Java static members in a subclass accessed via a generic parent

This seems like a newbish question, but the last time I worked with Java, the language didn't have generics. I have a class hierarchy (names changed to be as generalized as possible):

public abstract class AbstractBase { .... }
public class ConcreateSubA extends AbstractBase { .... }
public class ConcreateSubB extends AbstractBase { .... }
...
public class ConcreateSubZZ9PluralZAlpha extends AbstractBase { .... }
...

I'm trying to clean up some legacy code, and there's one place where a ton of repetitive duplication can all be factored out into a single routine via generics. (I'm thinking generics because when this routine is called, it needs to operate on only one of the concrete classes.)

The routine looks like

public <Thing extends AbstractBase> void someFunc()
{
    another_function_call (Thing.concreteSpecialToken);
    // could also be
    //   another_function_call (Thing.concreteSpecialToken());
    // if methods are more feasible than fields

    // Cannot use
    //   another_function_call (Thing().concreteSpecialToken());
    // because creating instances of these types is a Major Operation[tm]
}

I'm leaving out about a zillion lines, but that's the important part: someFunc() is type parametric (it actually takes arguments but none of them are Things so no inference). Eventually I need to fetch a special token and this is where I'm getting fuzzy.

The tokens are huge-ass unique strings for each concrete class. They're class-based, not instance-based. The actual token value is declared as a private static final field in each subclass.

So I need to use the public methods/fields of a base class to (eventually) get to the private static field of a subclass. Obviously I can't declare an abstract static method in the base, because that makes no sense. If the data were instance-based, then this would be trivial, with