Chevy traverse 2010 air conditioning button blinks then shuts off. What could be causing this??

Have it checked for DTC'S -diagnostic trouble codes . That will tell you what's up . Dealer or someother qualified repair shop .
DTC P0532: Air Conditioning A/C Refrigerant Pressure Sensor Circuit Low Voltage
DTC P0533: Air Conditioning A/C Refrigerant Pressure Sensor Circuit High Voltage
DTC P0645: A/C Clutch Relay Control Circuit
Circuit System Description
When the A/C switch is pressed, the HVAC control module sends a serial data message to the ECM. This input will request the engine control module (ECM) to ground the A/C compressor clutch relay control circuit, which will switch the A/C CMPRSR CLUTCH relay on. With the relay contacts closed, battery voltage is supplied to the A/C compressor clutch assembly.

DTC B0228 02 : Recirculate Position Feedback Circuit Short to Ground
DTC B0228 05 : Recirculate Position Feedback Circuit Short to Battery or Open
DTC B0228 61 : Recirculate Position Circuit Actuator Stuck
DTC B0413 02 : Temperature Control 1 Feedback Circuit Short to Ground
DTC B0413 05 : Temperature Control 1 Feedback Circuit Short to Battery or Open
DTC B0413 61 : Temperature Control 1 Feedback Circuit Actuator Stuck Open
DTC B0423 02 : Temperature Control 2 Feedback Circuit Short to Ground
DTC B0423 05 : Temperature Control 2 Feedback Circuit Short to Battery or Open
DTC B0423 61 : Temperature Control 2 Circuit Actuator Stuck