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 (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

[math]\displaystyle{ M(\lambda)=M(\lambda)^0\supseteq M(\lambda)^1\supseteq M(\lambda)^2\supseteq\cdots. }[/math]

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:

[math]\displaystyle{ \sum_{i\gt 0}\text{Ch}(M(\lambda)^i) = \sum_{\alpha\gt 0, s_\alpha(\lambda)\lt \lambda}\text{Ch}(M(s_\alpha \cdot \lambda)) }[/math]

where [math]\displaystyle{ \text{Ch}(\cdot) }[/math] denotes the formal character.

References

• Beilinson, A. A.; Bernstein, Joseph (1993), "A proof of Jantzen conjectures", in Gelʹfand, Sergei; Gindikin, Simon, I. M. Gelʹfand Seminar, Adv. Soviet Math., 16, Providence, R.I.: American Mathematical Society, pp. 1–50, ISBN 978-0-8218-4118-1, http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf, retrieved 2011-06-15 
• Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, 94, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4678-0, https://www.ams.org/bookstore-getitem/item=GSM-94 
• Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3