  We report on 3--4 $\mu$m slit spectroscopy of 13 Seyfert 2 nuclei. The 3.3
$\mu$m polycyclic aromatic hydrocarbon (PAH) emission is used to estimate the
magnitudes of compact ($<$ a few 100 pc) nuclear starbursts and to resolve the
controversy over their energetic importance in Seyfert 2 nuclei. For three
selected Seyfert 2 nuclei that have been well studied in the UV, the magnitudes
of the compact nuclear starbursts estimated from the 3.3 $\mu$m PAH emission
(with no extinction correction) are in satisfactory quantitative agreement with
those based on the UV after extinction correction. Based directly on our 3--4
$\mu$m slit spectra, the following two main conclusions are drawn: (1) except
in one case, the observed nuclear 3--4 $\mu$m emission is dominated by AGN and
not by starbursts, and (2) compact nuclear starbursts are detected in 6 out of
13 Seyfert 2 nuclei, but cannot dominate the energetics of the galactic
infrared dust emission in the majority of the observed Seyfert 2 galaxies. For
several sources for which Infrared Space Observatory spectra taken with larger
apertures and/or soft X-ray data are available, these data are combined with
our 3--4 $\mu$m slit spectra, and it is suggested that (3) extended (kpc scale)
star-formation activity is energetically more important than compact nuclear
starbursts, and contributes significantly to the infrared luminosities of
Seyfert 2 galaxies, and (4) the bulk of the energetically significant extended
star-formation activity is of starburst type rather than quiescent normal disk
star-formation; the extended starbursts are responsible for the
superwind-driven soft X-ray emission from Seyfert 2 galaxies. Finally, it is
implied that more powerful AGNs tend to be related to more powerful compact
nuclear starbursts.
