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$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary.

Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. Let $B\subset G$ be the Borel subgroup of upper triangular matrices. Let $H\subset B$ be the Cartan subgroup of diagonal matrices. Let $H_1\subset B$ be another Cartan subgroup which is also contained in $B$.

(Remark. Probably in the real case have to assume in addition that $H_1$ is conjugate to $H$ by an element of $G$.)

Question. Does there exist an element $b\in B$ such that $$b\cdot H\cdot b^{-1}=H_1?$$

Does this fact (if true) have any generalizations to real (or complex) semi-simple Lie groups? A reference would be helpful.

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    $\begingroup$ All Levi subgroups in $B$ are conjugate. This is a result of Mostow. $\endgroup$
    – YCor
    Dec 8 at 13:59
  • $\begingroup$ Thank you. Do you have a reference? Is it true for any real semi-simple Lie group? It seems to be the final answer. $\endgroup$
    – asv
    Dec 8 at 14:07
  • $\begingroup$ It's a result on algebraic groups, possibly non-semisimple, since it has to be applied in $B$. Alternatively there's conjugacy of Cartan subgroups in real solvable Lie algebras which does the job (it works for $H^0$, but $H$ is the centralizer of $H^0$ so it works). The latter is in Bourbaki. $\endgroup$
    – YCor
    Dec 8 at 14:17
  • $\begingroup$ Did you mean Lie groups rather Lie algebras (referring to Bourbaki)? $\endgroup$
    – asv
    Dec 8 at 14:20
  • $\begingroup$ Bourbaki does the job for Lie algebras and the Lie group statement follows (passing to the centralizer, to get the statement about all of $H$). $\endgroup$
    – YCor
    Dec 8 at 20:10

1 Answer 1

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A. Borel, Linear Algebraic Groups, 2nd edition, Grad. Texts in Math., 126, Springer-Verlag, New York, 1991. xii+288 pp. ISBN:0-387-97370-2, p. 48 section IV, 11.3 Corollary

  1. The maximal tori in $G$ coincide with the maximal tori in the various Borel subgroups of $G$, and they are all conjugate.

  2. The maximal connected unipotent subgroups of $G$ are each the unipotent part of a Borel subgroup, and they are all conjugate.

(Throughout this chapter $G$ denotes any connected affine group, and all algebraic groups are understood to be affine.)

I think that this theorem is only correct over an algebraically closed field. Over an arbitrary field $k$, Borel p. 228 proves:

20.9 Theorem.

  • The minimal parabolic $k$-subgroups of G are conjugate under $G(k)$.
  • The maximal $k$-split tori of $G$ are conjugate under $G(k)$.
  • If $P$ and $P'$ are parabolic $k$-subgroups conjugate under $G(K)$, then they are conjugate under $G(k)$.

(Here $K$ is any algebraically closed extension field of $k$.)

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