Definition of locally symmetric space of reductive groups

Question

This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.

In my attempt to study Shimura varieties, I came across the definition of a locally symmetric space associated to connected reductive group $G/\mathbb{Q}$ as follows:

Let $A_G\subset G$ denote the maximal $\mathbb{Q}$-split torus in the center of $G$ and $A_\infty=A_G(\mathbb{R})$. Let $K_{\infty}\subset G(\mathbb{R})$ denote a maximal compact subgroup. If, for any topological group $\mathfrak{G}$, the connected component of the identity is denoted by $\mathfrak{G}^\circ$, then we define $$X=G(\mathbb{R})/A^\circ_\infty K_\infty^\circ.$$ Now, let $\Gamma\subset G(\mathbb{Q})$ be a subgroup such that $\Gamma\cap G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$ has fintie index in both $\Gamma$ and $G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$ for some faithful representation $G\hookrightarrow \operatorname{GL}_n$. The locally symmetric space is defined as $$ X(\Gamma)=\Gamma\backslash X. $$

My question is basically, why?

Intuitively what is this trying to achieve? As in, why are we quotienting out by $K_\infty^\circ$ and $A_\infty^\circ$ (maybe this is the least we need to do to ensure some kind of compactness-like result, or better cohomology groups)?

I do have some familiarity with moduli spaces of elliptic curves with level structures and I know that we can get examples using this construction (which helps construct Galois representations using certain modular forms), but when I see this construction, it just looks like generalising without any end goal in mind (which is obviously not true and why I said I should probably keep reading rather than ask this question — but I cannot help myself).

I am sure the question sounds vague, but I just don't have an image in mind when I think of these locally symmetric spaces and it would greatly help it someone could help fix that.

Answer 1

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a definition, that all such objects have the form $G(\mathbb{R}) / A_\infty^\circ K_\infty^\circ$ for a reductive group $G$. You can find this perspective, for instance, in Milne's "Shimura Varieties and Moduli".

From this perspective, defining a symmetric space as $G(\mathbb{R}) / A_\infty^\circ K_\infty^\circ$ is rather misleading. It's common to define it in this way in number-theory texts, because it avoids wading through large amounts of quite intricate (aka: beautiful) differential geometry before getting to the number-theoretic parts of the theory; but it makes the motivation highly non-obvious.

(Edit: Just to clarify, this is about symmetric spaces, not locally symmetric spaces. A locally symmetric space is the quotient of a symmetric space by a discrete subgroup of its automorphism group $G$, so once you understand the motivation for symmetric spaces, the motivation for locally symmetric spaces should hopefully be clear.)