If it's only been in the ground for a month, it's probably spending all it's energy making new roots. Also, palms like heat, and like to be planted when its hot. So planting it when it was still pretty cool may be a disadvantage. Give it a little time.

The vast majority of palms are not fast growers. So too, you JUST planted it. As said above, root growth must precede top growth...especially, with a species adapted to grow in very arid environments, where the roots need to seek ground water first. As long as you are warm, don't be afraid to water. I would hold off a bit before applying a fertilizer though.

It will grow in time as others have said. I used to live in the low desert where Mexican Fan Palms grow as natives. I was told they like Epson Salts. I live in a different zone now, but they grow like weeds here also. In fact we are constantly weeding out baby palms that sprout everywhere. First year they sleep, next year they creep, third year they leap!

as the others said, it will grow. do NOT give it fertilizer until it is growing, though… not sure how you dug the hole and what dirt you put it in, but research done on Washingtonias show that there is no advantage whatsoever to amending soil… and planting them in narrow holes in which the root ball barely fits in works the best… in fact, some palms will sit and sulk in large holes with amended soil a lot longer than if native soil has been used (roots hate to make transitions into different kinds of soils I guess) in a narrow hole. Just a thought fore the next one you plant.