When a workers comp doctor puts an employee on light duty, do we have to pay their current hourly rate or can we pay a lower rate when they are performing the light duty work?

From strictly a workers' comp viewpoint, you can offer a lower rate of pay, provided it isn't less than what the employee would earn if they were collecting workers' comp benefits. Under the Family Medical Leave Act, you cannot require an employee to accept light duty, even with the same rate of pay, if they prefer instead to stay out on the 12-week disability leave.  Under the Americans with Disabilities Act, lowering pay would be a last resort, i.e.: the same rate of pay is an undue hardship. All three of these laws could apply to a person who was disabled due to a work related injury or illness.