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Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit $$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$ also equal to $$\exp(\mathfrak{g})\cdot x=\{\mathrm{Ad}_{\exp(y)}x:y\in\mathfrak{g}\}?$$

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    $\begingroup$ Equivalently, do we have $G=EC_x$ where $E$ is the image of the exponential map, and $C_x$ is the stabilizer of $x$. $\endgroup$
    – YCor
    Commented Jan 10, 2019 at 11:55
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    $\begingroup$ Note that the centralizer of a regular semisimple element is a maximal torus in $G$. Is there a reason for limiting the question to regular elements or to the Lie group framework over $\mathbb{C}$? (Naturally the notion of "exponential" would have to be broadened.) $\endgroup$
    – Jim Humphreys
    Commented Jan 10, 2019 at 23:55
  • $\begingroup$ Since $\exp(\mathfrak g)$ is not a group in general, I think, do you really want just $\exp(\mathfrak g)$, or the smallest set containing $x$ and closed under the adjoint action of $\exp(\mathfrak g)$? $\endgroup$
    – LSpice
    Commented Jul 18, 2019 at 15:53

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