Baby, I've been here before
I know this room, I've walked this floor
I used to live alone before I knew you.
I've seen your flag on the marble arch
Love is not a victory march
It's a cold and it's a broken Hallelujah
Clearly W is telling M that she's been in this situation, perhaps with another lover or M. But the "I used to live alone..." seems to contradict this. Maybe it's W saying that she left a previous lover, was alone for a while, then tried again with M. W is disappointed by M; she thought this one would be different. But M has proven to be more in love with love than with W. The "flag on the marble arch" is a sign to the rest of the world boasting that he is in love, shouting it from the rooftops. But W sees this as an abuse of trust. Love is not for the sake of love, she says, but for another person. It may not always be easy, which is why it's "cold and broken."
There was a time you let me know
What's really going on below
But now you never show it to me, do you?
And remember when I moved in you
The holy dove was moving too
And every breath we drew was Hallelujah
M is just going through the motions of having a relationship, and W recognizes here that the relationship is just a shell of what it used to be. He used to tell her what he was feeling, but communication is failing and M is becoming ever more secretive. The next line could simply refer to sex, but if it does it signifies the singer is male and the intended audience is female. It very well could be; the song was written by a man, and I arbitrarily assigned gender at the onset of these shenanigans. If we were to imagine, just for a moment, that Leonard Cohen did have a deeper meaning in mind (*more on this later) and the line is completely symbolic. Perhaps W (going back to my assignations) is begging M to recall a time when she was important to him, enough to cause some change in him. To [metaphorically] move him. I must admit, the dove stumps me. Maybe just symbolizing the soul? I don't know. And to breathe "Hallelujah" with every breath means that they were happy, W and M, and every breath was a gift.