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Exact spectra of periodic samples are computed up to $ N=36 $.  Evidence of an extensive set of low lying levels, lower than the softestmagnons, is exhibited.  These low lying quantum states are degenerated in the thermodynamic limit;their symmetries and dynamics as well as their finite-size scaling are strongarguments in favor of N\'eel order.  It is shown that the N\'eel order parameter agrees with first-order spin-wavecalculations. A simple explanation of the low energy dynamics is given as wellas the numerical determinations of the energies, order parameter and spinsusceptibilities of the studied samples. It is shown how suitable boundaryconditions, which do not frustrate N\'eel order, allow the study of sampleswith $ N=3p+1 $ spins.  A thorough study of these situations is done in parallel with the moreconventional case $ N=3p $.Comment: 36 pages, REVTeX 3.0, 13 figures available upon request, LPTL  preprin

Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice