  Full-sky microwave surveys like the upcoming Planck satellite mission will
detect of order 10^4 galaxy clusters through their thermal Sunyaev-Zel'dovich
effect. I investigate the properties of the gravitationally lensing subsample
of these clusters. The main results are: (1) The combined sample comprises
>~70% of the complete sample. (2) It is confined to redshifts 0.2+-0.1, and to
masses (5+-3) x 10^14 solar masses. (3) Using a particular measure for the weak
lensing effect, viz. the aperture mass, cluster masses can be determined with a
relative accuracy of ~20% if their density profile is known. Consequently, the
mass function of the combined sample can accurately be measured. (4) For
low-density universes, I predict a sharp peak in the measured (aperture) mass
function near 5 x 10^14 solar masses and explain its origin, showing that the
peak will be absent in high-density universes. (5) The location of the peak and
the exponential decrease of the mass function on its high-mass side will allow
the determination of the amplitude of the dark-matter power spectrum on the
cluster scale and the baryon fraction in clusters, and constrain the thermal
history of the intracluster gas.
