cone whose closure contains the origin). This article determines the maximum length of an alcove in the intersection of the dominant cone with the Jantzen region.