Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $\tilde{\mathfrak{g}}=\mathfrak{g}((t))\oplus \mathbf{C}K\oplus \mathbf{C}d$ its Kac-Moody extension ($K$ is the level and $d$ is the energy operator). $\tilde{\mathfrak{h}}=\mathfrak{h}\oplus \mathbf{C}K\oplus \mathbf{C}d$ is the Cartan subalgebra.

Pick an element $\sigma$ of the Weyl group $W$ of order $k$. Then the Heisenberg subalgebra $\hat{\mathfrak{a}}\subset \tilde{\mathfrak{g}}$ is the subalgebra of fixed points of $\mathfrak{h}((t^{1/k}))\oplus \mathbf{C}K$ under the action of $\sigma$, where $\sigma$ acts on $t$ by $t^{1/k}\mapsto \exp(2\pi i/k) t^{1/k}$. For example, for $\sigma=e$ one gets the homogeneous Heisenberg subalgebra $\widehat{\mathfrak{a}}\cong\mathfrak{h}((t))\oplus \mathbf{C} K$, while for $\sigma$ the Coxeter element one gets the principal Heisenberg subalgebra. Let $\mathfrak{a}_+\subset\hat{\mathfrak{a}}$ be the subalgebra containing nonnegative powers of $t$.

Consider $L(\lambda)$, an irreducible integrable highest-weight representation of $\tilde{\mathfrak{g}}$ of highest weight $\lambda\in \tilde{\mathfrak{h}}^*$. What is known about the restriction of $L(\lambda)$ to the Heisenberg $\hat{\mathfrak{a}}$? For example, is it true that the space of invariants $L(\lambda)^{\mathfrak{a}_+}$ is finite-dimensional?

If $\mathfrak{g}$ is simply-laced, $\hat{\mathfrak{a}}$ the homogeneous Heisenberg and $L(\lambda)$ is the level 1 basic representation, then a theorem of Frenkel-Kac and Segal identifies the restriction with a direct sum of Fock modules over the root lattice; in particular, $L(\lambda)^{\mathfrak{a}_+}$ is one-dimensional. It is also one-dimensional for the principal Heisenberg (using the principal realization of Kac-Kazhdan-Lepowsky-Wilson).