A B737-500 AT FL320 DECLARED AN EMER AND DIVERTED DUE TO #2 ENG OVERHEAT WARNING. ENG SHUTDOWN AFTER CHKLISTS COMPLETED.

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CRUISING AT FL320 IN VMC; #2 ENG OVERHEAT LIGHT ILLUMINATED. I CALLED FOR THE ENG OVERHEAT CHKLIST. AFTER RETARDING ENG TO IDLE; REQUESTED LOWER ALT FROM ARTCC. CLRED TO FL290 AND STARTED DSCNT. OVERHEAT LIGHT STAYED ON NECESSITATING ENG SHUTDOWN. INITIATED DRIFT-DOWN AND REQUESTED AT LEAST FL190 AND IT WAS APPROVED. ENG WAS SHUT DOWN AND EMER DECLARED WITH ARTCC. CLRED DIRECT ZZZ AND PLT'S DISCRETION TO 11000 FT. EMER INFO PASSED TO ARTCC AND COM ESTABLISHED WITH DISPATCH; AND MAINT DISCUSSED SIT. DECIDED TO LAND AT ZZZ SINCE IT WAS NEAREST SUITABLE WITH AVAILABLE DRIFT-DOWN ALT AND TIME AVAILABLE TO COORD WITH CREW AFTER RUNNING ALL CHKLISTS AND COORDINATING WITH FLT ATTENDANTS AND BRIEFING FOR LNDG APCH. DSCNT; APCH; AND LNDG UNEVENTFUL. CABIN ADVISORY USED WITH FLT ATTENDANTS. PAX BRIEFED BY CAPT. APCH FLOWN BY CAPT. MAINT DISPATCH AND DUTY OFFICER BRIEFED BY CAPT. SUPPLEMENTAL INFO FROM ACN 650032: MAINT HAD DEFERRED ONE OF THE #2 ENG FIRE LOOPS PRIOR TO ACCEPTING THE ACFT. DURING CRUISE FLT AT FL320; THE #2 ENG OVERHEAT ANNUNCIATION ILLUMINATED. ALL APPROPRIATE CHKLISTS WERE PERFORMED IN THEIR ENTIRETY. THE ENG OVERHEAT ANNUNCIATION DID NOT EXTINGUISH AFTER OPERATING THE ENG AT IDLE FOR 30 SECONDS. CALLBACK CONVERSATION WITH RPTR ACN 650031 REVEALED THE FOLLOWING INFO: THE RPTR STATED THE MAINT ACTION TAKEN AND THE FAILED COMPONENT IS UNKNOWN. CALLBACK CONVERSATION WITH RPTR ACN 650032 REVEALED THE FOLLOWING INFO: NO MAINT RPT TO CREW AND HAS NO IDEA WHAT FIXED THE AIRPLANE.