Let $G$ a group, $X$ a $G-$set under the action $\cdot :G\times X\to X$. So, I know the definition of an action, but I don't understand which information the give us. For example, if $G=SL_2(\mathbb R)$ and $X=\hat{\mathbb C}=\mathbb C\cup\{\infty \}$, we have that
$$\begin{pmatrix}a&b\\c&d \end{pmatrix}\cdot z=\begin{cases}\frac{az+b}{cz+d}&cz+d\neq 0\\\infty &cz+d=0\\ \frac{a}{c}&z=\infty ,\ c=0\\ \infty &z=\infty ,\ c\neq 0.\end{cases}$$
So, I know that the orbits are $\{z\in\mathbb C\mid \Im(z)>0\}$, $\{z\in \mathbb C\mid \Im(z)<0\}$ and $\hat{\mathbb R}=\mathbb R\cup\{\infty \}$.
Know, what does this mean ? How can I interpret it ? That the action separate $\hat{\mathbb C}$ in 3 part ?
Best Answer
The fact that the upper half-plane is a full orbit of ${\rm SL}_2(\mathbf R)$ is really the first step in several directions. For example, the action of ${\rm SL}_2(\mathbf R)$ on the upper half-plane gives isometries for the hyperbolic metric on the upper half-plane. Being able to get from anywhere to anywhere else in the upper half-plane by a matrix in ${\rm SL}_2(\mathbf R)$ is roughly analogous to being able to get from anywhere to anywhere in $\mathbf R^2$ by translations. (There's a bit more going on, but I did say "roughly".) If you were going to do something geometric in $\mathbf R^2$, isn't it convenient to be able to set the "origin" anywhere you want? Anyone who is doing geometry in the hyperbolic plane finds it equally convenient that the upper half-plane is a single orbit for ${\rm SL}_2(\mathbf R)$.
What you know about the orbit of ${\rm SL}_2(\mathbf R)$ can be fruitfully combined with knowing the stabilizer subgroup of some point. A nice number to pick is $i$: its stabilizer subgroup is ${\rm SO}_2(\mathbf R)$. (Fun fact: the orbits of ${\rm SO}_2(\mathbf R)$ on the upper half-plane are the circles in the upper half-plane centered at $i$ for the hyperbolic metric on the upper half-plane, so $i$ as a hyperbolic center of a circle is not in the same place as the Euclidean center of those circles). Therefore the upper half-plane can be regarded as a coset space ${\rm SL}_2(\mathbf R)/{\rm SO}_2(\mathbf R)$. This is the first step in generalizing concepts defined initially on the upper half-plane (e.g., modular forms as functions on the upper half-plane) to other settings by using coset or double coset spaces of other groups.
Quite generally, being able to think of certain constructions as orbits for a group action can give important insights.
Geometry: spheres are orbits for the orthogonal group acting on $\mathbf R^n$. This leads to the general idea that "homogeneous spaces" of all kinds are orbits for the action of a Lie group on a space.
Algebra: all the roots of a polynomial (technically, of a separable polynomial) are the orbit for a Galois group acting on a suitable field extension.
Algebraic geometry: a "point" on a variety over a non-algebraically closed field is an orbit for the Galois group over the field of its algebraic closure. For example, the curve $x^2 + y^2 = 1$ viewed over $\mathbf R$ has in addition to its classical real points on the curve some additional points such as $\{(\sqrt{2},i),(\sqrt{2},-i)\}$ of "degree $2$" that are an orbit for ${\rm Gal}(\mathbf C/\mathbf R)$ acting on complex solutions to the equation $x^2+y^2=1$; the classical points have degree $1$. These additional points correspond to something concrete about the ring $A = \mathbf R[x,y]/(x^2+y^2-1)$: its maximal ideals are all of the form $$ \mathfrak m_{a,b} := \{f(x,y) \bmod x^2+y^2-1 : f(x,y) \in \mathbf R[x,y], f(a,b) = 0\} $$ for each point $(a,b) \in \mathbf C^2$ such that $a^2 + b^2 = 1$, but $(a,b)$ and its conjugate point $(\overline{a},\overline{b})$, which is different from $(a,b)$ if $a$ or $b$ is not real, define the same maximal ideal. Even a curve over $\mathbf R$ like $x^2 + y^2 = -1$, which classically looks like the empty set, is not empty: it has real points but none of degree $1$. For instance, $\{(i,0),(-i,0)\}$ is a real point on this curve of degree $2$. This corresponds to the fact that $\mathbf R[x,y]/(x^2+y^2+1)$ has maximal ideals $$ \mathfrak m_{a,b} := \{f(x,y) \bmod x^2+y^2+1 : f(x,y) \in \mathbf R[x,y], f(a,b) = 0\} $$ for each $(a,b) \in \mathbf C^2$ such that $a^2 + b^2 = -1$, with points in the same orbit of ${\rm Gal}(\mathbf C/\mathbf R)$ corresponding to the same maximal ideal. The curve $x^2 + y^2 = -1$ has no real solutions, but $\mathbf R[x,y]/(x^2+y^2+1)$ has maximal ideals, and those correspond to orbits of ${\rm Gal}(\mathbf C/\mathbf R)$ on the complex solutions of $x^2 + y^2 = -1$.
Number theory: Galois groups act on the ideals in the ring of integers of a number field, and the fact that all prime ideals lying over a given prime number form a single orbit for this action is really, really important in number theory.
Number theory again: you can create all primitive Pythagorean triples from the single triple $(3,4,5)$ by applying the integral orthogonal group of $x^2+y^2 - z^2$ to $(3,4,5)$. See here. More generally, the orbits for that group on the set of integral solutions of $x^2 + y^2 = z^2$ are the solutions with a common gcd for the three coordinates.
Number theory again: for a field $K$, ${\rm GL}_2(K)$ acts on $K \cup \{\infty\}$ by $(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\cdot z = (az+b)/(cz+d)$ $(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\cdot \infty = a/c$ if $c \not= 0$ and the value is $\infty$ if $c = 0$. (It would be cleaner to act on ${\mathbf P}^1(K)$ by $(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\cdot [x:y] = [ax+by:cx+dy]$.) That ${\rm GL}_2(K)$ has a single orbit on $K \cup \{\infty\}$ is not very interesting. If $K$ is a finite extension of $\mathbf Q$ and we use the subgroup ${\rm SL}_2(\mathcal O_K)$, its action on $K \cup \{\infty\}$ has finitely many orbits, which correspond to the ideal classes of the ideal class group of $K$. To a number theorist, those ideal classes are interesting. For example, $K \cup \{\infty\}$ is a single orbit for ${\rm SL}_2(\mathcal O_K)$ if and only if $\mathcal O_K$ has unique factorization.