# highest weight orbit characterization (reformulated and extended )

Edit 1: I think that the question was not stated clearly enough so modified it a little

Edit 2: I thought over the physics that lies behind this question which led me to reformulation of the original problem. Orbit itself in non physically significant. What really counts is its image in projective space!

Edit 3: I changed the title and added new part of the question (see below). It is directly related to the original problem ( which was answered by David Bar Moshe) so I decided not to make it a separate question.

Edit 4: I erased the second part of the question and made it into a separate one - see here. Sorry for this mess...

Setting:

Let $G_0$ be a compact, simply connected Lie group giving rise to a semi-simple Lie Group $G$ (its Lie algebra I denote by $\mathfrak{g}$). Let $V_{\lambda}$ be a finite dimensional complex vector space on which $\mathfrak{g}$ is irreducibly represented (with the highest weight $\lambda$ and highest weight vector $v_\lambda$).

Context:

In the article by Lichteinstein:

http://www.ams.org/journals/proc/1982-084-04/S0002-9939-1982-0643758-8/S0002-9939-1982-0643758-8.pdf

there is a quadratic criterion that enables one to say whether a given vector $v\in V_\lambda$ belongs to the orbit of $G$ through the highest-weight vector $v_\lambda$. It says that a given $v$ is on the orbit of our interest iff:

$\Omega (v\otimes v) =\langle 2\lambda+2\delta,2\lambda\rangle (v\otimes v)$ (that's the equation (1) in this article)

$\Omega$ is the representation of the second order Casimir operator (treated as a member of universal enveloping algebra of $\mathfrak{g}$) in $V_{\lambda}\otimes V_{\lambda}$.

$\delta=\frac{1}{2}\sum_{\alpha>0}\alpha$ (summation over all possitive roots of $\mathfrak{g}$)

$\langle\cdot ,\cdot \rangle$ - a standard inner product on Cartan algebra dual $\mathfrak{h}^*$ .

Question:

Let $\mathbb{P}V_\lambda$ be a complex projective space of $V_\lambda$ and let $\pi:V_\lambda\rightarrow \mathbb{P}V_\lambda$ be a canonical map to projective space. Define:

$O_{v_{\lambda}}$ - orbit of $G$ through $v_\lambda$

$O_{v_\lambda}^0$ - orbit of $G_0$ through $v_\lambda$

Is it true that $\pi(O_{v_\lambda}) = \pi(O_{v_\lambda}^0)$ ?

In physics articles I came across criterion stated in the "context" part is interpreted as a necessary and sufficient condition for a given $v$ to be precisely on the orbit of $G_0$ through $v_\lambda$. In the quantum mechanical context what really matters are images of vectors from $V_\lambda$ in associated projective space (phase and normalization do not play a role ) - that's why physically interesting object is associated with the image of actual orbit in projective space.

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The linked article uses the associated projective space, where the point corresponding to a vector yields the orbit in question. The language here has to be used more carefully, I think. –  Jim Humphreys Dec 15 '10 at 22:23
As far as I understand projective space is used in a proof of the theorem. In particular in order to justify that $\pi(\mathcal{W})$ $\subset \pi(\mathcal{O}$ $_{v_{\lambda} } )$. I am pretty sure that the theorem concerns the orbit of a natural action of $G$ in $V_\lambda$. Projective space is just a way to handle action of group elements of the form $e^H,\ H\in\mathfrak{h}$ which simply multiply $v_\lambda$ by a constant. –  Michal Oszmaniec Dec 15 '10 at 22:44
Sorry: I meant $\mathcal{W}\subset\mathcal{O}$$_{v_\lambda} rather than \pi(\mathcal{W})\subset\pi(\mathcal{O}$$_{v_\lambda})$ –  Michal Oszmaniec Dec 15 '10 at 22:49
Michael, can you please clarify what is the group G and its relation to G0 (what is meant by "gives rise") –  David Bar Moshe Dec 16 '10 at 10:42
Ah! sorry - $G$ is a complexification of $G_0$ –  Michal Oszmaniec Dec 16 '10 at 10:47

Let $H$ be the isotropy group of the highest weight ray $\pi(v_\lambda)$ in $G_0$ and $P$ the highest weight ray isotropy group in $G$. It is easy to see that $H$ is the centralizer of the torus generated by the coroots corresponding to the nonvanishing components of $\lambda$ in the weight basis, which is a consequence of the fact that $v_\lambda$ is annihilated by the all positive roots generators and by the negative root generators corresponding to the vanishing weight components of $\lambda$. By the same reasoning $P$ is a parabolic subgroup of $G$ whose Lie algebra is the union of the complexification of the $H$ Lie algebra and the Borel subalgebra of the positive root generators.

Thus, what we are really looking for is the standard isomorphism between $G_0/H$ and $G/P$ as real homogeneous spaces. A sketch of the proof is given nicely for the nondegenerate case by Hogreve,Muller,Potthoff,Schrader in Commun. Math. Phys. 131, 465-494 (1990) , (section 2.3). I'll try to repeat it here for completeness:

On one hand the real span of the Lie algebras of $G_0$ and $P$ is the whole of $G$, thus the $G_0$ orbit is open in the $G$ orbit, on the other hand it is closed by being compact, thus the orbits are the same.

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Thanks a lot for explanation and a reference. Although I am not able to fully follow your reasoning, parts of it coincide with by own "naive thoughts" on this problem (I have a really small experience in this field). Yet, at least I can see a direction in which I should go now. Btw. It seems that in the first part of the article I'm referring to covers more or less what you gave in your answer. –  Michal Oszmaniec Dec 16 '10 at 12:14
Just to be sure: by a non degenerate case you understand situation when $H=T$ , right?Could you advice me some introductory textbooks that could help me to understand me this part of the article in comm.math.phys? I am able to follow section 2.3 up to the introduction of Borel algebra. My background in this field is very tiny: I read some parts of: amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/… and amazon.com/… –  Michal Oszmaniec Dec 16 '10 at 23:44
There exists a huge amount of literature on the role of the spaces G/P in quantum theory. I can recommend the following article by: M. Bordemann, M. Forger, and H. Romer: projecteuclid.org/… with an emphasis on their Kahler geometry. Also, the following lecture notes treat the special cases of flag manifolds which capture most of their important properties. math.uregina.ca/~mareal/flag-coh.pdf You cann see also the N. Hurt book: "Geometric quantization in action" –  David Bar Moshe Dec 19 '10 at 10:58
I forgot the question in the first part of your last comment. Yes, the nondegenerate case refers to H = T. In the case of G0 = SU(n), the space G/P is the adjoint orbit of a nondegenerate Hermitain matrix with distinct eigenvalues. As an orbit of a highest weight vector, this case refers to the situation where the highest weight has nonvanishing components along all the fundamental weights. –  David Bar Moshe Dec 19 '10 at 11:24
The orbit structure through a non-Highest weight vector is governed by the Kostant-Sternberg theorem: Kostant, B. and Sternberg, B.: Symplectic projective orbits. New directions in applied mathematics (Hilton, P. J. and Young, G. S., eds.) New York: Springer, 1982, pp 81–84, which states that the orbit through a weight vector $\psi_\lambda$ is symplectic (and a consequence Kahler)iff the stabilizer of $\psi_\lambda$ is the same as the stabilizer of $\lambda$, which is true for highest weights and regular weights. –  David Bar Moshe Jan 25 '11 at 13:07