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.

hypersurfacein the convex cone of positive definite quadratic forms on $\mathbb{R}^n$, a hypersurface that is asymptotic to the boundary of the convex cone. Are you asking for noncompact Riemannian symmetric spaces $M = G/K$ for which there is a $G$-module $V$ such that there is an open, proper convex cone $C\subset V$ that is $G$-invariant and such that $K$ is the stabilizer of a point inside $C$ and $\dim V = 1+\dim M$? – Robert Bryant Mar 29 '12 at 15:28almostthe boundary components of the (Borel-Serre or Satake) compactifications. On these points I am over my head--but not too over. I am trying to work these questions out. – J. Martel Mar 29 '12 at 17:37