There are two safety switches as part of the starting circuit. One is the seat switch and the other is the brake switch. You need to check those switches to make sure they are functioning. If either switch is not working, it will not start with the key switch.
If you can start it and it runs without weight on the seat, then the problem could be the seat switch. The brake switch you will have to look at with a meter to see if it is working (open/closed status works properly).
If those switches checkout correctly, it could be the key switch. I have had to replace the key switch on my GT275 as it made intermittent contact when turned to the "start" position. You can trouble shoot that with a meter by looking at the contacts on the switch and wiggling the key with the key turned to "start." When you wiggle the key, it will show intermittent continuity - this means the internal switch contacts are worn.