Is there any way to determine the "effeciency" of Jantzen's sum formula?

Question

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a reductive algebraic group over $k$.

In order to determine the structure of the Weyl modules $V(\lambda)$, usually the best tool in specific situations will be Jantzen's sum formula, which states that $V(\lambda)$ has a filtration $$V(\lambda) = V(\lambda)^0\supseteq V(\lambda)^1\supseteq V(\lambda)^2\supseteq \cdots$$ such that $V(\lambda)^0/V(\lambda)^1\cong L(\lambda)$ and such that $$\sum_{i > 0}\operatorname{ch}V(\lambda)^i = \sum_{\alpha\in R^+}\sum_{0 < mp < \langle\lambda+\rho,\alpha^{\vee}\rangle}\nu_p(mp)\chi(s_{\alpha,mp}\cdot\lambda)$$ (notation as in Jantzen's Representations of Algebraic Groups. If any of it needs further explanation, let me know. The statement is Proposition II.8.19).

Now, if you use this formula and is lucky, this allows you to determine the submodule structure of $V(\lambda)$. But in many cases, you will end up with something not quite conclusive.
Some things that might help get something conclusive in more cases would be to have bounds (both upper and lower) on two things:
1. The length of the filtration
2. The number of improper inclusions (ie, for how many $i$ does it happen that $V(\lambda)^i = V(\lambda)^{i+1}$)

Is anything known about such bounds (other than the obvious ones)?

An example that illustrates the sort of problem one can get: Let $G = SL_3$, $p = 2$ and $\lambda = (2,2)$ (all weights will be written in terms of the fundamental weights).
Now we get a filtration where the sum is $\chi(3,0) + \chi(0,3) + \chi(0,0)$ and one can check that (writing $\psi(\mu) = \operatorname{ch}L(\mu)$) that this is then $\psi(3,0) + \psi(0,3) + 3\psi(0,0)$, which means that we do not yet quite know how many times the trivial module occurs in $V(\lambda)$.
There are various ways to still do this, and it turns out that it only occurs once (one can for example argue by dimensions). This means that the filtration one has looks like $$V(\lambda) = V(\lambda)^0 \supseteq V(\lambda)^1\supseteq V(\lambda)^2 \supseteq V(\lambda)^3\supseteq 0$$ where $V(\lambda)^2 = V(\lambda)^3 = L(0,0)$ (the module $V(\lambda)^1$ has a quite interesting structure btw), so in some sense, we are overcounting the trivial module quite a bit, and probably this sort of example could get even worse if one was to take larger examples.

Answer 1

I think it's misleading to regard Jantzen's filtration in characteristic $p$ as a computational tool in finding the "structure" of a Weyl module, by which I guess you mean the composition factor multiplicities. (The full submodule structure is a daunting problem.)

It's instructive to compare the relatively easier situation in characteristic 0, for Verma modules and their composition factor multiplicities. Ultimately this was solved recursively in terms of the Kazhdan-Lusztig polynomials for the Weyl group $W$, though it's by no means easy to get explicit results in most cases. Here too there is a Jantzen filtration and related conjecture (not asserted in so many words by him), that canonical maps between Verma modules should respect the two Jantzen filtrations. Though Beilinson-Bernstein were able to prove this, it took a stronger version of their arguments for the Kazhdan-Lusztig Conjecture (which then follows from Jantzen's Conjecture). In any case, the results here are mainly of theoretical importance, showing the role of the Weyl group and Hecke algebra.

In characteristic $p$, Jantzen's filtration and sum formula are also primarily of theoretical importance. They bring out the key role of multiple affine Weyl groups, one for each power of $p$ relevant to the highest weight involved. Along with Steinberg's tensor product theorem, these ideas are probably the only deep ones known to hold for all primes and all highest weights. (Though in Jantzen's original formulation there were conditions of $p$, later work by Andersen in the sheaf cohomology context removed all conditions. See Andersen's later paper Jantzen's filtration of Weyl modules, Math. Z. 194 (1987). 127-142, and its references.)

The sheaf cohomology viewpoint goes pretty far here theoretically, but again it leaves the computational problems far behind. Andersen and I wrote loosely related papers with the same title On the structure of Weyl modules, mine in Comm. Algebra 12 (1984). Here I proposed a type of filtration in an arbitrary Weyl module which would involve sheaf cohomology groups in all degrees, which Andersen partly confirmed (while running into problems with torsion). This approach might have the advantage of making sense of why Luszxtig's conjecture for weights in a certain region relative to $p$ should become true for large enough $p$, in terms of the interaction of many affine Weyl groups. But there are more quetions than answers here.