Linear symmetric spaces are spaces with ''orthogonal complements''?

Question

The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.

I have only recently been made aware that $S_n$ is a so-called $linear$ symmetric space. Apparently this means the following. The collection of (marked) nondegenerate lattices in $n$-space corresponds to the collection $P_n$ of positive definite $n\times n$ symmetric real matrices. The space $P_n$ forms an open convex cone in $\mathbb{R}^N$, $N=n(n-1)/2$. The unimodular elements in $P_n$ consists precisely of $S_n$. Thus $S_n$ is a hypersurface in the cone $P_n$. More specifically $S_n$ is a 'section' of the cone, ie. $S_n$ is homeomorphic to an open ball $B$. The significance is this: taking the closure $\bar{B}$ of the ball gives a compactification. To extend the homeomorphism (if we can) now to the closure gives a compactification of $S_n$.

But still, I do not know what the ''linearity'' of $S_n$ $really$ means.

I would like to know how it relates to the following very remarkable property of unimodular lattices:

Take an unmarked rank $n$ unimodular lattice $\Gamma$. (ie. $\Gamma$ is an element in $S_n / SL_n(\mathbb{Z})$). The remarkable fact is this: for a given rational subspace $W$ in $\Gamma \otimes \mathbb{R}$ there is a CANONICAL complementary rational subspace $W^o$ --( of course, $W^o$ is just the orthogonal complement of $W$ in $\Gamma \otimes \mathbb{R}$).

My question is then how do we see the LINEARITY of $S_n$ as responsible for the fact that rational subspaces have CANONICAL rational complements? Any remarks on what a ''linear'' symmetric space really $is$ would be appreciated.

Answer 1

Definition of "linear symmetric space" appears in the book of Borel and Ji "Compactifications of symmetric and locally symmetric spaces", p. 286. Their definition is not very precise, but can be restated as follows: A symmetric space $X$ is called linear if there exists a convex domain $D$ in the projective space such that $(X, Isom(X))$ is equivariantly diffeomorphic to $(D, Proj(D))$, where $Isom(X)$ is the group of isometries of $X$ and $Proj(D)$ is the group of projective transformations of $D$. With this definition, there are very few irreducible linear symmetric spaces, see e.g. http://www.math.umn.edu/~garrett/m/v/classical_domains.pdf Whatever the property of lattices you need (I still do not understand which one since it is stated only for $SL(n, {\mathbb R})$) you can probably check case-by-case. I think, the main advantage of such spaces is that fundamental domains of lattices (used in the reduction theory) could be described by linear inequalities in this case.

Update: Below is my interpretation of what "complementary" lattice is in the general setting. First, consider $SL(n)$. Then every rational linear subspace $V$ in ${\mathbb Q}^n$ has a "canonical" orthogonal complement $V^\perp$ (also defined over rationals) once we have fixed a rational positive-definite quadratic form. (The complementary lattice is then obtained by taking the integer points in $V^\perp$.) This amounts to choosing a rational point $x$ in the symmetric space $X$ of $SL(n, {\mathbb R})$, i.e., points which is fixed by rational Cartan involution $\theta=\theta_x$. From the viewpoint of the symmetric space $X$, the space $V$ is a point $p_V$ on the ideal boundary of $X$, more precisely, rational Tits building $Y$ sitting inside of the Tits boundary of $X$. Applying $\theta$ to $p_V$ we obtain an antipodal (rational) point $q\in Y$, which is nothing but $p_{V^\perp}$. This interpretation goes through verbatim for every semisimple algebraic group $G$ defined over ${\mathbb Q}$ provided that $G$ contains a rational Cartain involution (one that preserves $G({\mathbb Q})$ and, hence, the corresponding rational Tits building). Existence and uniqueness of such involutions (up to conjugation in $G({\mathbb Q})$) was studied by several people, like I.Satake "On the rational structures of symmetric domains", (parts I and II which appeared in 1989, 1991), A.Helmnick "On the classification of k-involutions", http://www4.ncsu.edu/~loek/research/class.pdf and, I think, many others. Google "rational Cartan involution" and "rational points of symmetric space" to find more references. Jim Humphries who comments frequently on MO can surely provide much more information than I do. The bottom line is that linearity of the symmetric space is irrelevant for the construction of the "canonical" complementary lattice.

One more remark: Various rational flags that you have read about are nothing but various cells of the rational Tits building. I suggest you read a bit more about spherical (Tits) buildings if you want to understand compactifications of locally-symmetric spaces. Lizhen Ji has a survey "Buildings and their applications in geometry and topology". Asian J. Math. 10 (2006), no. 1, p. 11--80, which could be useful here.

Answer 2

Finally, i understand the answer! It is Hermann's Convexity Theorem -- and Harish-Chandra's canonical embedding, which proves that EVERY symmetric space (of noncompact type) is conformally equivalent to a complete and convex riemannian manifold. E.g., every symmetric space is Linear.

Available at: https://math.berkeley.edu/~jawolf/publications.pdf/paper_045.pdf