The example sentence is
- Lisa is upset about not being invited to the party
There are two clauses.
The main clause
- Lisa is upset about
has a psych predicate adjective upset (which requires an auxiliary be),
and a gerund complement clause indicating the source of the upset, namely
- (Lisa('s)) not being invited to the party
(Note that the subject of not being invited has to be Lisa.)
The subject is deleted under identity with the subject of be upset.
So we'll never know whether it was sposta be Lisa or Lisa's; both are possible.
(they're called the
POSS-ing complementizers, respectively.)
OK, so what is the purpose of being in the complement?
Well, it turns out that the complement is a passive construction,
and passive constructions require some form of the auxiliary verb be,
followed by a past participle.
part. Being is a form of be, and invited is a past participle.
That explains why it's a form of be, but it doesn't explain why being was the form chosen.
The reason is that the predicate in the main clause,
takes a gerund complement, and the gerund form of be is being.
See the Verb Phrase Study Guide for further details.