User cannot keep login in browser

Linux+OsCommerce website recently changed SSL. [login to view URL] The page shows customer user name after login. But login disappeared after going to another page. For example, in IE9, go to [[login to view URL]][1]. 1) login using <test@[login to view URL]>. It shows customer name test test in upper right side and it has 'https://' in address bar. 2) Click on any product or click on 'Home', the upper right section becomes login box and user name disappeared. Now it becomes 'http://' in address bar. 3) click on shopping cart, it comes back to test test with https in address bar. I think it's a session/cookie issue. The website did not have this problem before migrating to new server with new SSL cert. The problem exists in IE9 and Firefox 4.0. It may or may not happen in IE8, for example, ie8 on my winxp does not have this issue. Browing to any page the logged in user is always there (upper right section). But the problem shows up in IE8 on Windows server 2003.