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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 $\rho$?

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The answer is yes. The Jordan block decomposition of the generic nilpotent in $SL_m$ on a representation is the same as the decomposition of any representation under the principal $SL_2$ (which is a map of $SL_2$ to $SL_n$ which sends the generic nilpotent in $SL_2$ to a generic one in $SL_n$; people usually have a particular one in mind, but they are all the same up to conjugation by Jacobson-Morozov).

This can be extracted from the formula for the character of the principal $SL_2$ usually called the "quantum Weyl dimension formula"

$$\chi(V_\lambda)=\frac{\prod_{\alpha\in \Delta^+}q^{\langle\rho+\lambda,\alpha\rangle}-q^{-\langle\rho+\lambda,\alpha\rangle}}{\prod_{\alpha\in \Delta^+}q^{\langle\rho,\alpha\rangle}-q^{-\langle\rho,\alpha\rangle}}$$

One "only" needs to expand this out in terms of the characters of the irreducible $SL_2$ reps $\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}$.

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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

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

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

Victor Ginzburg, Perverse sheaves on a Loop group and Langlands' duality

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