UPDATE: Norm Coleman has conceded, rendering most of the below discussion moot.
The Minnesota Supreme Court has ruled unanimously (.pdf) that neither due process nor equal protection were violated during the conducting of that state's recount, and that therefore an election certificate should be given to the candidate determined to have the most votes, which was Al Franken.
Minnesota Governor Tim Pawlenty had suggested yesterday that he would sign the election certificate if the Supreme Court ordered him to. As the Court's decision does not appear to contain an explicit order to Pawlenty to sign the certificate, he may have some wiggle room to elect not to.
It's also possible that Coleman will concede; his chances of success in federal court are considiered low, and are probably lower still given the strength and unanimity of the decision in Minnesota. Moreover, as we have indicated many times, even if a federal court ruled in Coleman's favor and ordered a different procedure for re-counting the ballots, it is unlikely -- perhaps extremely unlikely -- that he would gain enough votes to overtake Franken.
More important than what Pawlenty does may be what the U.S. Senate chooses to do. Harry Reid could try and seat Franken, probably on a provisional basis, even if Pawlenty does not sign the certificate. Alternatively, John Cornyn could try to filibuster his seating even if Pawlenty has signed the certificate. My crude reckoning at this stage is that a filibuster would be slightly more likely than not to be broken, with one or more moderate (Olympia Snowe) or process-oriented (Chuck Grassley) Republicans being candidates to defect. But the spectacle of an attempt to filibuster Franken's seating would be intense and would carry plenty of risks and rewards for both sides, and might not be the the sort of thing that the usually risk-averse Reid would want to undertake. On the other hand, Democrats very well might need Franken's vote on health care, especially if they hope to pass a policy with a public option.