Let $\mathcal{N}(\mu,\sigma^2)$ denote the Gaussian distribution on $\mathbb{R}$:

$$ \mathcal{N}(\mu,\sigma^2)(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$

A Gaussian distribution is defined by its mean $\mu\in\mathbb{R}$ and its standard deviation $\sigma>0$. Thus the upper-half plane is the parameter space for the normal family of distributions. Let $z = \mu+i\sigma$ denote an element of the upper-half plane, to be interpreted as the parameters for the Gaussian $\mathcal{N}(\mu,\sigma^2)$.

Let $G\in SL(2,\mathbb{R})$, the group of two by two real matrices with unit determinant. Let $g = \begin{pmatrix}a & b\\ c&d\end{pmatrix}$ be an element of $G$. This group acts on the upper-half plane by fractional linear transformations:

$$g\cdot z = \frac{az+b}{cz+d}.$$

Thus $G$ acts on Gaussians via $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}(Re(g\cdot z),\, (Im(g\cdot z))^2).$$

We now describe this action in more detail. If $g = \begin{pmatrix}a & b\\ 0&d\end{pmatrix}$, then $g$ acts on a Gaussian as follows: $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac a d \mu + \frac b d,\, \frac {a^2} {d^2} \sigma^2\right).$$

For an arbitrary $g$, it follows that $$\begin{align*} Re(g\cdot z) &= \frac{ac(\mu^2+\sigma^2) + bd + (ad+bc)\mu}{(c\mu+d)^2+c^2\sigma^2}\\ Im(g\cdot z) &= \frac{\sigma}{(c\mu+d)^2+c^2\sigma^2}, \end{align*} $$ and the action is $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac{ac(\mu^2+\sigma^2) + bd + (ad+bc)\mu}{(c\mu+d)^2+c^2\sigma^2},\left(\frac{\sigma}{(c\mu+d)^2+c^2\sigma^2}\right)^2\right).$$

$G$ has the Iwasawa decomposition $KAN$. Thus $g = k(g)a(g)n(g)$, where $$\begin{align*} k(g) &= \begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\end{align*}\\ a(g) = \begin{pmatrix}e^t & 0\\ 0&e^{-t}\end{pmatrix}\\ n(g) = \begin{pmatrix}1 & u\\ 0&1\end{pmatrix}, $$ where $\theta$, $t$, and $u$ are analytic functions of the coefficients of $g$ (see Keith Conrad's notes).

It follows that $N$ acts on a Gaussian by moving its mean: $$ n(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left( \mu + u\,, \sigma^2\right),$$ and $A$ acts on a Gaussian by dilating the mean and standard deviation: $$ a(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left( e^{2t}\mu\,, e^{4t}\sigma^2\right).$$

$K$'s action is much more interesting:

$$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac{\cos\theta\sin\theta(\mu^2+\sigma^2) - \cos\theta\sin\theta + (\cos 2\theta)\mu}{((\sin\theta)\mu+\cos\theta)^2+(\sin^2\theta)\sigma^2}, \left(\frac{\sigma}{((\sin\theta)\mu+\cos\theta)^2+(\sin^2\theta)\sigma^2}\right)^2\right).$$

The standard normal $\mathcal{N}(0,1)$ is fixed by $K$. For any other Gaussian $\mathcal{N}(\mu,\sigma^2)$, moving through values of $\theta$ will cause oscillations of the mean and standard deviation, creating a "wobbling" of the Gaussian.

In particular, if $\mu=0$ and $\theta = \pi/2$, then $$ k(g)\cdot \mathcal{N}(0,\sigma^2) = \mathcal{N}\left(0,\frac 1 {\sigma^2}\right). $$

The right-hand side is the Fourier transform of $\mathcal{N}(0,\sigma^2)$. For an arbitrary $\mathcal{N}(\mu,\sigma^2)$, with $\theta = \pi/2$ we have: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(-\frac \mu {\mu^2+\sigma^2},\left(\frac \sigma {\mu^2+\sigma^2}\right)^2\right), $$ which means that for any $z=\mu+i\sigma$ on the upper semi-circle of radius $\rho$: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(- \frac \mu {\rho^2} ,\frac {\sigma^2} {\rho^4}\right). $$

Now define $$ g_\mu = \begin{pmatrix}1 & \mu\\ 0&1\end{pmatrix} \begin{pmatrix}0 & -1\\ 1&0\end{pmatrix}\begin{pmatrix}1 & -\mu\\ 0&1\end{pmatrix}. $$ Then $$g_\mu\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\mu,\frac 1 {\sigma^2}\right).$$

Questions: What are some physical interpretations of this? What are some good resources for exploring similar links among probability and geometry?

thatparticular space: after all, it acts on any space in bijection with $\mathbb{R}\times\mathbb{R}_{>0}$ just by transport of the action on the upper half plane $\mathfrak{H}$. And the space of pairs $(\mu,\sigma)$ doesn't seem to have anything to do with the complex structure on $\mathfrak{H}$ through which the SL(2,Z) action is defined. Of course, I'd be happy to be proved wrong. $\endgroup$