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Mathematics(Homogeneous spaces) Moduli Space of Lattices
[0] [0] Johann Birnick
[2024-03-05 02:26:45]
[ lie-groups group-actions integer-lattices homogeneous-spaces automorphic-forms ]
[ https://math.stackexchange.com/questions/4875262/homogeneous-spaces-moduli-space-of-lattices ]

I'm looking at the "moduli space of $n$-dimensional lattices", which should be the double quotient $$ \mathcal{M} = \text{GL}_n(\mathbb{Z}) \backslash \text{GL}_n(\mathbb{R}) / (\mathbb{R}^\times \cdot O_n(\mathbb{R})) $$ or I guess equivalently: $$ \mathcal{M} = \text{SL}_n(\mathbb{Z}) \backslash \text{SL}_n(\mathbb{R}) / SO_n(\mathbb{R}) $$ It should already help to understand the thing without the $\text{SL}_n(\mathbb{Z})$ quotient $$ \mathcal{N} = \text{GL}_n(\mathbb{R}) / (\mathbb{R}^\times \cdot O_n(\mathbb{R})) $$ or again equivalently: $$ \mathcal{N} = \text{SL}_n(\mathbb{R}) / SO_n(\mathbb{R}) $$ I'm trying to understand the geometry of this. More explicitly, I'm looking for:

  1. An "easy-to-understand" (or rather, easy-to-represent-on-a-computer) space with which I can identify this.
  2. A natural geometry on this space. By "geometry" I mean "give me as much as possible". At least a metric, ideally it has very nice properties, and a Lie bracket or what not (I don't know much about this).

For example, in the case $n=2$, we can identify $\mathcal{N}$ with the complex upper half plane, or equivalently the complex unit disk, and the geometry is given by the hyperbolic metric, overall that's called the hyperbolic plane. Also in this case, if we furthermore proceed to $\mathcal{M}$ by quotiening out by $\text{SL}_n(\mathbb{Z})$ on the left, we just get the sphere $S^2$. (Or I guess one point is missing.) People who have seen modular forms are familiar with this example.

In general, what I got from other posts, is that I think $\mathcal{N}$ should be a "homogeneous space", but I don't know much about this, and I don't know anything about $\mathcal{M}$ in the case $n > 2$.

In the end I want to work with these spaces on a computer, so I need as explicit descriptions as possible of these spaces and geometries. If you can point me to some resources or even explain a little bit, I would be very grateful!

EDIT: I'm mainly looking for the Riemannian metric on $\mathcal{N}$ or $\mathcal{M}$. If you could explain me what it is or point me to resources where it is explained that would be very helpful!

These two $\mathcal M$s are different quotients (just think about their dimensions). The second quotient classifies unimodular lattices. Your two spaces $\mathcal N$ are also different. They are both symmetric spaces, the second one is of noncompact type. - Moishe Kohan
Thanks. I was unsure about the two $\mathcal{M}$s, but how can the $\cal{N}$s be different? Consider the natural map $SL_n (R) / O(n) \to GL_n(R) / R^\times O(n)$. For surjectivity, just write a coset $A \cdot R^\times O(n)$ as $A/det(A) \cdot R^\times O(n)$. For injectivity, assume $A_1 A_2^{-1} \in R^\times O(n)$ with $A_1, A_2 \in SL_n(R)$ . Then $det(A_1)/det(A_2) = 1$ so $A_1 A_2^{-1} \in O(n)$. Where is my mistake? - Johann Birnick
Also, which space classifies lattices then? I think actually that classifying unimodular lattices is the same as classifying lattices in my sense, where I assume lattices to be the same even when they are scaled versions of each other. - Johann Birnick
Oh, I did not notice that you are dividing by $\mathbb R^\times$. - Moishe Kohan
@MoisheKohan Where can I read what the Riemannian metric on $\mathcal{N}$ or $\mathcal{M}$ is? - Johann Birnick
(1) The standard reference for symmetric spaces is Helgason's book. - Moishe Kohan
He wrote three books with "Symmetric Spaces" in the title... - Johann Birnick
(1) Differential geometry, Lie groups and symmetric spaces - Moishe Kohan