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added a comment in the second statement to clarify it.
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Ilia Smilga
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Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian subalgebra made of hyperbolic elements); $\mathfrak{a}^+$ an open Weyl chamber of $\mathfrak{a}$; and $a_0$ some element of $\mathfrak{a}^+$.

I would like to have the following property:

Let $g \in G$. Suppose that $\operatorname{Ad} g$ fixes $a_0$. Then $\operatorname{Ad} g$ fixes the whole subspace $\mathfrak{a}$.

(In other words, the stabiliser of $a_0$ is equal to the stabiliser of $\mathfrak{a}$, commonly called $L$ or $MA$.)

Since all Cartan subspaces are conjugate to each other, an equivalent statement would be:

Any two distinct open Weyl chambers in $\mathfrak{g}$ are (not necessarily lying in the same Cartan subspace) are disjoint.

This looks like it should be a classical result, but I can neither prove it myself nor find a reference. Can anyone help me?

If the proof is longer than a couple of lines, I would rather quote it than rewrite it, so a reference would be appreciated!

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian subalgebra made of hyperbolic elements); $\mathfrak{a}^+$ an open Weyl chamber of $\mathfrak{a}$; and $a_0$ some element of $\mathfrak{a}^+$.

I would like to have the following property:

Let $g \in G$. Suppose that $\operatorname{Ad} g$ fixes $a_0$. Then $\operatorname{Ad} g$ fixes the whole subspace $\mathfrak{a}$.

(In other words, the stabiliser of $a_0$ is equal to the stabiliser of $\mathfrak{a}$, commonly called $L$ or $MA$.)

Since all Cartan subspaces are conjugate to each other, an equivalent statement would be:

Any two distinct open Weyl chambers in $\mathfrak{g}$ are disjoint.

This looks like it should be a classical result, but I can neither prove it myself nor find a reference. Can anyone help me?

If the proof is longer than a couple of lines, I would rather quote it than rewrite it, so a reference would be appreciated!

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian subalgebra made of hyperbolic elements); $\mathfrak{a}^+$ an open Weyl chamber of $\mathfrak{a}$; and $a_0$ some element of $\mathfrak{a}^+$.

I would like to have the following property:

Let $g \in G$. Suppose that $\operatorname{Ad} g$ fixes $a_0$. Then $\operatorname{Ad} g$ fixes the whole subspace $\mathfrak{a}$.

(In other words, the stabiliser of $a_0$ is equal to the stabiliser of $\mathfrak{a}$, commonly called $L$ or $MA$.)

Since all Cartan subspaces are conjugate to each other, an equivalent statement would be:

Any two distinct open Weyl chambers in $\mathfrak{g}$ (not necessarily lying in the same Cartan subspace) are disjoint.

This looks like it should be a classical result, but I can neither prove it myself nor find a reference. Can anyone help me?

If the proof is longer than a couple of lines, I would rather quote it than rewrite it, so a reference would be appreciated!

Source Link
Ilia Smilga
  • 1.6k
  • 9
  • 20

Are two distinct Weyl chambers always disjoint?

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian subalgebra made of hyperbolic elements); $\mathfrak{a}^+$ an open Weyl chamber of $\mathfrak{a}$; and $a_0$ some element of $\mathfrak{a}^+$.

I would like to have the following property:

Let $g \in G$. Suppose that $\operatorname{Ad} g$ fixes $a_0$. Then $\operatorname{Ad} g$ fixes the whole subspace $\mathfrak{a}$.

(In other words, the stabiliser of $a_0$ is equal to the stabiliser of $\mathfrak{a}$, commonly called $L$ or $MA$.)

Since all Cartan subspaces are conjugate to each other, an equivalent statement would be:

Any two distinct open Weyl chambers in $\mathfrak{g}$ are disjoint.

This looks like it should be a classical result, but I can neither prove it myself nor find a reference. Can anyone help me?

If the proof is longer than a couple of lines, I would rather quote it than rewrite it, so a reference would be appreciated!