Window Pivot?

Thanks for posting. The white plastic block is called a pivot shoe. As you surmised, it connects to the bottom corners of the sash via pivot pins. We carry many different kinds of pivot shoes, which you can view here. To know which you need, please remove an undamaged version (if possible) and send us a clear picture. We also need to know its width and thickness. That will narrow down your options. The video below shows how to remove and replace pivot shoes.