Although it is well understood that selection shapes the polymorphism pattern in Drosophila, signatures of classic selective sweeps are scarce. Here, we focus on Drosophila mauritiana, an island endemic, which is closely related to Drosophila melanogaster. Based on a new, annotated genome sequence, we characterized the genome-wide polymorphism by sequencing pooled individuals (Pool-seq). We show that the interplay between selection and recombination results in a genome-wide polymorphism pattern characteristic for D. mauritiana. Two large genomic regions (>500 kb) showed the signature of almost complete selective sweeps. We propose that the absence of population structure and limited geographic distribution could explain why such pronounced sweep patterns are restricted to D. mauritiana. Further evidence for strong adaptive evolution was detected for several nucleoporin genes, some of which were not previously identified as genes involved in genomic conflict. Since this adaptive evolution is continuing after the split of D. mauritiana and Drosophila simulans, we conclude that genomic conflict is not restricted to short episodes, but rather an ongoing process in Drosophila.