 
Summary: pFILTRATIONS AND THE STEINBERG MODULE
HENNING HAAHR ANDERSEN
Let k be an algebraically closed eld of characteristic p > 0. Denote by G a
connected and simply connected reductive algebraic group over k. Fix a maximal
torus T in G and let X = X(T ) be the set of characters of T . In X we choose a
chamber X + and call its elements the dominant weights.
For each 2 X + we have a simple module L(), a Weyl module (), a dual
Weyl module r(), and an indecomposable tilting module T (). All these modules
have as their unique highest weight. Moreover, L() is the unique simple quotient
(resp. submodule) of () (resp. r()), and () (resp. r()) occurs as the rst
(resp. last) subquotient in a Weyl (resp. good) ltration of T ().
In this paper we study pltrations of Gmodules. Let M be a nite dimensional
Gmodule. Recall that a good ltration of M is a sequence of submodules
0 = M 0 M 1 M r = M ()
such that M i =M i 1 ' r( i ) for some i 2 X + . If instead the quotients M i =M i 1
have the form L( 0
i
)
r( 1
i ) (p) for some i = 0
