So it does specify that you can't make an Action before you stand up from prone after all, right? So you can't make any Move action, be it 0 movement or full, while prone? Glad that we agree on that.
Ok, p.11 will give the first quote we need:
While Prone, the player loses his tackle zones and may do nothing before standing up at a cost of three squares of his movement when he next takes an Action.
Next one will be that p.23:
Unless otherwise stated in the skill description, the following rules apply to all skills:
6. Only Extraordinary skills work when a player is Prone or Stunned.
And the last one will be Hypnotic Gaze skill description, p. 65:
The player may use hypnotic gaze at the end of his Move Action on one opposing player who is in an adjacent square.
So, in order to use Hypnotic Gaze the player has to make a Move action first, and in order to make a Move action he needs to stand up. And p.23 is overruled by the skill description.
Thus you can't use Hypnotic Gaze while being prone and not stand up.