Don't use a button, use a link and set the href attribute to the URL of your PDF file. The browser will handle the file download for you, honoring the user's browser preferences.
<a href="your/file.pdf" />
If you need the link to look like a button, you can style it using CSS. See for example this SO thread.
But the user's browser can block the new window from being created. You might want to simply generate a download link:
$('<a>Click here to download PDF</a>').attr('href', filename).appendTo(that.$el);
And have the user click the link to initiate the file download.
@beNerd, you should've probably included that in your original question... anyway, edited the answer. No silver bullet.
This will work, but the responding server will also need to set the Content-Disposition header to "attachment" if you want to force a download: stackoverflow.com/questions/8875949/
@Ben, true, but I think it makes more sense to let the browser make the decision. That's what user in most cases expects, and they can Save link as... if they want to force a download themselves.