D01 Controller described conflict event with two aircraft turning final; the reporter indicated attempts to clarify a TCAS RA alert cluttered the frequency; delaying instructions to the conflicting aircraft.

Narrative

I was working FR2; all finals combined; visual approaches to Runway 5L/R. The Number 1 Air Carrier was at 90; 090 heading base leg to the Visual Approach Runway 35L. The aircraft was restricted to 90 due to a military fighter inbound to BKF AFB for an overhead Visual Approach to Runway 32. The B190 was on a 320 heading to intercept the Runway 35R localizer and cleared for the Visual Approach Runway 35R. The second company Air Carrier X was a 170 heading; at 110; and assigned a descent to 90. I believe Air Carrier X had just left 110 when he advised me he was responding to a TCAS RA at his 12 o'clock. I looked and advised I didn't observe any traffic at his 12 o'clock the closest was the military fighter now in the break at BKF below 7200 MSL; about 10 o'clock and 2nm. After some discussion we determined it must have been the military fighter; but I commented I was surprised he received an RA on traffic over 3000ft below him. As the pilot responded I observed that I needed to turn the number 1 Air Carrier to intercept his final; but Air Carrier X was still talking. The number 1 Air Carrier had previously reported the airport in sight. As soon as Air Carrier X un-keyed; I issued the number 1 Air Carrier a left turn direct to DYMON; the Runway 35L FAF; and issued a visual approach clearance. He had previously been assigned 170KTS; but I felt he would still go through the final. The B190 appeared to be established on the Runway 35R final and just leaving 10000 MSL. I issued traffic to the B190. He didn't just say 'in sight' or 'not in sight'; but I thought I eventually heard in sight. I instructed him to maintain visual separation; and that the first Air Carrier was going to overshoot his final. The B190 responded for either the first or second time; in sight and acknowledged the; maintain visual separation. The B190 advised that he was going to widen out reference the first Air Carrier. I advised that was fine. I was again surprised; this time by how little he widened out; and that he continued his descent. I had advised the first Air Carrier to expedite his turn to join his final reference a B190 behind and too his right for Runway 35R. All else was normal. No complaint or comment was received by either pilot on the frequency. Recommendation; rather than discussing the RA incident with the downwind Air Carrier X; just acknowledge the event and continue on. This is hindsight speaking; I know that because I didn't observe any traffic at his specified position; I wanted him to know that there really wasn't an aircraft at his 12 o'clock. I think quite a few pilots mostly without military backgrounds don't realize they can get a false return; just like we do with our RADAR. Because of the position and speed of the military fighter in the overhead at BKF; Air Carrier X probably didn't get any altitude; and received an RA. This is conjecture on my part; but the bottom line; don't get distracted by the unusual.