Hosting the webplayer on a https url will cause the default web templates to fail loading correctly. You need to make sure that all content referenced by the web page come from https addresses. Either you can modify the html page to get the UnityObject.js from "https://ssl-"... or if you provide your own UnityObject.js file that does not access any http:// links. Also there is a link to an image that should also be replaced from "http://" to "https://ssl-".
For your convenience, Unity 3.5.x Default SSL WebPlayer Template can be installed into your Unity WebPlayer Templates so you can build a https:// compatible player with very little setup.