Huygens' principle -- Wave equations in $\mathbb{R}^n$ seem similar for different $n$, but behave quite differently depending on the parity of $n$: waves in odd dimensional spaces never look back, while waves in even dimensional spaces linger. A more vivid depiction can be found in this related MSE post
.. when you switch a light bulb on and then off (in 3D), there will be a light wave traveling with the speed on light and behind the wave, there will be total darkness. But when you throw a rock into a pond (with a 2D surface), there will be lots of waves traveling outwards from where the rock hit the water and, in theory, the water will never a still again.
There are several ways to explain the difference. One is by looking at the solutions to the PDE directly, one uses complex analysis, one interprets waves in a case as a "projection" of another case..
Question
In this post, I'm looking for a representation theoretic explanation. Indeed, the wave equations look pretty similar for all $n$, and indeed all properties of waves are entailed in the equations. The only (hopeful) difference I notice is the symmetries that involve: the representation theory of $\frak{so}(n,1)$ is quite different based on the parity of $n$! There's already a hint in this answer without details. I hope someone would point out a more detailed account that show Huygens' principle alone this line. To me, this will be more satisfactory as an intuitive explanation.
