Let $G/K$ be a symmetric space of a non-compact type, i.e. $G$ is a semi-simple connected Lie group, and $K$ is its maximal compact subgroup. Helgason in his book "Differential geometry and symmetric spaces" defines the Weyl group $W(G,K)$ as follows.

Let $\mathfrak{g}=Lie(G), \mathfrak{k}=Lie(K)$. Let $$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$ be the Cartan decomposition. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$. Let $M$ and $M'$ be respectively the centralizer and the normalizer of $\mathfrak{a}$ in $K$. Clearly $M$ is a normal subgroup of $M'$. Then $$W(G,K):=M'/M$$ is called the Weyl group of $G/K$ and is known to be finite.

Question. Is there a direct relation between $W(G,K)$ and the Weyl group $W$ of $G$ (or may be better to say of the compact form of $G$)? E.g. can one say that $W(G,K)\subset W$ in some natural way?

A reference would be very helpful.

  • 2
    $\begingroup$ You need to be more careful about notation here (abundant in this subject). For instance, your $\mathfrak{a}$ seems to be a maximal abelian subspace of $\mathfrak{p}$. Usually it's best to use subscripts 0 to contrast with the complexified Lie algebra. Anyway, this "Weyl group" is a finite crystallographic reflection group and is included (often properly) in the abstract Weyl group $W$ attached to the root system of the complexified Lie algebra. (What do you mean by "Weyl group $W$ of $G$"?) On the other hand, the "roots" need a lot of bookkeeping here. See Knapp's book. $\endgroup$ Nov 9, 2014 at 23:36
  • $\begingroup$ You are right, I corrected the $\mathfrak{a}$, though not the 0 subscript. By $W$ I meant the Weyl group of a compact form $U$ of $G$ which, as far as I understand, coincides with the Weyl group of the complexified Lie algebra. (For example if $G=SO_0(p,q)$ then $U=SO(p+q)$.) $\endgroup$
    – asv
    Nov 10, 2014 at 9:26
  • $\begingroup$ Could you please specify what book by Knapp contains the result. According to MathSciNet, Knapp has 14 books... $\endgroup$
    – asv
    Nov 10, 2014 at 9:29
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    $\begingroup$ Knapp's relevant book is Lie Groups Beyond an Introduction (Birkhauser, 1996). There are other books, for example an older one by Vogan. Keep in mind that Helgason is focused especially on symmetric spaces rather than just Lie groups and their Lie algebras; but your question involves mostly the latter. Helgason also leaves some of the pieces like this one for the reader to assemble, so it's well worth consulting other sources too. $\endgroup$ Nov 10, 2014 at 14:22

1 Answer 1


The answer is in the book you quote (1978 version) on page 325.

  • $\begingroup$ Thank you. Do I understand correctly that, according to your book, $W(G,K)$ is a quotient of a subgroup of the Weyl group $W$, but not necessarily a subgroup? $\endgroup$
    – asv
    Nov 27, 2014 at 10:43
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    $\begingroup$ W(G,K) is the homomorphic image of a subgroup og W(G). The tables on pages 532-534 in the book list both of these groups in terms of the Dynkin diagrams for all simple G. W(G,K) is actually the Weyl group of another group H. For example W(E(6), F(4)) is the Weyl group of SL(3,C). See middle of page 534 for this example. Helgason $\endgroup$ Nov 27, 2014 at 13:15

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