用户: Aripriner/Langlands Program/Shimura Varieties/Locally symmetric spaces
1Locally symmetric spaces
Let be a semi-simple algebraic group over , for example , or a special orthogonal or special unitary group. Locally symmetric spaces for are "nice" enough spaces whose cohomology is related to automorphic representations of .
For simplicity, we will assume that is connected. Let be a maximal compact subgroup of , and let . If is a discrete subgroup of such that (or equivalently ) is compact and that acts properly and freely on (this holds for example if is torsion free), then there is a classical connection between the cohomology of and automorphic representations of , called Matsushima’s formula.
Here we have a nice way to produce discrete subgroups of . We say a subgroup of is a arithmetic group if there is a closed embedding of algebraic groups such that, setting , we have that is of finite index in and in . If is small enough, then it acts properly and freely on , so the quotient is a real analytic manifold.
We actually would like to see automorphic representations of , so we will use adelic versions of . Let be an open compact subgroup of , where is the adele of finite places. Let