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*