$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition: Two finite subgroups $H_1,H_2$ of $G$ are said to be almost conjugate in $G$ if $|H_1\cap [g]_G| = |H_2\cap [g]_G|$ for all $g\in G$, where $[g]_G=\{hgh^{-1}: h\in G\}$ is the conjugacy class of $g$ in $G$.
This is equivalent to
- the left regular representations $L^2(G/H_1)$ and $L^2(G/H_2)$ are equivalent (as $G$-modules).
- there is a bijection $\varphi:H_1\to H_2$ such that $h$ and $\varphi(h)$ are conjugate in $G$ for all $h\in H_1$.
General question: Which simply-connected compact simple Lie groups admits almost conjugate but not conjugate finite subgroups?
Here is what I know so far. From a classical example by Gassmann, we have the following almost conjugate (but not conjugate) subgroups of $\SO(6)$: \begin{equation} \begin{aligned} H_1 &= \left\{ {I}_{6}, \; \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &&&& 1\\ &&&&& 1 \end{smallmatrix} \right), \; \left( \begin{smallmatrix} 0&&1&\\ &0&&1 \\ 1&&0&\\ &1&&0\\ &&&& 1\\ &&&&& 1 \end{smallmatrix} \right), \; \left( \begin{smallmatrix} 0&&&1\\ &0&1 \\ &1&0\\ 1&&&0\\ &&&& 1\\ &&&&& 1 \end{smallmatrix} \right) \right\}, \\ % % % H_2 &= \left\{ {I}_{6}, \; \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &&&&1 \\ &&&&&1 \end{smallmatrix} \right), \; \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&1&&&\\ &&&1&&\\ &&&&0&1\\ &&&&1&0 \end{smallmatrix} \right), \; \left( \begin{smallmatrix} 1&&&\\ &1& \\ &&0&1&\\ &&1&0\\ &&&&0&1&\\ &&&&1&0 \end{smallmatrix} \right) \right\}. \end{aligned} \end{equation}
Their preimages $\widetilde H_1,\widetilde H_2$ of the $2$-cover $\textrm{Spin}(6)\to \SO(6)$ are almost conjugate (but not conjugate) in $\textrm{Spin}(6)\simeq \SU(4)$. Moreover, $\widetilde H_1,\widetilde H_2$ are almost conjugate and non-conjugate subgroups of
- $\SU(n)$ for any $n\geq4$ via the standard embedding $\SU(4)\hookrightarrow \SU(n)$,
- $\textrm{Spin}(n)$ for $n\geq 6$ via $\textrm{Spin}(6)\hookrightarrow \textrm{Spin}(n)$,
- $\Sp(n)$ for $n\geq4$ via $\SU(4)\hookrightarrow U(4)\hookrightarrow \Sp(n)$,
- ${E}_n$ for $n=6,7,8$ via $\SO(6)\hookrightarrow \SO(10)\hookrightarrow {E}_6\hookrightarrow {E}_7\hookrightarrow {E}_8$, and
- ${F}_4$ via $\textrm{Spin}(4) \hookrightarrow \textrm{Spin}(9)\hookrightarrow {F}_4$.
I hope this reasoning is correct.
One can easily see from the classification of finite subgroups (up to conjugation) of $\SU(2)$ that almost conjugate subgroups of $\SU(2)$ are necessarily conjugate.
The only remaining simply-connected compact simple Lie groups are $\SU(3)$, $\Sp(2)\simeq \textrm{Spin}(5)$, $\Sp(3)$, and $G_2$.
Main question: Are there almost conjugate and non-conjugate finite subgroups of $\SU(3)$ (resp. $\Sp(2)$, $\Sp(3)$, and $G_2$)?
The interest on this group theoretical question is its application to inverse spectral geometry. Namely, the manifold $G/H_1$ and $G/H_1$ endowed with the restriction of a bi-invariant metric on $G$ are isospectral, that is, their Laplace-Beltrami operators have the same spectra.
