text:
\section{Introduction}
Let $G$ be a compact, connected Lie group, and  $(M, \omega)$ a compact symplectic manifold with a Hamiltonian $G$-action. By choosing a $G$-invariant $\omega$-compatible almost complex structure on $M$, we can define a $G$-equivariant $\Z_2$-graded spinor bundle $S^\pm_{M}$. If the Hamiltonian $G$-space $M$ is pre-quantizable and has a $G$-equivariant pre-quantum line bundle $L$, we define a $\Z_{2}$-graded Hilbert space by
\[
\mathcal{H}^\pm = L^{2}(M, S^\pm_{M}\otimes L)
\]
and a $G$-equivariant Spin$^{c}$-Dirac operator
\[
D^\pm : \mathcal{H}^\pm \to \mathcal{H}^\mp.
\]
Attributed to Bott, the \emph{quantization} of $(M, \omega)$ can be defined as the equivariant index
\[
Q(M, \omega) = \mathrm{Ind}(D) =  [\mathrm{ker}(D^+)] - [\mathrm{ker}(D^-)] \in R(G).
\]

The goal of this paper is to generalize the quantization process to the \emph{quasi-Hamiltonian $G$-space} introduced by Alekseev-Malkin-Meinrenken \cite{AMM98}. The q-Hamiltonian $G$-space, arising from infinite-dimensional  Hamiltonian loop group space, differs in many respects from Hamiltonian $G$-space. In particular, the moment map takes values in the group $G$ and the 2-form $\omega$ doesn't have to be closed or non-degenerate. Consequently,  the two key ingredients in defining $Q(M, \omega)$: the spinor bundle $S_{M}$ and pre-quantum line bundle $L$ might not exist in general.

Given a q-Hamiltonian $G$-space $(M, \omega)$, we use the spin representation of loop group to construct twisted spinor bundles $S^\mathrm{spin}$ on $M$, and  the Hilbert space of Wess-Zumino-Witten model to construct twisted pre-quantum bundles $S^\mathrm{pre}$. Both of them are bundles of Hilbert spaces and play the same roles as the spinor bundle and pre-quantum line bundle for Hamiltonian $G$-spaces. We analogously define a Hilbert space
\[
\mathcal{H}:= \big[L^{2}(M, S^{\mathrm{spin}}\otimes S^{\mathrm{pre}})\big]^{G}.
\]

One key in the construction of Dirac operators on $\mathcal{H}$ is the algebraically defined cubic Dirac operator. It was introduced by Kostant for finite-dimensional Lie group, and extensively studied for infinite-dimensional loop group by various people.  Our strategy is to construct a Dirac operator as a combination of algebraic cubic Dirac operators and geometric Spin$^{c}$-Dirac operators. To be more precise, we choose an open cover of $M$ using the symplectic cross-section theorem for q-Hamiltonian $G$-spaces,  so that every open subset $U$ has the geometric structure:
\[
U \cong G \times_{H} V,
\]
where the slice $V$ is a Hamiltonian $H$-space. Accordingly, the tangent bundle $TU$ splits equivariantly into ``vertical direction" and ``horizontal direction". We define a suitable Dirac operator on $U$ so that it acts as the Spin$^{c}$-Dirac operator on the vertical part $V$ and the cubic Dirac operator for loop group on the horizontal part $G/H$. Using partition of unity, we obtain a global Dirac operator $\mathcal{D}$ on $\mathcal{H}$ by patching together Dirac operators on the open sets $U$.  The main result of this paper is that we show the Dirac operator $\mathcal{D}$ has a well-defined index given by positive energy representations of loop group.

\subsection*{Acknowledgements}
The author received many useful suggestions as well as warm help from Eckhard Meinrenken. The author also would like to thank Nigel Higson and Mathai Varghese  for inspiring conversations.

\section{Loop group and positive energy representation}
We first give a brief review on loop groups and their representations. We use \cite{PS86} as our primary reference.

\subsection{Loop group and central extension}
Let $G$ be a compact, simple and simply connected Lie group, and fix a ``Sobolev level" $s > 1$. We define $LG$ the \emph{loop group}  as the Banach Lie group consisting of maps $S^{1} \to G$ of Sobolev class $s +\frac{1}{2}$ with the group structure given by pointwise multiplication. The Lie algebra $L\mathfrak{g} = \Omega^{0}(S^{1}, \mathfrak{g})$ is given by the space Lie algebra $\mathfrak{g}$-valued 0-forms of Sobolev class $s + \frac{1}{2}$ and  $L\mathfrak{g}^{*} = \Omega^{1}(S^{1}, \mathfrak{g})$  the space of $\mathfrak{g}$-valued 1-forms of Sobolev class $s-\frac{1}{2}$. Integration over $S^{1}$ gives a natural non-degenerate pairing between $L\mathfrak{g}$ with $L\mathfrak{g}^{*}$.

