"Google Software Update would like to control this computer..." What?

I've received feedback from the Chromium team and the issue is now closed as they stated it's a legitimate Google request. See message from the Chromium team below. Ironically, it opens up more questions. Any additional comments can be seen on the issue page at the Chromium project.

Message from Chromium team (Oct. 17, 2013):

You're right that Chromium does not autoupdate. And the autoupdater used by Google Chrome does not have a GUI element. This "Google Software Update" GUI app is the updater used by some other Google properties, like the AppEngine SDK, Picasa, etc.

Here's the original support issue (#307568) that I opened.

I decided to post this answer to expand on a comment on the accepted answer, because it provides important details as to why the dialog appears, according to one of the Chromium development team members. The details, as they appear in the issue thread, are as follows:

The Google Software Updater runs install scripts for each of the Google products that you have installed: Earth, Chrome, Drive, etc. Each of these scripts are unique because each product has to handle its own install flow.

Some of these scripts use osascript, which is how you can make AppleScript calls from a shell environment. The typical use case is for an installer script to check and see if the product it is updating is already running. An example would be:

$ osascript -e 'tell application "System Events" to exists application process "Google Chrome"'

Unfortunately in Mavericks these calls now make this dialog appear.

So that explains why the dialog appears. Based on that information, the permissions request seem legitimate, which is why I have decided to grant the requested access.

I am only using Google Drive and there is no Chrome on my machine so I suspect this message, which I received also after installing Maverick, is related to Google Drive and not Chrome. In any case this is not what I want, so, deny.