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 Template:Harvs.

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.

References

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