Hecke operator and Hecke points

The following is from the introduction of [1].

Let \(G\) be a connected almost simple simply connected linear algebraic group defined over \(\mathbb{Q}\) such that \(G(\mathbb{R})\) is non-compact. Let \(\Gamma \subseteq G(\mathbb{Q})\) be a congrence subgroup. The theorem of Borel and Harish-Chandra implies that \(\Gamma\) is a lattice in \(G(\mathbb{R})\), i.e. \(\Gamma \backslash G(\mathbb{R})\) has finite volume. Denote \(d\mu\) to be the probability Haar measure on \(\Gamma \backslash G(\mathbb{R})\).

The points \(T_{a} x\) are called Hecke points. [1] and [2] is about the equidistribution of these points in \(\Gamma \backslash G(\mathbb{R})\).

Lifts of codes

Consider the map \(\pi_{p} : \mathbb{Z}^{n} \rightarrow \mathbb{F}_p^{n}\) and then consider the set \[\begin{align} \mathcal{L}_{p}^{k} = \{ \beta_{p} \pi_{p}^{-1}(C) \ | \ C \subseteq \mathbb{F}_p^{n} , \dim_{\mathbb{F}_p} C = k\}, \end{align}\] where \(\beta_{p}\) is a constant chosen so that each lattice in \(\mathcal{L}_{p}^{k}\) has covolume \(1\) in \(\mathbb{R}^{n}\).

We can now consider this set a subset of \[\begin{align} SL_n(\mathbb{Z}) \backslash SL_n(\mathbb{R}) = \{ g^{T} \mathbb{Z}^{n} \ | \ g \in SL_n(\mathbb{R})\}. \end{align}\]

Then, this set turns out to be the same as \(T_{a}x\) for \(x = I_{n}\) and \(a = \left[ \begin{smallmatrix} p^{\frac{k-n}{n}}I_{k} & 0 \\ 0 & p^{\frac{k}{n}} I_{n-k} \end{smallmatrix} \right]\).

\(S\)-arithmetic spaces

See [4] for a nice exposition.

References