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:
:$\sum_\text(M(\lambda)^i) = \sum_\text(M(s_\alpha \cdot \lambda))$
: where $\text(\cdot)$ denotes the formal character.

References

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*{{Citation , last1=Jantzen , first1=Jens Carsten , title=Moduln mit einem höchsten Gewicht , publisher=Springer-Verlag , location=Berlin, New York , series=Lecture Notes in Mathematics , isbn=978-3-540-09558-3 , doi=10.1007/BFb0069521 , mr=552943 , year=1979 , volume=750

Lie algebras
Representation theory