用户: Aripriner/Langlands Program/Shimura Varieties/Locally symmetric spaces

1Locally symmetric spaces

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, for example $S{L}_n,S{p}_{2n}$, or a special orthogonal or special unitary group. Locally symmetric spaces for $G(\mathbb{R})$ are "nice" enough spaces whose cohomology is related to automorphic representations of $G$.
For simplicity, we will assume that $G(\mathbb{R})$ is connected. Let $K_{\infty }$ be a maximal compact subgroup of $G(\mathbb{R})$, and let $X=G(\mathbb{R})/K_{\infty }$. If $\Gamma$ is a discrete subgroup of $G(\mathbb{R})$ such that $\Gamma \backslash G(\mathbb{R})$ (or equivalently $\Gamma \backslash X$) is compact and that $\Gamma$ acts properly and freely on $X$ (this holds for example if $\Gamma$ is torsion free), then there is a classical connection between the cohomology of $\Gamma \backslash X$ and automorphic representations of $G(\mathbb{R})$, called Matsushima’s formula.
Here we have a nice way to produce discrete subgroups of $G(\mathbb{R})$. We say a subgroup $\Gamma$ of $G(\mathbb{R})$ is a arithmetic group if there is a closed embedding of algebraic groups $G\subset G{L}_N$ such that, setting $G(\mathbb{Z})=G(\mathbb{Q})\cap G{L}_N(\mathbb{Z})$, we have that $\Gamma \cap G()\mathbb{Z}$ is of finite index in $\Gamma$ and in $G(\mathbb{Z})$. If $\Gamma$ is small enough, then it acts properly and freely on $X$, so the quotient $\Gamma \backslash X$ is a real analytic manifold.
We actually would like to see automorphic representations of $G(\mathbb{A})$, so we will use adelic versions of $\Gamma \backslash X$. Let $K$ be an open compact subgroup of $G({\mathbb{A}}_f)$, where ${\mathbb{A}}_f$ is the adele of finite places. Let $$\begin{array}{r}{M}_K=G(\mathbb{Q})\backslash X\times G({\mathbb{A}}_f)/K\end{array}$$