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

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:

References