if the GT driving the compressor is an industrial type GT, not an Aircraft Derivative, the black start of the GT is just using the DC auxiliaries instead of the AC auxiliaries.
So... the emergency DC Lube oil pump is used AND there is an alternator (convert DC to AC) to drive the sparkplugs
when the speed reaches about 40% the main lube oil pump and cooling water pump and fin-fan coolers (packaged units) are driven by the auxiliary gearbox and normal operation resumes.
google black-start and plenty of documents show up.
saludos.
a.