I am struggling with understanding representations of parabolic subgroups of algebraic groups. We work over complex numbers. The case that I have in mind is $SL(V)$ where $V$ is a vector space of dimension $n$.
I present a couple of examples in the way that I understand them. I would like to know how far from understanding them correctly I am.
We can identify $SL(V)$ with $n\times n$ matrices with determinant $1$. I fix the torus of diagonal matrices, the Borel $B$ of upper triangular; for every subset $I$ of simple roots, let $P_I$ be the standard parabolics whose algebra is generated by the Borel $\mathfrak{b}$ and the opposites of the simple roots NOT in $I$. So $P_I$ can be identified with "block upper triangular" matrices whose structure is defined by $I$. If $I$ is the whole set of simple roots, then $P_I = B$. If $I = \emptyset$ then $P_I = SL(V)$.
As far as I understand, weights for $SL(V)$ (which correspond to partitions with at most $n-1$ parts), define representations for the standard parabolics. I would like to understand better how this correspondence works.
I start with the Borel. Let $\lambda = (\lambda_1 , ... , \lambda_{n-1})$ be a partition and let $\lambda_n = - (\lambda_1 + \cdots + \lambda_n)$. If $\mathbf{F} = (0 = F_0 \subsetneq F_1 \subsetneq \cdots \subsetneq F_n = V)$ is the flag fixed by the Borel (so for upper triangular matrices $F_j = \langle e_1,...,e_j \rangle$ where the $e_i$'s are the standard basis of $V = \mathbb{C}^n$), then the irreducible representation of $B$ defined by $\lambda$ should be $$ F_1^{\otimes \lambda_1} \otimes (F_2/F_1)^{\otimes \lambda_2} \otimes \cdots \otimes (F_{n-1}/F_{n-2})^{\otimes \lambda_{n-1}} \otimes (F_n/F_{n-1})^{\otimes \lambda_n}. $$ Is this correct?
Now, what about maximal parabolics, so that $I$ consists of a single simple root, or equivalently a single node of the Dynkin diagram? I understand that if $I$ contains the $k$-th node, then the semisimple part of the Levi corresponding to $P_I$ is the product $SL_k \times SL_{n-k}$ of block diagonal matrices (whose type is obtained by erasing the marked node from the Dynkin diagram) - this is consistent with what happens in the case of the Borel, where one marks all the nodes of the Dynkin diagram: indeed the Levi in that case is the torus, which has trivial semisimple part. In this case, if $\lambda$ is a partition, what is the corresponding representation for $P_I$? I would expect to write it as a representation for $SL_k \times SL_{n-k}$ tensor something that depends on the degree which is acted on by the center of the Levi (something like the factor with $\lambda_n$ in the case of the Borel), but I am not quite able to write it down.