how to execute Button click event when user press on ctrl+left arrow

Your problem here, as I think you've already become aware, is that with Controls on the Form, the Form is never going to have 'Focus, so: the PreViewKeyDown EventHandler is never going to fire.

Use the KeyDown Event instead:

private void Form1_KeyDown(object sender, KeyEventArgs e)
{
    if (Control.ModifierKeys == Keys.Control)
    {
        switch (e.KeyCode)
        {
            case Keys.Left:
                Console.WriteLine("control-left");
                button1.PerformClick();
                break;
            case Keys.Right:
                Console.WriteLine("control-right");
                break;
        }
    }
}

I suggest you use Console.WriteLine while debugging rather than using 'MessageBox.Show: it's less "intrusive."

Finally, I'd like to note that simulating a Button Click using a key-combination like this is a very unusual design choice, and one that may confuse users. It would be better, imho, to have a common method that would be called by both the Button being clicked, and by the key-combination.