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

${\displaystyle M(\lambda )=M(\lambda )^{0}\supseteq M(\lambda )^{1}\supseteq M(\lambda )^{2}\supseteq \cdots .}$

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:

${\displaystyle \sum _{i>0}{\text{Ch}}(M(\lambda )^{i})=\sum _{\alpha >0,s_{\alpha }(\lambda )<\lambda }{\text{Ch}}(M(s_{\alpha }\cdot \lambda ))}$

where ${\displaystyle {\text{Ch}}(\cdot )}$ denotes the formal character.

References

• Beilinson, A. A.; Bernstein, Joseph (1993), "A proof of Jantzen conjectures" (PDF), in Gelʹfand, Sergei; Gindikin, Simon (eds.), I. M. Gelʹfand Seminar, Adv. Soviet Math., 16, Providence, R.I.: American Mathematical Society, pp. 1–50, ISBN 978-0-8218-4118-1, archived from the original (PDF) on 2015-07-09, 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, MR 2428237
• 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, MR 0552943