I also think that you should have symmetrical peaks. Although I am not familiar with aqueous GPC, I suspect that your standards are sulfonated by reaction of the polystyrene standards (which typically have a polydispersity in the 1.01 - 1.06 range) with H2SO4. The polydispersity should not increase appreciably after sulfonation. Although looking at the Polymer Labs site (http://www.polymerlabs.com/gpc/aquaapps.htm
), there does appear to be some tailing in the standards. I suppose it then becomes a question of how much is acceptable. If these are the display samples, then tailing might be somewhat common.
I would approach your situation in the following manner.
1. Make sure your pump is working correctly (ie check valves aren't stuck). This can be done by simply collecting the fluid coming out into a small graduated cylinder (or by weighing it after a fixed interval knowing the density of the fluid) and confirming that it matched your set flow rate.
2. Organic GPC columns are notorious for "drying out", which is sort of like the channeling effect observed if you have ever done preparative column chromatography in organic chemistry. If your GPC hasn't been used in a while, call up Waters or Polymer Labs and ask their technical service department if there is a test you can do to confirm if your column is OK.
3. If the solution isn't in 1 or 2, then I would look at your standards (unless you have a set of home-made standards from a previous student).