Google Calendar – Send Gmail Calendar Invite from Alternate Email

The instructions at http://www.tekgrl.com/accepting-google-calendar-invites-to-a-non-gmail-account/ worked for me, taking into the account the comments at the end. Here's a summary:

Google has finally fixed this, all you have to do is enable a setting in your Google Calendar to be able to respond to any calendar invite sent to any of your Gmail alternate addresses. Here’s how you do it:

1. [from the comments] Set up the alternate email address a@example.com for your Google account. The option in Google Calendar below may not show up unless you have. Here is Google’s help article for setting up alternate addresses.
2. [from the comments] Wait some necessary time for the Google Calendar to recognize you've added the Alternate Email from step above.
3. In Google Calendar (on a@gmail.com account`) click the gear icon and choose Settings.
4. Select your primary calendar in the Calendars Configuration menu item on the left pane.
5. In the “General Notifications” section for your primary calendar, check the box “Allow me to respond to event invitations forwarded from these addresses.”

Note that invitations already sent before you set this up won't work. Only new invitations will work properly.

In the new Google Calendar settings interface, the requisite "respond to forwarded events" setting described by Stevoisiak's answer is found under the General Notifications category of settings accessed by selecting your main calendar in the left-side menu.

Note, it can take some time before the setting shows up in GCal after you've added and verified your alternate email address.