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

M

(

λ

)

=

M

(

λ

)

0

⊇

M

(

λ

)

1

⊇

M

(

λ

)

2

⊇

⋯

.

{\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:

∑

i

>

0

Ch

(

M

(

λ

)

i

)

=

∑

α

>

0

,

s

α

(

λ

)

<

λ

Ch

(

M

(

s

α

⋅

λ

)

)

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

where

Ch

(

⋅

)

{\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., vol. 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, 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.