What is the enchantment on the Arch-Mage robe?
Most Unique items that have a unique enchantment on them cannot be disenchanted, to stop people from adding overpowered enchantments to high level armor.
You can see here what the enchantment actually entails.
The Archmage's Robes is quest reward given to the player after they become the Archmage of The College of Winterhold.
You can't disenchant certain unique items and artifacts, this robe being one of them.