For a finite dimensional simple Lie algebra $\mathfrak{g}$, I am familiar with the following definition of the dual representation: Given a finite dimensional representation of $\mathfrak{g}$ with the highest weight $\omega$, the dual representation is the one with the highest weight $-w_0 (\omega)$ where $w_0$ is the unique element of the Weyl group of $\mathfrak{g}$ with the highest length. (Equivalently, $w_0$ is the unique element that maps the positive roots to the negative roots.)

What is the analogue of this definition for an integrable highest weight representation of the affine Lie algebra $\hat{\mathfrak g}$ (the affine extension of $\mathfrak g$)? My confusion is about the element of the Weyl group of the highest length, given that the Weyl group for an affine Lie algebra is infinite.

I am assuming all the underlying fields to be complex.

If this question is not meaningless then it's probably something already in the literature, I will be happy with a reference. Thank you.


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