Franklin, James

Homomorphisms between Verma modules in characteristic p.

J. Algebra 112 (1988), no. 1, 58-85.

The composition series of Verma modules and homomorphisms between Verma modules in the case of a complex semisimple Lie algebra were studied by Verma and by Bernstein, Gelfand and Gelfand. The author studies homomorphisms between the Verma modules in characteristic p.

Let Vµ be the Verma module corresponding to an integral weight µ. The main results show:
For a positive root r and an integral weight µ, there exists a nonzero homomorphism Vµ-dr -> Vµ (d in Z+) provided that
(µ+ρ)hr = Mpe+d for some M in Z, e in Z+, and d < pe/N, where N is almost always 1, but may be 2 or 3 for certain high roots, and ρ is half the sum of the dominant roots.

Reviewed by A. S. Dzhumadildaev

Mathematical Reviews