The high speed does't work on the a/c controls

check fuse and relay but most likely fan speed resistor or motor failed

Not a fuse problem ! GM instrument cluster problems 2003 to 2006 . Stepper motor that controls the gauge problem .
How to repair GM Instrument Cluster Speedometer Gauge Stepper

Cruise control fuse # 31 10amp located in the under hood fuse / relay box an HZDFL fuse # 40 15amp located in the center console fuse box . If you pull the plastic side panel off of the right side of the console there is a fuse box inside . If the fuse are good the control module could be bad . Could also be turned off because the ABS has a problem .
DTC P1554 - do you know what a diagnostic trouble code is ?

The powertrain control module (PCM) has the ability to inhibit cruise control operation if conditions exist that would make cruise operation undesirable. The following conditions may inhibit cruise control operation:
• The engine speed is too high or too low.
• The vehicle speed is too high or too low.
• The traction control has been active for longer than 1 second.
• The vehicle acceleration or deceleration rate is too high

If the ABS light is on ,there are DTC'S stored in the system . Your best bet , take it to a GM dealer or someother qualified repair shop.

RR/HVAC fuse 30amp located in the under hood fuse box. You can check it but highly doubt it's the problem . More then like it's the blower motor control module that's bad .
Circuit/System Description
The auxiliary blower motor is a variable speed fan. The HVAC control module controls the fan speed by sending a pulse width modulated signal to the auxiliary blower motor control module. The auxiliary blower motor control module varies the duty cycle across the auxiliary blower motor in relation to the pulse width modulated signal. The auxiliary blower motor speed can be adjusted from 10 percent to 90 percent duty cycle (PWM), 90 percent is high speed fan.
Watch this video for testing purpose's only , this is a front blower motor control module . Same way your back one works . GM is GM Chevy , Buick ,GMC , Cadillac .

Check with a Dealer for a Recall on the Cruise unit and park the car outside until you do. There were fire hazards for some year models. Get your Vin# checked to see if the upgrade was missed. The Dealers installed an inline fuse to repair these.

Of course it does(I think this is a GM engine,found in the Grand Am and other midsize cars).GM usually puts the inline filter behind the rocker panel under the car.

The fan speed not working is due to a burned out resistor that is readily available at a GM dealer. The lights being out could be a bad fuse.

Hi Jim, If the vehicle is fitted with climate control it will have electronic management. However it can be wired by someone knowing electrical circuitry to by pass the control. It will save you some cash but will need manual control. Make sure whoever does the job knows what he or she is doing. Have a happy new year. Regards John

Trace wire from fan to ground if pulling relay doesn't stop fan.

hi..

hi this may be the connection to the blower motor got disconnected so no power is going to the motor. check weather the power is comig.

if the power is coming to the motor check the terminals to which it is been conected. this may be the problem please check and let me know