I am reading Kac's book on infinite dimensional lie algebras. In the last chapter, he starts with a highest weight module of an affine lie algebra $\mathfrak{g}(A)$, and uses it to define tau functions and Hirota bilinear equations. Roughly speaking starting from the basic representation of an affine lie algebra with highest weight $v_{\lambda}$, the space of tau functions is given by the orbit of the vacuum vector under the action of a certain automorphism group $G$ of the basic representation $L(\Lambda_0)$. Using this formalism, one can obtain the exact system of differential equations of the integrable hierarchy. For example, when $\mathfrak{g}(A)=gl_{\infty}$ and $\mathfrak{g}(A)=\widehat{\mathfrak{sl}_2}$ one gets the KP and KdV hierarchies respectively. The wavefunction $\psi(t,z)$ is a fundamental object in integrable hierarchies (for ex. the work for Jimbo-Date-Kashiwara-Miwa). In the case of the KdV hierarchy,

$$ \frac{\partial \psi}{\partial t_n} = (L^{n+1/2})_{+} \psi, \quad L\psi = z\psi $$

where $L$ is the Lax operator. Has there been any progress in understanding this object from a representation theory point of view.

In particular:

  1. Given a highest weight module of a certain affine lie algebra, how can you describe the space of wave-functions?
  2. Can you generate a system of differential equations such as the one above from the algebraic point of view?

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