Jantzen filtration

In representation theory, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtrations were introduced by.

Jantzen filtration for Verma modules

If M is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration
It has the following properties:

M¹=N, the unique maximal proper submodule of M
The quotients Mⁱ/Mⁱ⁺¹ have non-degenerate contravariant bilinear forms.
The Jantzen sum formula holds: