This is very common on all AE-1 variants and the closely related AV-1.
The simple answer is that your camera needs a thorough CLA service (Clean, Lubricate, Adjust). In most cases all that's happened is that the lubricants put in there thirty plus years ago have dried out and gummed up. If you're unlucky then gears or levers have worn, but there are plenty more of these cameras in good condition for free/cheap if that's happened (although they'll also be due a CLA if not done recently).
At the same time, you can get the camera's gooey foam light seals and mirror buffers replaced; most places charge extra for this one-off service, but sometimes they'll do it for free if you politely point out that you can take the camera elsewhere.
Freshly serviced, your camera can last for many more years.
If in the UK then I can recommend
Colchester Camera Repair, they're Canon specialists and absolutely excellent, but you may have your own preferred repairer. Any competent camera repairer can do this job very easily.
Don't try to force anything on your camera until you get it repaired, and try to get the job done asap, as sometimes this fault weakens internal parts due to springs being held in stressed positions for longer than designed.
DIY repair is possible but quite tricky, but if you're prepared to risk it going wrong then the experience is worthwhile. You'll find various "how-to" guides online although I haven't got any bookmarked at present to refer you to.
I hope that I've answered your question, please don't forget to rate my answer.