At home, I like being invited to events and meetings on occasion. It is part of developing a relationship. That the elected official can meet the officers and membership of the organization and feel they have several contacts is very important. I like to see and meet with the people. I want to know about their concerns. I want to be able to answer questions if they are important to the individuals or group. (I would always rather speak for myself than have someone else carry the message though that might not always be possible.)
In the legislative setting, if the invitations have not gone unanswered, there should be a reunion of sorts. The elected official already knows the group and its leaders. Isn't that a better way to start a conversation than if there has been no contact?
In the case of (name of group)issues I am, based on my record, a "go to" person for their efforts. They should identify natural allies and specifically work to develop firm associations. Once the group has identified it's like-minded legislators then the legislator and the group can work together to improve government.
I appreciated being asked.