Ideals in enveloping algebras of affine Kac-Moody algebras

Let L be an affine Kac-Moody algebra, with central element c, and let λ∈C. We study two-sided ideals in the central quotient Uλ(L):=U(L)/(c−λ) of the universal enveloping algebra of L, and prove:
Theorem 1. If λ≠0 then Uλ(L) is simple.
Theorem 2. The algebra U0(L) has just-infinite growth, in the sense that any proper quotient has polynomial growth.
As an immediate corollary, we show that the annihilator of any nontrivial integrable highest weight representation of L is centrally generated, extending a result of Chari for Verma modules.
We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals.