The global database name is made up of the values set for the initialization parameters db_name plus db_domain for each database. If db_domain is not set, then the global database name is simply the value set for db_name.
You can find out what the global name is for your database by executing the following query from SQL*Plus:
select * from global_name;
It is also possible to alter the global name of an existing database with the following command:
alter database rename global_name to <db_name.db_domain>;
In your case, if the parameter db_domain is set, then the global name for your database would be db_name.db_domain. If you do not have db_domain set, then the global_name would just be set to the value of db_name (which is usually the SID).