Learn Russian: Этот вопрос не касается моего авторитета. - This issue does not concern my authority.

The verb касаться (to concern / to touch upon) strictly requires its object to be in the genitive case. In Russian grammar, verbs that express reaching, touching, or achieving often require the genitive, rather than the accusative direct object you might expect from English.

Is моего авторитета in the genitive because of the negative не (Genitive of Negation)?

No, though that is a very logical guess! While negated verbs in Russian often force an otherwise accusative object into the genitive, the verb касаться takes the genitive all the time. It would still be моего авторитета even if the sentence were positive.

Касаться usually means "to touch". How does it mean "to concern" here?

Касаться literally means "to touch", but it is very frequently used in a figurative sense to mean "to touch upon", "to relate to", or "to concern". Because the subject вопрос (issue/question) is an abstract concept, it cannot physically touch anything, making the figurative meaning the only natural translation.

Could we use the perfective verb коснуться in this sentence?