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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_{i0}\text{Ch}(M(\lambda)^i) = \sum_{\alpha0, s_\alpha(\lambda) : where \text{Ch}(\cdot) denotes the formal character.
References
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