  HIRES, a high resolution spectrometer, is one of the first five instruments
foreseen in the ESO roadmap for the E-ELT. This spectrograph should ideally
provide full spectral coverage from the UV limit to 2.5 microns, with a
resolving power from R$\sim$10,000 to R$\sim$100,000. At visual/blue
wavelengths, where the adaptive optics (AO) cannot provide an efficient
light-concentration, HIRES will necessarily be a bulky, seeing-limited
instrument. The fundamental question, which we address in this paper, is
whether the same approach should be adopted in the near-infrared range, or
HIRES should only be equipped with compact infrared module(s) with a much
smaller aperture, taking advantage of an AO-correction. The main drawbacks of a
seeing-limited instrument at all wavelengths are: \textit{i)} Lower
sensitivities at wavelengths dominated by thermal background (red part of the
K-band). \textit{ii)} Much higher volumes and costs for the IR spectrograph
module(s). The main drawbacks of using smaller, AO-fed IR module(s) are:
\textit{i)} Performances rapidly degrading towards shorter wavelengths
(especially J e Y bands). \textit{ii)} Different spatial sampling of extended
objects (the optical module see a much larger area on the sky). In this paper
we perform a trade-off analysis and quantify the various effects that
contribute to improve or deteriorate the signal to noise ratio. In particular,
we evaluate the position of the cross-over wavelength at which AO-fed
instruments starts to outperform seeing-limited instruments. This parameter is
of paramount importance for the design of the part of HIRES covering the
K-band.
