Let's take a finite group $G$ (I'm mostly interested in the case of nilpotent $G$) and an irreducible representation $V$. I'm interested if in this case it is possible to introduce any analogs of the following notions in Lie theory: highest weight vector, Cartan subalgebra, Borel subalgebra.
This question is motivated by the fact that in many respects finite groups behave as compact Lie groups. I would be interested in driving this analogy as far as possible.
I think the biggest obstruction is a lack of highest-weight vectors, which comes down to a lack of Borel or unipotent subgroups in $G$. In Lie theory, a unipotent subgroup (like the group of upper triangular matrices with 1's along the diagonal) is characterised by having the trivial representation as its unique simple representation. For representations of finite groups over $\mathbb{C}$, this only happens for the trivial group.
The closest thing I have seen to a Cartan subalgebra is in the Okounkov-Vershik approach to the representation theory of the symmetric groups. They define a maximal commutative subalgebra $A_n \subseteq \mathbb{C}[S_n]$, which is kind of like a Cartan subalgebra. The subalgebra $A_n$ is spanned by the Jucys-Murphy elements $J_1, \ldots, J_n$, where $J_k$ is the sum of transpositions $$ J_k = (1, k) + \cdots + (k-1, k).$$ They then show that any irreducible $\mathbb{C}S_n$ representation $V$ breaks up into a sum of one-dimensional "weight spaces", by looking at the simultaneous eigenspaces of the $J_k$. From this they deduce all of the usual results about the representations of $\mathbb{C}S_n$ in terms of standard tableaux and so on.