several things could be causing the problem... a loose drive belt or related tensioner pulley(s) loosing a bearing; loose/broken hydro pedal links/rods; low/burned hydro fluid; or an overheated damaged hydro tranny. Also, the tranny's cooling fan could be trashed and/or the tranny top is covered with clippings/trash, minimizing cooling.
The cub 1000 series hydros are generally factory sealed but - with some difficulty - can be drained/refilled with a quality 20-50w synthetic motor oil. Do a search for "LT1050 hydro oil change" to get more how to detail.
If you have been subjecting the machine to heavy pull loads with less than full engine RPM (lots of hills, etc.), those hydros will quickly overheat, not only burning the oil but also damaging internal pump/motor parts = $$$$ replacement. Not good.
Hopefully, the machine may still be under warranty - if it was not mis-treated. M/W, compressed air or an electric leaf blower is best to keep deck and tranny top clean after each use - avoid using the garden hose or a power washer.
Like said, once the drive belt, hydro fan and overall cleanliness of the cooling fins are ruled out, you're pretty much looking at the trans-axle itself. This Hydro-Gear brand hydrostatic transaxle can be serviced, but you typically have to remove it from the machine to be able to dump the old oil out the top vent. Flip it back, right side up and refill with heavy-duty and/or synthetic 15w40 or 20w50 motor oil. For the sealed, small capacity units, I'd use a synthetic 20w50.