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Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\subset \mathfrak{t}_\mathbb{C}^\ast$ be the associated root system (where $_\mathbb{C}$ denotes complexification). Let now $\mathfrak{g}'\subset \mathfrak{g}$ be a compact subalgebra with the property that $\mathfrak{t}':=\mathfrak{g}'\cap \mathfrak{t}$ is maximal abelian in $\mathfrak{g}'$. Then the root system $\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})\subset {\mathfrak{t}'_\mathbb{C}}^\ast$ can be regarded as a subset of $\mathfrak{t}_\mathbb{C}^\ast$ by identfying ${\mathfrak{t}'_\mathbb{C}}^\ast$ with the elements in $\mathfrak{t}_\mathbb{C}^\ast$ that are supported on $\mathfrak{t}'_\mathbb{C}$.


How large is the intersection $\;\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})\cap\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})$ compared to $\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})$ ?


Edit: The initial question was whether one has $\;\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})\subset\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})$, but this is rarely true, see the comment by Jeffrey. However, it still seems to me that the intersection should not be empty, at least, and the number of elements in it should be about half the number of elements in the smaller root system.

Background: The original setting where my question comes from is the following. A compact connected Lie group $G$ acts smoothly on a smooth manifold $M$. Let $T\subset G$ be a maximal torus in $G$, which then also acts on $M$, and let $\mathfrak{t}$ be its Lie algebra. For a point $p$ in $M$ with stabilizer group $G_p$ and associated stabilizer algebra $\mathfrak{g}_p$ it is easy to show that $\mathfrak{t}_p:=\mathfrak{t}\cap \mathfrak{g}_p$, which is the Lie algebra of the stabilizer group $T_p$ of $p$ with respect to the $T$-action, is maximal abelian in $\mathfrak{t}$ (i.e., the identity component of $T_p$ is a maximal torus in $G_p$). I would like to know under which circumstances the root system $\Delta({\mathfrak{g}_p}_\mathbb{C},{\mathfrak{t}_p}_\mathbb{C})$ is contained in $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})$.

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    $\begingroup$ This is very far from being true. Every compact group embeds in U(n) for some n, but most root systems don't embed in type A. $\endgroup$ – Jeffrey Adams Aug 10 '17 at 19:22
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    $\begingroup$ As Jeff Adams points out, this will be far from true. (I wonder if it can be made clearer without the "compact" set-up, just as a question about root systems in the Bourbaki sense? Or is the compact origin essential?) $\endgroup$ – Jim Humphreys Aug 10 '17 at 20:16
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    $\begingroup$ The root system $\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})\subset {\mathfrak{t}'_\mathbb{C}}^\ast$ should not be regarded as a subset of $\mathfrak{t}_\mathbb{C}^\ast$. $\endgroup$ – Mikhail Borovoi Aug 11 '17 at 0:10
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    $\begingroup$ Instead, the embedding $\frak t'\hookrightarrow \frak t$ induces a map $\frak t_{\Bbb C}^*\to (\frak t'_{\Bbb C})^*$, and you may ask whether the image of $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})$ contains $\Delta(\mathfrak{g}'_\mathbb{C},\mathfrak{t}'_\mathbb{C})$ or contains in a large subset of it (in some sense). $\endgroup$ – Mikhail Borovoi Aug 11 '17 at 0:19
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    $\begingroup$ Therefore, I suggest that you edit your question again... $\endgroup$ – Mikhail Borovoi Aug 11 '17 at 0:20

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