# Jantzen filtration

(Redirected from Jantzen conjectures)

In algebra, 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 (1979).

## 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

${\displaystyle M(\lambda )=M(\lambda )^{0}\supseteq M(\lambda )^{1}\supseteq M(\lambda )^{2}\supseteq \cdots .}$

It has the following properties:

• M(λ)1 is the maximal proper submodule of M(λ)
• The quotients M(λ)k/M(λ)k+1 have non-degenerate contravariant bilinear forms.
• ${\displaystyle \sum _{i>0}{\text{Ch}}(M(\lambda )^{i})=\sum _{\alpha >0,s_{\alpha }(\lambda )<\lambda }{\text{Ch}}(M(s_{\alpha }(\lambda )))}$
(the Jantzen sum formula)