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Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?

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Representations of Nilpotent Lie Groups and Their Applications. L. Corwin, F. Greenleaf. Chapter 3. –  Reimundo Heluani Apr 9 '13 at 19:15
This has been an influential book, with MathSciNet linking about 150 citations of it. The main theme is harmonic analysis on nilpotent Lie groups, with Kirillov's coadjoint orbit method being an essential model. The full reference is Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge Studies in Advanced Mathematics, 18. Cambridge University Press, Cambridge, 1990. (But there is no Part II. After Larry Corwin's premature death, a 1993 Rutgers conference was held in his memory.) –  Jim Humphreys Apr 9 '13 at 23:10
@ Jim, thanks , I will follow your references in library –  Hassan Jolany Apr 9 '13 at 23:13
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2 Answers

up vote 2 down vote accepted

Let me first say, that I am not an expert, but was interested in the same question recently, so I would also be happy if someone provides more info on the question. Let me collect some facts, which I know.

1) As far as I understand the general classification of orbits is in certain sense "wild" problem.

2) Classification up to n=7 can found in Coadjoint orbits of the group $UT(7,K)$ (2006) , further papers by A.Panov and his students give partial results on general $n$, e.g. these ones Involutions in $S_n$ and associated coadjoint orbits Diagram method in research on coadjoint orbits (2009)

3) There is a lots of recents studies which are "related" to the question. Especially in the case of ground field is finite. In such a case people are greatly interested in understanding representation theory of U(n,F_q) and in particular of the "orbit method" approach to it, and hence in coadjoint orbits. See some comments at mathoverflow question: Finite Unipotent Groups: References.

4) If you restrict to ground field to be finite, then it is worth to mention several facts: a) number of adjoint and coadjoint orbits is the same b) it is the same with the number of conjugacy classes in the group c) hence the same as number of irreps d) it is related to interesting combinatorics, see e.g. paper by A.A. Kirillov, A. Melnikov On a Remarkable Sequence of Polynomials and other papers by these and other authors.

For the finite field, these MO questions, related: Finite Unipotent Groups: References, Representation theory of p-groups in particular upper tringular matrices over F_p Irreducible representations of the unitriangular group.

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Suppose that $G=\operatorname{GL}_n(\mathbb{C})$. Since two $n\times n$ complex matrices are conjugate if and only if they have the same Jordan canonical form, a classifying invariant of your orbits is the Jordan canonical form. In particular, every orbit contains an upper-triangular matrix and a lower-triangular matrix. So, you obtain no fewer orbits by restricting to those containing upper (or lower)-triangular matrices.

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It is also possible that I have misinterpreted your question. Are you asking about the orbits of the subgroups of upper and lower-triangular matrices? If this is the case, you might first think about the Borel subgroup of $\operatorname{GL}_n(\mathbb{C})$ consisting of the upper-triangular invertible matrices. This is a solvable Lie group, and considerable work has been done concerning the structure of their homogeneous spaces. Consider the book Lie Groups and Lie Algebras III by Onishchik and Vinberg. –  Peter Crooks Apr 9 '13 at 19:16
@PDC: I think the question refers to coadjoint orbits for the unipotent group. So you're not allowed to conjugate by elements of GL_n. The complete characterization is in the reference pointed above in the comments. –  Reimundo Heluani Apr 9 '13 at 19:40
Yes I am agree with Reimundo –  Hassan Jolany Apr 9 '13 at 19:41
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