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}.