Jantzen filtration

In algebra, 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

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

It has the following properties:

M(λ)¹ is the maximal proper submodule of M(λ)
The quotients M(λ)^k/M(λ)^k+1 have non-degenerate contravariant bilinear forms.
$\sum_{i>0}\text{Ch}(M(\lambda)^i) = \sum_{\alpha>0, s_\alpha(\lambda)<\lambda}\text{Ch}(M(s_\alpha(\lambda)))$

(the Jantzen sum formula)

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
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, http://books.google.com/books?id=8GCP4Ng6risC
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 552943