  We present the results of attempts to detect the ellipticity of dark matter
halos using galaxy-galaxy weak lensing with SDSS data. We use 2,020,256
galaxies brighter than r=19 with photometric redshifts (divided into colour and
luminosity subsamples) as lenses and 31,697,869 source galaxies. We search for
and identify several signal contaminants, which if not removed lead to a
spurious detection. These include systematic shear that leads to a slight
spurious alignment of lens and source ellipticities, intrinsic alignments (due
to contamination of the source sample by physically-associated lens source
pairs), and anisotropic magnification bias. We develop methods that allow us to
remove these contaminants to the signal. We split the analysis into blue
(spiral) and red (elliptical) galaxies. Assuming Gaussian errors as in previous
work and a power-law profile, we find f_h=e_h/e_g=0.1+/-0.06 for red galaxies
and -0.8+/-0.4 for blue galaxies using 20-300 kpc/h, averaged over luminosity.
Inclusion of the more realistic non-Gaussian error distributions and of the NFW
density profile (which predicts much smaller ellipticity of the shear for
scales above the scale radius) yields 0.60+/-0.38 for ellipticals and
-1.4+1.7-2.0 for spirals. While there is no concrete detection of alignment in
either case, there is a suggestion in the data of a positive alignment in the
brightest lens sample of ellipticals. Our results appear to be mildly
inconsistent with a previously reported detection by Hoekstra et al. (2004),
but more data and further tests are needed to clarify whether the discrepancy
is real or a consequence of differences in the lens galaxy samples used and
analysis methods.
