LT-equivariant Index from the Viewpoint of KK-theory

Let $T$ be a circle group, and $LT$ be its loop group. We hope to establish an index theory for infinite-dimensional manifolds which $LT$ acts on, including Hamiltonian $LT$-spaces, from the viewpoint of $KK$-theory. We have already constructed several objects in the previous paper \cite{T}, including a Hilbert space $\mathcal{H}$ consisting of "$L^2$-sections of a Spinor bundle on the infinite-dimensional manifold", an "$LT$-equivariant Dirac operator $\mathcal{D}$" acting on $\mathcal{H}$, a "twisted crossed product of the function algebra by $LT$", and the "twisted group $C^*$-algebra of $LT$", without the measure on the manifolds, the measure on $LT$ or the function algebra itself. However, we need more sophisticated constructions. In this paper, we study the index problem in terms of $KK$-theory. Concretely, we focus on the infinite-dimensional version of the latter half of the assembly map defined by Kasparov. Generally speaking, for a $\Gamma$-equivariant $K$-homology class $x$, the assembly map is defined by $\mu^\Gamma(x):=[c]\otimes j^\Gamma(x)$, where $j^\Gamma$ is a $KK$-theoretical homomorphism, $[c]$ is a $K$-theory class coming from a cut-off function, and $\otimes$ denotes the Kasparov product with respect to $\Gamma\ltimes C_0(X)$. We will define neither the $LT$-equivariant $K$-homology nor the cut-off function, but we will indeed define the $KK$-cycles $j^{LT}_\tau(x)$ and $[c]$ directly, for a virtual $K$-homology class $x=(\mathcal{H},\mathcal{D})$ which is mentioned above. As a result, we will get the $KK$-theoretical index $\mu^{LT}_\tau(x)\in KK(\mathbb{C},LT\ltimes_\tau \mathbb{C})$. We will also compare $\mu^{LT}_\tau(x)$ with the analytic index ${\rm ind}_{LT\ltimes_\tau\mathbb{C}}(x)$ which will be introduced.