# Definitions and notation for lattices

Recall that a lattice $L$ in $\mathbb{R}^{d}$ is the $\mathbb{Z}$-span of a basis $\left\{ v_{1},...,v_{d}\right\}$ of $\mathbb{R}^{n}$. Equivalently, letting $g$ be the matrix with rows $v_{i}$ so that $g\in\mathrm{GL}_{d}\left(\mathbb{R}\right)$, the lattice $L$ is $\mathbb{Z}^{d}\cdot g$. Given $g,h\in\mathrm{GL}_{d}\left(\mathbb{R}\right)$, it is easily seen that we have the equality of lattices $\mathbb{Z}^{d}\cdot g=\mathbb{Z}^{d}\cdot h$ if and only if $g\in\mathrm{GL}_{d} \left(\mathbb{Z}\right) \cdot h$. Thus, we can parametrize the space of all lattice by $\mathrm{GL}_{d}\left(\mathbb{Z}\right) \backslash\mathrm{GL}_{d} \left(\mathbb{R}\right)$.

We say that two lattices $L_{1},L_{2}$ are homothetic and write $L_1 \sim L_2$ if there exists some $0\neq c\in\mathbb{R}$ such that $L_{1}=cL_{2}$. Since usually we do not consider homothetic lattices as different, we would like to have a smaller space that do not distinct between such lattices. We consider two approachs – one by taking the quotient space and the other by taking a representative from each homothetic class.

#### Quotient space:

Given $g,h\in\mathrm{GL}_{d}\left(\mathbb{R}\right)$ such that $g=ch$ for some $0\neq c\in\mathbb{R}$ we get that $\mathbb{Z}^{d}\cdot g\sim\mathbb{Z}^{d}\cdot h$, thus to represent a class of homothetic lattices we may consider elements in $\mathrm{PGL}_{d}\left(\mathbb{R}\right)$ instead of $\mathrm{GL}_{d}\left(\mathbb{R}\right)$. As in the previous case, two such elements $g,h$ define the same lattice class if and only if $g\in\mathrm{PGL}_{d} \left(\mathbb{Z}\right) \cdot h$, thus we can paramerize the class of lattices up to homothety by $\mathrm{PGL}_{d}\left(\mathbb{Z}\right) \backslash\mathrm{PGL}_{d} \left(\mathbb{R}\right)$.

#### Set of representatives – unimodular lattices:

Let $L=\mathbb{Z}\cdot g \leq\mathbb{R}^{d}$ be a lattice for some $g\in GL_d(\mathbb{R})$. The covolume of the lattice is defined to be $covol \left( L \right) :=vol\left( \mathbb{R}^{d} / L \right)$ which is the volume of a fundamental domain of $L$ in $\mathbb{R}^{d}$ and is equal to $\left|\det\left(g\right)\right|$. We call a lattice unimodular if its covolume is one, or equivalently it can be written as $\mathbb{Z}^{d}\cdot g$ for some $g\in\mathrm{SL}_{d} \left(\mathbb{Z}\right)$. Given a $d$-dimensional lattice $L$ of covolume $c$, we obtain that $\sqrt[d]{c}L$ is a unimodular lattice homothetic to $L$. Thus, every homothetic class contains a unimodular lattice which is easily seen to be unique. As in the previous cases, we can parametrize the space of $d$-dimensional unimodular lattices by $\mathrm{SL}_{d}\left(\mathbb{Z}\right) \backslash\mathrm{SL}_{d} \left(\mathbb{R}\right)$ and by the argument above $\mathrm{SL}_{d}\left(\mathbb{Z}\right)\backslash\mathrm{SL}_{d}\left(\mathbb{R}\right)\cong\mathrm{PGL}_{d}\left(\mathbb{Z}\right)\backslash\mathrm{PGL}_{d}\left(\mathbb{R}\right)$ under the natural map $\mathrm{SL}_{d}\left(\mathbb{Z}\right)\cdot g\mapsto\mathrm{PGL}_{d}\left(\mathbb{Z}\right)\cdot g$ for $g\in\mathrm{SL}_{d}\left(\mathbb{R}\right)$. We shall denote this space by $X_{d}$.

Remark: The argument that the map above is surjective was due the fact that every positive element in $\mathbb{R}$ has a $d$-root. When a similar construction is done over other fields, e.g. $p$-adic fields, this is no longer the case.

In each of the groups $\mathrm{GL}_{d}\left(\mathbb{R}\right) ,\;\mathrm{PGL}_{d} \left(\mathbb{R}\right)$ and $\mathrm{SL}_{d}\left(\mathbb{R}\right)$ we have the respective diagonal group with nonnegative entries. Namely $A_{full} :=\left\{ diag\left(a_{1},...,a_{d}\right)\;\mid\;a_{i}>0\right\} \leq\mathrm{GL}_{d}\left(\mathbb{R}\right)$ $\overline{A}_{full} :=A_{full}/\mathbb{R}\leq\mathrm{PGL}_{d} \left(\mathbb{R}\right)$ $A :=A_{full}\cap\mathrm{SL}_{d}\left(\mathbb{R}\right).$

For $\bar{t}\in\mathbb{R}^{d}$ we denote $a\left(\bar{t}\right)= diag\left(e^{t_{1}},...,e^{t_{d}}\right) \in A_{full}$, and we note that the map $\bar{t}\mapsto a\left(\bar{t}\right)$ defines an isomorphism $A_{full}\cong\mathbb{R}^{d}$. The restriction to $\mathbb{R}_{0}^{d}=\left\{ \left(t_{1},...,t_{d}\right)\in\mathbb{R}^{d}\;\mid\;\sum_{1}^{d}t_{i}=0\right\}$
induces an isomorphism $A\cong\mathbb{R}_{0}^{d}$. Finally the natural map $A\to A_{full}\to\overline{A}_{full}$ is an isomorphism. We will also use the isomorphism $\mathbb{R}^{d-1}\cong\overline{A}_{full}$ defined by $\bar{t}\mapsto\left[a\left(1,t_{1},...,t_{d-1}\right)\right]$ for $\bar{t}\in\mathbb{R}^{d-1}$.

We shall be interested in the diagonal orbits in their respective spaces of lattice. Letting $\pi_{1}:\mathrm{GL}_{d}\left(\mathbb{Z}\right) \backslash\mathrm{GL}_{d} \left(\mathbb{R}\right)\to\mathrm{SL}_{d}\left(\mathbb{Z}\right)\backslash\mathrm{SL}_{d}\left(\mathbb{R}\right)$ be the normalization to covolume 1 map and $\pi:\mathrm{GL}_{d}\left(\mathbb{Z}\right)\backslash\mathrm{GL}_{d}\left(\mathbb{R}\right)\to\mathrm{PGL}_{d}\left(\mathbb{Z}\right)\backslash\mathrm{PGL}_{d}\left(\mathbb{R}\right)$ be the quotient map, it is easy to see that for any lattice $x\in\mathrm{GL}_{d} \left(\mathbb{Z}\right)\backslash\mathrm{GL}_{d} \left(\mathbb{R}\right)$ we have $\pi_{1}\left(xA_{full}\right) =\pi_{1}\left(x\right)A$ $\pi\left(xA_{full}\right) =\pi\left(x\right)\bar{A}_{full}$.