I think the window artifacts you see here are caused by serious over exposure. Digital cameras do not tolerate over exposure well.
The artifacts described here:
http://www.dpreview.com/learn/?/Glossary/Optical/chromatic_aberration_01.htm
are intensified by over exposure. Be sure to read all the whole article.
Filters can cause reflections and artifacts as well, as other point out.
So two different things may be going on.
Was there anything you could do without moving? No, unless you weren't shooting RAW. RAW always helps in this type of situation. To balance outdoor light from a large window you need to use fairly high levels of strobe light. The idea would be to expose for the window (using the highest shutter speed compatible with strobe syncing) and light the room with the strobe. The strobe intensity is adjusted such that the indoor light and window light fall within the dynamic range of your camera. In this case that would be a rather high amount of strobe light.
The purple tint could just be a color balance issue. Or, it could be an artifact from the over exposure, or from filter reflections. The tint looks uniform so you should be able to compensate using any number of image editing software packages.
The lighting conditions from your seat were very difficult. I doubt any camera/lens/filter combination could cope with the dynamic range inside the gym.