Existence and uniqueness of $SL(n,\mathbb R)$ left-invariant Borel probability measure on $SL(n,\mathbb R)/SL(n,\mathbb Z)$.

Question

Consider the action of $SL(n,\mathbb R)$ on the homogeneous space $SL(n,\mathbb R)/SL(n,\mathbb Z)$ by left translation. Are there any good refences where I can find the proof of the existence and uniqueness of the left $SL(n,\mathbb R)$-invariant Borel probability measure on $SL(n,\mathbb R)/SL(n,\mathbb Z)$.

If there is some powerful theorem addressing this for general $G$ and $G/\Gamma$, please let me know how $SL(n,\mathbb R)$ and $SL(n,\mathbb R)/SL(n,\mathbb Z)$ satisfy the hypothesis of that theorem.

Answer 1

For the $SL(n,\mathbb{R})$ case, this is the content of the fact that $SL(n,\mathbb{Z})$ is a lattice. There are two well-known approaches; one is by way of building good fundamental domains ("Siegel sets") and the other is Margulis' unipotent orbit argument.

As a starting point Witte Morris' Introduction to Arithmetic Groups is probably appropriate (available at https://arxiv.org/abs/math/0106063). See Chapter 7 there. The references there would point toward generalizations too I presume.

Answer 2

Generally, if $G$ is a unimodular locally compact topological group, and $\Gamma$ is a unimodular closed subgroup of $G$, there is a unique up to scalar left $G$-invariant Borel measure on the quotient space $\Gamma \backslash G$. This measure $\mu_{G/\Gamma}$ has the property that $$\int\limits_G f(g)dg = \int\limits_{\Gamma \backslash G} \int\limits_{\Gamma} f(\gamma g)d \gamma d\mu_{G/\Gamma}(g) \tag{$f \in L^1(G)$}$$ In your case, $G = \operatorname{SL}_2(\mathbb R)$ is unimodular, and $\Gamma = \operatorname{SL}_2(\mathbb Z)$ is a discrete and hence unimodular topological group, so the existence and uniqueness of the measure on $\Gamma \backslash G$ is no problem. The more interesting question is why this measure space $\Gamma \backslash G$ has finite measure.

An indirect, but enlightening way of seeing this is to let $K = \operatorname{SO}_2(\mathbb R)$, and identify $G/K$ with the upper half plane $\mathbb H = \{ a+bi : b > 0\}$. This is an identification of topological spaces and $\Gamma$-actions. Since $K$ is compact and hence unimodular, $G/K$ has a right $K$-invariant measure, and one can check (painfully) that this measure coincides with the hyperbolic measure $\frac{dx dy}{y^2}$ on $\mathbb H$.

Let $D$ be the closure of the usual fundamental domain for $\Gamma$ acting on $\mathbb H$. It has finite hyperbolic measure on $\mathbb H$, and for every $z \in \mathbb H$, there exists a $\gamma \in \Gamma$ such that $\gamma.z \in D$. Identifying $D$ with a subset of $G/K$, we therefore also have $\mu_{G/K}(D) < \infty$. Moreover, Since $K$ is compact, the preimage $E = \{ g \in G : gK \in D\}$ has finite measure, since if $f$ is the characteristic function of $E$, we have

$$\operatorname{meas}(E) = \int\limits_G f(g)dg = \int\limits_{G/K} \int\limits_K f(gk)dkd\mu_{G/K}(k) = \operatorname{meas}(K) \mu_{G/K}(D) < \infty.$$

On the other hand, we can look at this finite number in a different way:

$$\operatorname{meas}(E) = \int\limits_G f(g)dg = \int\limits_{\Gamma \backslash G} \bigg(\sum\limits_{\gamma \in \Gamma} f(\gamma g) \bigg))d\mu_{G/\Gamma}(g).$$

Since for every $g \in G$, there is at least one $\gamma \in \Gamma$ such that $\gamma g \in E$, we have $\sum\limits_{\gamma \in \Gamma} f(\gamma g) \geq 1$ for every $g \in G$, and therefore

$$\operatorname{meas}(E) \geq \int\limits_{\Gamma \backslash G} d\mu_{G/\Gamma}(g).$$

This forces $\Gamma \backslash G$ to have finite measure, and it also implies an upper bound on each sum $\sum\limits_{\gamma \in \Gamma} f(\gamma g)$, except possibly for $g$ in a set of measure zero.