# Representation theoretic definition of wavefunctions of an integrable hierarchy?

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?