The polarized fluorescence spectra of the be plane of alpha-sexithienyl (alpha T-6) single crystal at T = 4.2 K have been measured and analyzed. The spectra show a sharp component superimposed on a broad component. The origin of fluorescence at 18332 cm(-1) with a rather narrow linewidth (FWHM = 8 cm(-1)) is completely polarized along b. We assign it to a shallow X-trap 18 cm(-1) below the bottom of the 1 B-1(u) exciton band. The 1 B-1(u) molecular electronic excited level splits in the crystal in four Davydov components (a(g), b(g), a(u), and b(u)) of which a(u) and b(u) are dipole allowed. The calculated exciton band structure based on the Ewald sums, assuming the Feint dipole-dipole approximation, predicts two lowest degenerate Davydov components : an a,, b polarized component and a forbidden as component. The fluorescence with the intrinsic sharp vibronic progressions is analyzed in terms of ground state totally symmetric modes which are in excellent agreement with Raman scattering data of the single crystal. Sharp c polarized false origins are identified at 18 167, 18 084, and 18 026 cm(-1) and discussed either in terms of X-traps or Herzberg-Teller vibronic coupling with the second higher 2 B-1(u) molecular level polarized along the in plane short axis. A broad c polarized component is attributed to aggregates.