This can happen in the following circumstances:
Order authorization has failed
Please login to your Google checkout account and verify that the credit authorization has been accepted.
Order is in 'Customer review in progress' status
It seems that the order is getting flagged for investigation as potentially fraudulent order. Here's Google's description of this issue - Fraud protection. This document states that this is normally resolved in 4-6 hrs so I would suggest trying to re-download again after that period or contacting Google to get the order processed.
Mystery reason - lots of google posts for the following stuck positions
downloading
authorizing
installing
One user has reported this post as resolving the installing stuck position - http://www.google.com/support/forum/p/Android+Market/thread?tid=47f4f0c5...
If you have purchased the app and are having this problem, you can create an account here and email me the username. I can give you access to download the app from this site.