2001 Jaguar S-type 4.0 V8

May be symptom not cause.

Which light and exactly what do you mean by stays on?

Plausible causes include the light stalk and the ignition flap but you could hardly give much less information.

the anti-thief light in the window that should flash when alarm is set. Also the battery light stays on and the traction control warning appears and car shuts off.

Load test battery, test items I posted, check powers & grounds.

Plug it in and get the codes. It will have thrown some for that lot.

yes , but most likely a string of false one's too if it is a failing ALT or BAT.
start with the volts .

If it's not the batt or alt (or connections for them), expect to spend a LOT of hours finding the fault.

Quite possibly. With the battery light on permanently then that alone indicates a possible alternator problem, not charging the battery properly, easy enough to check with a multimeter.
The DSC warning coming on and the car shutting off could also be a symptom of a bad battery voltage, but I would think it would show bad starting ability/slower cranking before it showed up this lot.
Unless of course the OP is jump starting it off another car/battery and hasn't said yet?

Terse posts tend not to help get solutions.