  We report and discuss JHKs photometry for Sgr dIG, a very metal-deficient
galaxy in the Local Group, obtained over 3.5 years with the Infrared Survey
Facility in South Africa. Three large amplitude asymptotic giant branch
variables are identified. One is an oxygen-rich star that has a pulsation
period of 950 days, that was until recently undergoing hot bottom burning, with
Mbol~-6.7. It is surprising to find a variable of this sort in Sgr dIG, given
their rarity in other dwarf irregulars. Despite its long period the star is
relatively blue and is fainter, at all wavelengths shorter than 4.5microns,
than anticipated from period-luminosity relations that describe hot bottom
burning stars. A comparison with models suggests it had a main sequence mass
Mi~5 times solar and that it is now near the end of its AGB evolution. The
other two periodic variables are carbon stars with periods of 670 and 503 days
(Mbol~-5.7 and -5.3). They are very similar to other such stars found on the
AGB of metal deficient Local Group Galaxies and a comparison with models
suggests Mi~3 times solar. We compare the number of AGB variables in Sgr dIG to
those in NGC6822 and IC1613, and suggest that the differences may be due to the
high specific star formation rate and low metallicity of Sgr dIG.
