Pilot flames are commonly used to extend combustor operability limits and suppress combustion oscillations in low-emissions gas turbines. Combustion oscillations, a coupling between heat release rate oscillations and combustor acoustics, can arise at the operability limits of low-emissions combustors where the flame is more susceptible to perturbations. While the use of pilot flames is common in land-based gas turbine combustors, the mechanism by which they suppress instability is still unclear. In this study, we consider the impact of a central jet pilot on the stability of a swirl-stabilized flame in a variable-length, single-nozzle combustor. Previously, the pilot flame was found to suppress the instability for a range of equivalence ratios and combustor lengths. We hypothesize that combustion oscillation suppression by the pilot occurs because the pilot provides hot gases to the vortex breakdown region of the flow that recirculate and improve the static, and hence dynamic, stability of the main flame. This hypothesis is based on a series of experimental results that show that pilot efficacy is a strong function of pilot equivalence ratio but not pilot flow rate, which would indicate that the temperature of the pilot products as well as the combustion intensity of the pilot flame play more of a role in oscillation stabilization than the length of the pilot flame relative to the main flame. Further, the pilot-flame efficacy increases with pilot-flame equivalence ratio until it matches the main-flame equivalence ratio; at pilot equivalence ratios greater than the main equivalence ratio, the pilot-flame efficacy does not change significantly with pilot equivalence ratio. To understand these results, we use large-eddy simulation (LES) to provide a detailed analysis of the flow in the region of the pilot flame and the transport of radical species in the region between the main flame and pilot flame. The simulation, using a flamelet/progress variable-based chemistry tabulation approach and standard eddy viscosity/diffusivity turbulence closure models, provides detailed information that is inaccessible through experimental measurements.