My understanding is that the syndicate would be committing an act of extortion.
Under the RICO Act, IF the politician conspired with the syndicate or had something like an 'understanding' with the syndicate then they (any and all involved) could also be charged with the crime (extortion), racketeering and conspiracy.
Additional crimes/laws to consider:
- Campaign finance violations
- Money laundering
- Compelled speech (1st Amendment)