TITLE: Dimensions of Jordan blocks associated to representations
QUESTION [5 upvotes]: Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks of size $m_1,\dots,m_k$.
Is it possible to describe the partition $m=m_1+m_2+\dots+m_k$, say in terms
of the highest weight vector associated to an irreducible representation $\rho$?

REPLY [3 votes]: As Ben has mentioned, this question can be best formulated in terms of restrictions
of finite-dimensional simple highest weight modules over a complex semisimple Lie algebra to a principal $sl_2$-subalgebra.
A classical reference:

Kostant, Bertram, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81, 1959, 973-1032, doi:10.2307/2372999 (collected papers version: https://doi.org/10.1007/b94535_11)

More refined information about these restrictions is contained in the $q$-analogues of weight multiplicities introduced by Lusztig,

Lusztig, George, Singularities, character formulas, and a q-analog of weight multiplicities. Asterisque 101-102, 208-229, Numdam

This is explained in Sec 4 of Ginzburg's paper,

Victor Ginzburg, Perverse sheaves on a Loop group and Langlands' duality, arXiv:alg-geom/9511007