Commercial Fixed Wing
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 1.5
Situational Awareness; Communication Breakdown
ATC Issue All Types; Conflict Airborne Conflict
Air Traffic Control Separated Traffic; Air Traffic Control Issued New Clearance
Human Factors
I had just taken the position. One of the aircraft I had was Aircraft X. This aircraft was at 39;000 feet still in the adjacent sector's airspace with a crossing at a intersection at 25;000 feet. I am assuming that the sector had handed off the aircraft with a discretionary descent; but this was not explicitly stated in the briefing. Shortly after sitting down; the aircraft reported leaving 39;000 feet for the crossing. Right after he said this; Conflict Alert activated with Aircraft Y who was in their airspace at 38;000 feet. This was the first I knew of this aircraft. Once CA (Conflict Alert) activated I issued a vector 30 degrees right for the traffic and coordinated. He said he had turned Aircraft Y; but the vector turned out to be insufficient.  We lost separation with approximately 4.8 miles and less than 1;000 feet.
Aircraft Y landing PHL I gave a pilot's discretion descent from 390 to 330 and switched to IRL who issued the crossing restriction of COFAX at FL250.  He did not start his descent until around JST.  Aircraft W was SW from PSB to HVQ at 380.  I saw the confliction and intended to turn the Aircraft X sooner but got busy with other traffic in the sector.  I turned aircraft X but not enough and it took him too long to start and complete the turn.
ZOB Center Controllers working adjacent sectors reported one of them gave a descent clearance to an aircraft with another aircraft on a conflicting route.