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(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing filtration
 * $$M(\lambda)=M(\lambda)^0\supseteq M(\lambda)^1\supseteq M(\lambda)^2\supseteq\cdots.$$

It has the following properties:
 * M(λ)1=N(λ), the unique maximal proper submodule of M(λ)
 * The quotients M(λ)i/M(λ)i+1 have non-degenerate contravariant bilinear forms.
 * The Jantzen sum formula holds:
 * $$\sum_{i>0}\text{Ch}(M(\lambda)^i) = \sum_{\alpha>0, s_\alpha(\lambda)<\lambda}\text{Ch}(M(s_\alpha \cdot \lambda))$$
 * where $$\text{Ch}(\cdot)$$ denotes the formal character.