# Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

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?

• I would have posted this question in the mathematics section, not the physics one. – DanielC Oct 17 '19 at 10:00
• @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... – LFH Oct 17 '19 at 10:17