You have to use the plural form of an adjective after the definite article. Den is the article, and generelle is the plural form.
It really isn't the plural form, but the plural form has become identical with the definite singular form.
I thought that direktør could mean manager also. General manager is a much more common phrase than general director. I have no idea what a general director is.
No, i have never heard of anyone with that title. The leader of a company are usually just "direktøren". A movie director is "instuktøren".
I would say that den generelle direktør corresponds to E. "the general manager". Typically, you'd see it in a football context: Jorge Valdano er klubbens generelle direktør (JV is the club's general manager). "Director-general", on the other hand, is generaldirektør in Danish: DRs nye generaldirektør, Maria Rørbye Rønn (DR's new director-general, MRR).
@Pips: Funny you should say that because I answered Director General and it marked it as wrong. I believe it's a term we also use in the UK.
It is not plural, but the definite singular form of the adjective does ressemble its plural form.
Would the word "hoved" also fittable for general? (I know that f.x. "hoved rengøring" means "general cleaning" and I think that would be the one I'd do in the spring- putting all the winter stuff away, cleaning windows and all the corners in my flat etc. Am I right?
I'd translate 'hoved' rather with 'main' instead. Gives you a better idea of the usage.
See also the name of Copenhagen's main railway station: Københavns Hovedbanegård -- although this is usually translated as "Copenhagen Central" (a pity in a way, as cities' main railway stations are by no means always centrally located!).
It has been noted by several actual Danes that no Dane would ever use this phrase: "Den generelle direktør," as there is no such title at all in the Danish language, even though when literally translated it means the General Manager or Director; and we do have general managers in English --- especially for professional sports teams. But the same position på dansk must be a title other than "den generelle directør". I believe this phrase is here to teach us that when the adjective is specific, as in this case, it takes the plural form. Therefore, please correct this phrase so that it teaches us the correct Dansk terms and title in the Danish language and not just a literal translation of the English.
Well, the title IS used: e.g.
-- Real-præsident Florentino Perez vil ansætte Hierro som erstatning for Jorge Valdano som klubbens generelle direktør"
-- "Arbejdsloven fastsætter yderligere begrundelsen for at afskedige den generelle direktør"
but not commonly, it's true (most references seem to be to the general managers of foreign football teams, or of other set-ups outside Denmark).