Good question. He seems to lead a very secluded life. Other than Sherlock he has no friends at all and usually spends Christmas at home and not with his parents as we could see in ASiB. And being even more clever than Sherlock I suppose he would not be able to stand being surrounded by goldfish all day.

I a not sure about the connection to HLV apart from the fact that he warns Sherlock about getting involved, something he usually avoids, and then must realise that he is already involved because he loves his brother.

Also, it set up the scenario where Mycroft only communicates with him as a figment of his imagination, instead of real life.  And it puts the whole focus on Sherlock to solve the crime.

And, perhaps, Mycroft knows about Mary, and perhaps Mary suspects, and he didn't want to be face to face with her.

To me, Mycroft's refusal to the wedding invitation is mainly for plot reasons.
Imagine Sherlock and him being together on the wedding. It would have ended probably with both of them leaving the party early - which would have screwed up the end which left Sherlock behind as the lonely wolf.

Mycroft would have solved the case.

zeratul wrote:

Well as we learned in the mini episode, Sherlock doesn't like such events, so I think Mycroft also doesn't. Sherlock is just there, because he loves John and he was asked to be his best men (as he also says on the phone). I think Mycroft just didn't like to go there.

Exactly. John is important enough to Sherlock to endure this kind of event he doesn't like. Mycroft doesn't feel that way.

I am actually a bit surprised John invited Mycroft. Does he consider Mycroft a friend? That speaks to his closeness to Sherlock.

It goes along with the subtexual theme of a three-way wedding: Sherlock's brother is invited because it's Sherlock's wedding too. 

Actually, I am surprised at how many people John had that he invited, considering that he's previously said he was all alone without Sherlock.