Summary: FILTRATIONS ON G 1 TMODULES
HENNING HAAHR ANDERSEN \Lambda AND KANEDA MASAHARU \Lambda\Lambda
Let G be an almost simple and simply connected algebraic group defined and split
over the prime field F p . Choose a split maximal torus T in G and a Borel subgroup
B containing T . We denote the kernel of the Frobenius homomorphism on G (resp.
B) by G 1 (resp. B 1 ).
Recall that the representation theory for G is closely related to the corresponding
theory for G 1 T , see [Ja2]. This paper will mainly be concerned with the latter
theory. More precisely, we study filtrations in the G 1 T setup analogous to those
introduced by the first author in [An3] involving tilting modules for G.
To explain our results we need a little more notation. Let X denote the character
group of T and set R (resp. R + , resp. S) equal to the root system for (G; T ) (resp.
the set of positive roots relative to B + , the Borel subgroup opposite to B, resp. the
set of simple roots in R + ). Write W for the Weyl group.
For each – 2 X we have a standard G 1 Tmodule Z(–) with highest weight –.
It is sometimes called a baby Verma module and it is defined as the G 1 T module
induced by the 1dimensional B 1 Tmodule –. In fact, for each w 2 W we have
such a standard G 1 Tmodule Z w (–) (obtained by replacing B by wBw \Gamma1 ). By using
certain deformations of these modules it was shown in [AJS, Section 6 ] how one may
construct Jantzen filtrations of Z w (–) and prove the corresponding sum formulas.