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This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting on some Hilbert space $\mathcal{H}$. Given a Lie algebra element $A$, I will refer to its induced representation as $\hat {A}$.

Given a state $|\psi_0\rangle$, I can define the sub Lie algebra \begin{align} \mathfrak{h}=\left\{A\in\mathfrak{g}\,\big|\,\hat{A}|\psi_0\rangle=0\right\}\,. \end{align} Due to the fact that $\mathfrak{g}$ is semi-simple, its Killing form $K$ is non-degenerate, so I can define the unique orthogonal complement \begin{align} \mathfrak{h}_{\perp}=\left\{A\in\mathfrak{g}\,\big|\,K(A,B)=0\forall B\in\mathfrak{h}\right\}\,. \end{align} Clearly, I have $\hat{A}|\psi_0\rangle=\hat{A}_{\perp}|\psi_0\rangle$, where $A_{\perp}$ is the orthogonal projection onto $\mathfrak{h}_{\perp}$ of $A$.

I would like to pull back the Hilbert space inner product onto the Lie algebra, i.e., my parametrization $U(g)|\psi_0\rangle$ describes a sub manifold in Hilbert space. At the identity, i.e., at $|\psi_0\rangle$ the tangent space is given by \begin{align} \mathcal{T}_{|\psi_0\rangle}\mathcal{M}=\{\hat{A}|\psi_0\rangle\,|\,A\in\mathfrak{g}\}=\{\hat{A}|\psi_0\rangle\,|\,A\in\mathfrak{h}_{\perp}\}\,. \end{align} In concrete examples, I found the relationship \begin{align} G(A,B)=2\mathrm{Re}\langle\psi_0|\hat{A}^\dagger\hat{B}|\psi_0\rangle=-K(A_{\perp},B_{\perp})\,, \end{align} i.e., the pull back of the Hilbert space inner product (real part of it) is given by the negative Killing form evaluated on the orthogonal complement of the stabilizer Lie algebra $\mathfrak{h}$ associated to $|\psi_0\rangle$.

My intuition is that this should be true in general (under certain conditions, such as $|\psi_0\rangle$ being highest weight or similar. To prove this or find the relevant conditions, I would need to compute the expression \begin{align} G(A,B)=2\mathrm{Re}\langle\psi_0|\hat{A}^\dagger\hat{B}|\psi_0\rangle=-\langle\psi_0|\hat{A}_{\perp}\hat{B}_{\perp}+\hat{B}_{\perp}\hat{A}_{\perp}|\psi_0\rangle\,, \end{align} where I used that the representations of generators must be anti-Hermitian.

That's the point where I am stuck. Any suggestions how to proceed or references that discuss this issue?

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  • $\begingroup$ I would have posted this question in the mathematics section, not the physics one. $\endgroup$ – DanielC Oct 17 '19 at 10:00
  • $\begingroup$ @DanielC: Well, when searching I found this question (physics.stackexchange.com/questions/221851/…) which is slightly related and similar in style. The respective question in the math section didn't receive many answers, so I thought that physics section is more promising... $\endgroup$ – LFH Oct 17 '19 at 10:17

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