# Formula (5.2-4) in Frenkel's Langlands Correspondence for Loop Groups

The context starts at page 130 of this book. Not only am I not sure how to derive the formula, I'm not even sure how to interpret it:

On the left side, we seem to interpret $x$ as a function, by somehow using coordinates $y_\alpha$. (How exactly?) On the right hand side, we seem to interpret it as a matrix and first conjugate and project to get an element in $\mathfrak{n}_+\subset \mathfrak{g}$, then multiply by $-x$ to arrive at something which is not usually an element of $N_+$ and doesn't seem to be a function on $N_+$ in an obvious way?

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You might want to include the formula itself so that people don't have to have a copy of the book in order to answer your question. –  Qiaochu Yuan Aug 26 '12 at 9:43
In fact I think the entire context is required to understand what you say about coordinates $y_\alpha$, which don't occur in the formula itself. Here's an online version of the book. –  joriki Aug 26 '12 at 10:34