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 (eds.), I. M. Gelʹfand Seminar (PDF), Adv. Soviet Math., vol. 16, Providence, R.I.: American Mathematical Society, pp. 1–50, ISBN 978-0-8218-4118-1
Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, vol. 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, vol. 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3, MR 0552943