You can easily publish important email messages coming into your Gmail to a Slack channel using Google Apps Script. Your Slack channel has a Webhook URL and you can send a POST HTTP request to this URL with the email message in JSON format as the payload parameter.

You can archive the message, or apply a specific Gmail label, to prevent duplicate posts in Slack for the same message. Credit: @matsuyoro

var postUrl = 'https://hooks.slack.com/services/xxxxxxxxxxxxxxxx';
var postChannel = '#labnol';

function postGmailToSlack() {
  var threads = GmailApp.getInboxThreads();

  for (var m in threads) {
    sendHttpPost(threads[m].getFirstMessageSubject() + ' link:' + threads[m].getPermalink(), 'GMAIL');
  }
}

function sendHttpPost(message, username) {
  var jsonData = {
    channel: postChannel,
    username: username,
    text: message,
  };
  var payload = JSON.stringify(jsonData);
  var options = {
    method: 'post',
    contentType: 'application/json',
    payload: payload,
  };

  UrlFetchApp.fetch(postUrl, options);
}