My lab results show very low Immunoglobulin A, I have symptoms of autoimmune disease. Does the immunoglobulin play a role? How?

Genetics. There's a statistical link between low / absent IgA and a proclivity to a couple of the autoimmune diseases. Here the knowledge stops. If you have no solid lab evidence to support a diagnosis of lupus, and you don't have physical findings for scleroderma, polymyositis, dermatomyositis, rheumatoid arthritis, or autoimmune hepatitis, don't let anxiety over this trouble you.