FLC OF LR36 SIGHTS TFC WHICH APPEARS TO BE BELOW WHEN ASKED TO EXPEDITE CLB. THE OTHER ACFT RECEIVES A TCASII ALERT. UPON CALLING CTR AS REQUESTED IT IS INDICATED THAT THERE WAS NO REAL PROB.

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WHILE CLBING ENRTE; ZID ASSIGNED FL330. DUE TO THE ACFT WT AND ABOVE NORMAL TEMP AT ALT MY ACFT (AN LR36) WAS CLBING ABOUT 1200-1500 FPM AT THE TIME OF THE OCCURRENCE. AS WE WERE GOING THROUGH FL276 ATC TOLD US TO EXPEDITE THROUGH FL280. WE HAD JUST BARELY ADJUSTED OUR PITCH ATTITUDE TO COMPLY WITH ATC INSTRUCTION; WHEN SIMULTANEOUSLY ATC ASKED OUR ALT; ALSO WE NOTICED AN MD80 APCHING FROM R TO L APPROX FL270. WE CALLED LEAVING FL280 WITHIN 5 SECONDS FROM THE INITIAL ATC REQUEST TO EXPEDITE OUR CLB. INDIANAPOLIS NEVER MENTIONED THE MD80 TFC TO US. ABOUT 1 HR LATER WHILE ON ZJX; WE WERE GIVEN A TELEPHONE NUMBER AND TOLD TO CALL ZID REGARDING A POSSIBLE AIRSPACE DEV. I CALLED AT MY DEST; AND WAS TOLD THAT INITIALLY THEY THOUGHT THAT THERE WAS A PLTDEV; BUT AFTER CAREFULLY LOOKING AT THE SIT THERE WERE NO INFRACTIONS. NOW I WAS A LITTLE CONFUSED; SO I ASKED WHAT SIT ARE WE DISCUSSING. I WAS TOLD THAT THE MD80 WAS ASSIGNED CLB TO FL270. APCHING FL270 THE MD80 RECEIVED A TCASII RA THAT TOLD THEM TO DSND. THERE WERE SEVERAL REASONS WHY THE TCASII ACTIVATED. ASSUMING THE MD80 WAS CLBING AT A RATE FASTER THAN US THEN THAT WOULD CAUSE AN RA. THE MD80 FLT PATH CROSSED OURS WITHOUT THE MINIMUM SEPARATION. ATC NEVER CALLED THE MD80 TFC; NOR TOLD US TO EXPEDITE UNTIL IT WAS TOO LATE. THE MD80 OR US SHOULD HAVE BEEN ASSIGNED A LOWER ALT UNTIL SAFE SEPARATION WAS ACHIEVED.