CRIRES^+ transmission spectroscopy of WASP-127b. Detection of the resolved signatures of a supersonic equatorial jet and cool poles in a hot planet

General circulation models of gas giant exoplanets predict equatorial jets that drive inhomogeneities in the atmospheric physical parameters across the planetary surface. We studied the transmission spectrum of the hot Jupiter WASP-127\,b during one transit in the K band with CRIRES$^+$. Telluric and stellar signals were removed from the data using SYSREM. The planetary signal was investigated using the cross-correlation technique. After detecting a spectral signal indicative of atmospheric inhomogeneities, we employed a Bayesian retrieval framework with a 2D modelling approach tailored to address this scenario. We detected strong signals of H$_2$O and CO, which exhibited not one but two distinct cross-correlation peaks. The double-peaked signal can be explained by a supersonic equatorial jet and muted signals at the planetary poles, with the two peaks representing the signals from the planet's morning and evening terminators. We calculated a jet velocity of $7.7\pm0.2$ km~s$^{-1}$ and derive distinct atmospheric properties for the two terminators as well as the polar region. Our retrieval yields a solar C/O ratio and metallicity and challenges previous studies of WASP-127b's atmosphere. It provides tentative evidence for the morning terminator to be cooler than the evening terminator by $-175^{+133}_{-117}$K and shows that the muted signals from the poles can be explained by either significantly lower temperatures or a high cloud deck. The presence of a clear double-peaked signal highlights the importance of taking planetary 3D structure into account during interpretation of atmospheric signals. The supersonic jet velocity and lack of signal from the polar regions, representing a detection of latitudinal inhomogeneity in a spatially unresolved target, showcases the power of high-resolution transmission spectroscopy for the characterisation of global circulation in exoplanet atmospheres.