We work over an algebraically closed field of characteristic 0.
Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathfrak{p}\supset\mathfrak{m}$ be a parabolic subalgebra, respectively a Levi subalgebra.
There is the adjoint action of $\mathfrak{m}$ on $\mathfrak{g}$ and I would like to know how it decomposes into simples. More precisely, what are the highest weights appearing?
Does anyone know the answer or a reference? (Does it depend on the characteristic of the field?)



adjoint action of a Levi subalgebra
Not sure that there would be a closed formula as Jim has pointed out. You can easily calculate it in examples because you can determine highest weight vectors in $g$ for $m$. Indeed, if $\alpha\_1, \ldots \alpha\_k$ are simple rooots for $m$, you just need to find those roots $\beta$ of $g$ such that no $\beta + \alpha\_i$ is a root of $g$.


This will give you all non-trivial summands. The trivial summands sit in the Cartan and are easy to find as well.


And, if you are lazy or rich or both, just buy [this book](http://rads.stackoverflow.com/amzn/click/0840502281) 


