The Verlinde algebra description of the Thom class for loop-group representations

Let $G$ be the compact Lie group under consideration, with loop group $\mathbb{L}G$, rank $r$, and dual Coxeter number $h$. Let $V_k$ denote the Verlinde algebra of level $k$, let $D_{\mathbb{L}G}$ be the associated equivariant space, and write its central grading using a formal variable $z$:

$$K_{\mathbb{L}G}^*(D_{\mathbb{L}G})=\bigoplus_n K_{\mathbb{L}G}^{*,n}(D_{\mathbb{L}G})z^n.$$

For the parabolic subgroups $\mathbb{H}_I$ of $\mathbb{L}G$, consider the colimit of their equivariant K-theory groups $\operatorname{colim}_I K_{\mathbb{H}_I}(pt)$.

The Verlinde algebra isomorphism conjecture. For every homogeneous degree $z^n$ with $n\neq 0$, the map

$$\bigoplus_{k\geq 0}V_kz^{k+h}\;\bigoplus_{k\geq 0}V_kz^{-(k+h)}\cong \operatorname{colim}_I K_{\mathbb{H}_I}(pt)\longrightarrow K_{\mathbb{L}G}^{r+1}(D_{\mathbb{L}G})=\bigoplus_n K_{\mathbb{L}G}^{*,n}(D_{\mathbb{L}G})z^n$$

is an isomorphism.

The claim identifies the nonzero homogeneous pieces of the equivariant K-theory of $D_{\mathbb{L}G}$ with the positive- and negative-level Verlinde algebras. The source describes this as suggested by the construction of classes from positive- and antidominant-weight representations; the parser supplies no evidence resolving the claim.