SQL Server: Multiple Availability Groups, one cluster, how do I keep the primary cluster node as primary?
Converted from comments:


  Are you referring to the core cluster resources? If yes, the answer is no, failover of the cluster role for the availability group would not impact the core cluster resources. The core cluster resources will of course fail over to another node if the current owner is rebooted or loses connectivity, but that behavior is not impacted by what anything that happens with the AG resources. – Tony Hinkle