Filter the correct language on WordPress archive pages based on the lang $_GET param.

functions.php

  <?php
  /**
   * Language Detection
   * @param $query,
   * @uses pre_get_posts filter
   * @return the modified query
   */

  function kdk_exclude_category( $query ) {
    if ( ($query->is_tag() && $query->is_main_query()) ||
         ($query->is_search() && $query->is_main_query()) ||
         ($query->is_author() && $query->is_main_query()) ||
         ($query->is_date() && $query->is_main_query()))
    {
        // Get the $GET Param
        if(isset($_GET["lang"]))
        {
          $langcode = $_GET["lang"];
          $query->set( 'category_name', $langcode );
        }

        return;
    }
  }
  add_action( 'pre_get_posts', 'kdk_exclude_category' );

This should be placed in your theme's functions.php file
The language code is the parent category (eg: en - de etc..) so we basically filter posts per category.