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 V_µ 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 V_µ-dr -> V_µ (d in Z⁺) provided that
(µ+ρ)h_r = Mp^e+d for some M in Z, e in Z⁺, and d < p^e/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