FORMATION OF FGT CAUSED FLC OF LGT TO HAVE TCASII RA TO DSND. DEVIATING DOWN 500 FT CLRED THE RA.

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ACFT WAS BEING FLOWN BY THE CAPT TRAINEE FROM THE L SEAT. WE HAD JUST LEVELED AT FL280 ABOUT 2 MINS PRIOR TO THE EVENT. CHICAGO ADVISED THAT WE HAD 3 FIGHTERS AT 2 O'CLOCK AND LEVEL AT FL290. SHORTLY THEREAFTER WHILE WE WERE TRYING TO VISUALLY ACQUIRE THE TFC THE TCAS SOUNDED AN ALERT FIRST FOR TFC (TA) THEN IMMEDIATELY FOLLOWED BY CONFLICT RESOLUTION ADVISORY (RA). THE STUDENT LOOKED AT ME FOR INSTRUCTIONS AND I SAID 'DO IT NOW!' (THE ADVISORY WAS 'DSND; DSND NOW' WITH THE RED BAND EXTENDING TO 1500 FPM DSCNT). VISUAL CONTACT HAD NOT BEEN ESTABLISHED. AN IMMEDIATE PUSHOVER WAS COMMENCED AND THE ADVISORY TERMINATED AT ABOUT FL274. THE ACFT WAS LEVELED AT ABOUT FL275. AT THIS TIME THE INTRUDER SHOWED 1400 FT ABOVE AND VERY CLOSE AT 12 O'CLOCK. DURING THE DEV I TRANSMITTED 2 ADVISORIES TO CHICAGO ARTCC TELLING THEM OF THE ACTIONS WE WERE TAKING. APPARENTLY THEY WERE BUSY TRYING TO RESOLVE THE ALERTS GENERATED AT THEIR FACILITY AND DID NOT ACKNOWLEDGE EITHER XMISSION AND PROFESSED NO KNOWLEDGE OF MY CALLS. THE ACFT WAS RETURNED TO FL280 AFTER ATC WAS FINALLY CONTACTED. COMPANY DISPATCHER WAS NOTIFIED AND CHICAGO CENTER WAS CALLED. APPARENTLY THE FGT WERE NOT AT THE CORRECT ALT AND I WAS INSTRUCTED TO SUBMIT A NEAR MISS RPT.