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Irreducible riemannian symmetric spaces come in pairs: one compact and one not compact, usally called the noncompact dual. The compact symmetric spaces include spheres, complex and quaternionic projective spaces, compact simple Lie groups, Grassmannians of different denominations,... They all have established names and to a large extent an established notation. I'm presently writing a paper where many low-dimensional riemannian symmetric spaces make their appearance, both compact and noncompact, and I have found that I do not know the names to some (indeed, most) of them.

Of course, the noncompact duals of the spheres are the hyperbolic spaces, and I am guessing (someone will surely correct me if I'm wrong) that the noncompact duals of the complex and quaternionionic projective spaces are called the complex and quaternionic hyperbolic spaces, respectively. I am also guessing that these are denoted $\mathbb{C}H^n$ and $\mathbb{H}H^n$; although I do not remember where I have actually seen this notation before. But how about the noncompact duals of the other symmetric spaces?

Questions:

Is there an accepted terminology and/or notation for the noncompact dual of the grassmannians of real, complex, quaternionic, associative, special lagrangian,... subspaces? How about for the noncompact dual of a simple Lie group (other than $SU(2)$)?

Thank you.

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  • $\begingroup$ For the duals of projective spaces, one founds in Berger's 'Panoramic View of Riemannian Geometry' $\text{Hyp}_\mathbb{K}^n$, but no notation for the others. $\endgroup$
    – Thomas Richard
    Commented Apr 21, 2011 at 11:32
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    $\begingroup$ I do not think there are accepted names/notations for most symmetric spaces. Even the complex hyperbolic space is sometimes denoted as you suggest, and sometimes as $H^n_\mathbb C$. I find it less confusing when people simply write them as $G/K$ and say explicity what $G$, $K$ are; after all there ARE established notations for linear Lie groups. $\endgroup$
    – Igor Belegradek
    Commented Apr 21, 2011 at 12:53
  • $\begingroup$ Seconding the comments/answers that make more of a point of expressing the non-compact symmetric space as $X=G/K$, where $G$ is semi-simple and $K$ maximal compact, etc., presumably as already established in the context. The very fact that there seems to be no reliable naming scheme for symmetric spaces, while there is one for semi-simple and reductive Lie groups, operationally means that there's less to be disambiguated with the $G/K$ schema, I think. $\endgroup$
    – paul garrett
    Commented Jul 11, 2011 at 22:45

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The Riemannian noncompact duals of of Grassmannians and other Hermitian symmetric spaces possess names according to their realizations:

  1. As bounded models: Cartan domains (4 classical and two exceptional types), or,

  2. In the unbounded realization: Hermitian upper half space,Quaternionic upper half space and Siegel upper half space for the first three classical types.

  3. (Some of them ) As tube domains.

In addition, the symmetric cones are also named after their realizations.

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I'm not quite sure if these are the accepted names, but you find the list of irreducible Hermitian symmetric spaces (Cartan's list) in e.g. Helgason's "Differential geometry and symmetric spaces" (ChapIX, Sec4): The non-compact duals of $\mathbb{CP}^n$ and $\mathbb{HP}^n$ are refered to as Hermitian/quaternian hyperbolic space as you said. I have not found a symbol for them. Personally, I use $\mathbb{D}_n$ for the Hermitian hyperbolic space such that $\mathbb{D} = \mathbb{D}_1$ becomes the Poincare disc, but that might not be universally accepted ;)

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I cannot remember having seen the family $\mathrm{SL}(n;\mathbb{R})/\mathrm{SO}(n)$ named or denoted otherwise. More generally, people usually say that they consider a symmetric space of non-compact type $X=G/K$, but do not name them (except for rank one ones, for which your denomination is the only I know, while the notation varies: $\mathbb{C}\mathrm{H}^n$, $\mathrm{H}_{\mathbb{C}}^n$ probably are the most common).

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