# What are the simply-connected non-compact irreducible symmetric spaces?

Would someone be able to list (or provide a reference to) the simply-connected non-compact irreducible symmetric spaces of rank $\ge 1$(as quotients of Lie groups $G/H$)?

Any help would be appreciated!

• Wolf (2011), p. 286 sq..
– Francois Ziegler
Jul 26, 2018 at 0:25
• @FrancoisZiegler Question: When it says "duality interchanges compact and non-compact simply connected irreducible symmetric spaces further reducing the classification to the compact irreducible case", does this mean that a decomposition of a, say non-compact, symmetric space must have irreducible factors that are compact simple simply connected Lie groups? Jul 26, 2018 at 0:47
• I don’t get how this pertains to your question. You are assuming $M$ irreducible, so there is just one factor, which is noncompact. (Wolf explains the decomposition and duality on pp. 234-237.)
– Francois Ziegler
Jul 26, 2018 at 1:37
• @FrancoisZiegler No, I am assuming the factors of $M$ are irreducible ($M$ is a non-compact reducible simply connected symmetric space $M= \mathbb{R^n}\times\prod M_i \times \prod M_j^* \times \prod N_k \times \prod N_l^*$). I am asking if the irreducible factors of the non-compact space are compact by the so-called "duality". Jul 26, 2018 at 1:49
• Yes. Not “changed” in the sense of modifying $M$, only in the sense of bijecting them to something already-classified. As to “may (not must)” it was in reference to the “must” your very first comment above. E.g. $\smash{\mathbf R^2\times\mathrm S^2}$ is noncompact with a compact factor.
– Francois Ziegler
Jul 26, 2018 at 2:36

The reference from which I learned the lists of $G/K$'s of various sorts is S. Helgason's book "Differential Geometry and Symmetric Spaces".

(For my own mathematical purposes, explicit details about at least the "classical groups and domains" is very useful, so I do keep these things in my head.) Possibly dropping overt reference to the anomalous isogenies and edge cases and quibbles about notational conventions...

Type A: $SL_n(\mathbb R)/SO(n,\mathbb R)$, $SL_n(\mathbb C)/SU(n)$, $SL_n(\mathbb H)/Sp^*(n)$, $U(p,q)/U(p)\times U(q)$

Types B,D: $O(p,q,\mathbb R)/O(p)\times O(q)$, $O(n,\mathbb C)/SO(n,\mathbb R)$, $O^*_{2n}/U(n)$.

Type C: $Sp_n(\mathbb C)/Sp^*n$, $Sp_n(\mathbb R)/U(n)$, $Sp^*_{p,q}/Sp^*_p\times Sp^*_q$.

The not-well-known cases are: $Sp^*_{p,q}$ is (modeled by) the group of quaternion matrices preserving a quaternion hermitian form. Provably, these have signatures, much as Sylvester's inertia theorem for quadratic forms. And $O^*_{2n}$ is quaternion matrices preserving a skew-hermitian form.

Note that in all cases the three $\mathbb R$-algebras $\mathbb R, \mathbb C, \mathbb H$ play roles. In fact, a completely parallel thing happens for classical groups over $p$-adic fields and in other cases, as in A. Weil's "Algebras with involutions and classical groups", Indian (not Indiana) J. Math. 1960.