Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $\rho$?

share|improve this question

2 Answers 2

up vote 4 down vote accepted

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 $\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}$.

share|improve this answer

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.