We get the question all the time, so here goes:
A CVSG is an acronym for “call for the views of the Solicitor General.” This is a option for the Supreme Court when the United States is not a party to a cert petition but the interests of the federal government are implicated. It comes up when a party files a cert petition (say, the State of Michigan) where the respondent is not federal (say, the Bay Mills Indian Community) that involves the interpretation of federal statutes or some other federal interests. It comes up a lot (relatively speaking) in Indian law because the United States always has an interest (as trustee, etc.) in Indian cases, and have been involved in more Indian cases by far than any other party (even if you add up all the Supreme Court cases involving all Indian tribes collectively, I bet).
Since the Office of Solicitor General is influential on the Court, in part because it represents the United States and because it does so with remarkable candor about its positions, the SG’s brief (usually termed an “invitation brief”) is a strong indicator where the Court will go in terms of deciding whether to grant cert. In short, if the SG recommends denial, the Court very likely will deny.