Let $G$ be an almost simple algebraic group defined
over ${\Bbb F}_p$ for some prime $p$. Denote by $G_1$
the first Frobenius kernel in $G$ and let $T$ be a
maximal torus. In this paper we study certain
Jantzen type filtrations on various modules in the
representation theory of $G_1T$.
We have such filtrations on the baby Verma modules
$Z(\lambda)$, where $\lambda$ is a character of $T$.
They are obtained via a certain deformation of the
natural homomorphism from $Z(\lambda)$ into its
contravariant dual $Z(\lambda)^\tau$. Using the
same deformation we construct for each projective
$G_1T$-module $Q$ a filtration of the vector space
$F_\lambda(Q)=\text{Hom}_{G_1T}(Z(\lambda)^\tau, Q)$.
We then prove that this filtration may also be
described in terms of the above-mentioned
homomorphism
$Z(\lambda) \rightarrow Z(\lambda)^\tau$
and this leads us to a sum formula for our
filtrations.
When $Q$ is indecomposable with highest weight
in the bottom alcove (with respect to some special
point) we are able to compute the filtrations on
$F_\lambda(Q)$ explicitly for all $\lambda$. This
is then the starting point of an induction which
proceeds via wall crossings to higher alcoves. If
our filtrations behave as expected under such wall
crossings then we obtain a precise relation
between the dimensions of the layers in the
filtrations of $F_\lambda (Q)$ for an arbitrary
indecomposable projective $Q$ and the coefficients
in the corresponding Kazhdan--Lusztig polynomials.
We conclude the paper by proving that the above
results in the $G_1T$ theory have some analogues
in the representation theory of $G$ (where, however,
we have to work with representations of bounded
highest weights) and the corresponding theory for
quantum groups at roots of unity. These results
extend previous work by the first author. 2000 Mathematics Subject Classification:
20G05, 20G10, 17B37.