The pressure gradient will try to move the air from the high to the low at the surface, but coriolis will deflect it to the right in the northern hemisphere, so in theory, it will "neutralise" the convergence.
With a low pressure system the pressure gradient is already in towards the centre. Surface friction slows the flow and it turns across the isobars, meaning that you have more flow into the low, giving convergence and feeding the convection.
With a high pressure system, the pressure gradient is away from the high, and surface friction means more air is "sucked" from the high, giving divergence.