Note that $L\mathfrak{g}^{*}$ can be identified with the affine space of connections on the trivial principle $G$-bundle over $S^{1}$. The loop group $LG$ acts on $L\mathfrak{g}^{*}$ by gauge transformation
\begin{equation}
\label{gauge transformation}
g \cdot \xi = \mathrm{Ad}_{g}(\xi) - dg \cdot g^{-1}, \hspace{5mm} g \in LG, \xi \in L\mathfrak{g}^{*},
\end{equation}
where $dg\cdot g^{-1}$ is the pull-back of the right-invariant Maurer-Cartan form on $G$.

Let $\widehat{LG}$ be the basic central extension of $LG$, defined infinitesimally by the cocycle
\[
(\xi_1, \xi_2) \mapsto \oint d\xi_1 \cdot \xi_2, \hspace{5mm} \xi_1,\xi_2 \in L\mathfrak{g}^*.
\]
The coadjoint action of $LG$ on
\[
\widehat{L\mathfrak{g}}^* = L\mathfrak{g}^*\oplus \mathbb{R}
\]
is given by the formula
\[
g \cdot (\xi,k) = (\mathrm{Ad}_{g}(\xi) - k \cdot g^{-1}dg, k).
\]
One can view the action (\ref{gauge transformation})  as the coadjoint action on the affine hyperplane $L\mathfrak{g}^* \times \{1\} \subset \widehat{L\mathfrak{g}}^*$.

Fixing a maximal torus $T$, the choice of a set of positive roots $\mathfrak{R}_{+}$ for $G$ determines a positive Weyl chamber $\mathfrak{t}_{+}^*$. It is well-known that the orbits of coadjoint $G$-action on $\mathfrak{g}^{*}$ are parametrized by points in $\mathfrak{t}^*_+$. The set of coadjoint $LG$-orbits can be described as follow. Denote by $\alpha_{0}$ the highest root and
\[
\rho_{G} = \frac{1}{2}\sum_{\alpha \in \mathfrak{R}_{+}}\alpha.
\]
There is a unique ad-invariant inner product $\langle \cdot, \cdot \rangle_{\mathfrak{g}}$ on $\mathfrak{g}$, rescaled so that the highest root  of $\mathfrak{g}$ has norm $\sqrt{2}$. The \emph{dual Coxeter number} of $G$ is defined by
\[
h^{\vee} = 1 + \langle \rho_{G}, \alpha_{0} \rangle_\mathfrak{g},
\]
and the \emph{fundamental Weyl alcove} for $G$ is the simplex
\[
\mathfrak{A} = \{ \xi \in \mathfrak{t}_{+}\big| \langle \alpha_{0}, \xi \rangle_{\mathfrak{g}} \leq 1\} \subset \mathfrak{t} \subset \mathfrak{g}.
\]
Every coadjoint orbit of $LG$-action on $L\mathfrak{g}^*$ contains a unique point in $\mathfrak{A}$.

For any $\xi \in L\mathfrak{g}^*$, we define the \emph{holonomy map}
\[
\mathrm{Hol}: L\mathfrak{g}^* \to G
\]
the smooth map that sends $\xi$ to the holonomy of $\xi$ around $S^1$. This map satisfies the equivariance property
\[
\mathrm{Hol}(g \cdot \xi) = g(0) \cdot \mathrm{Hol}(\xi) \cdot g(0)^{-1}.
\]
It follows that the based loop group
\[
\Omega G = \{g \in LG \big| g(0) = e\}
\]
acts freely on $L\mathfrak{g}^*$.

The isotropy group $(LG)_{\xi}$ is isomorphic to $G_{\mathrm{Hol}(\xi)}$, and thus compact. It sets up a 1-1 correspondence between the set of coadjoint $LG$-orbits and conjugacy classes. Moreover $(LG)_{\xi}$ with $\xi \in \mathfrak{t}^*_+$ depends only on the open face $\sigma$ of $\mathfrak{A}$ containing $\xi$ and will be denoted by $(LG)_{\sigma}$. If we  introduce a partial order on open faces by setting $\tau \preceq\sigma$ if $\tau \subseteq \overline{\sigma}$, then one has that
\[
\sigma \preceq \tau \Rightarrow (LG)_{\tau} \subseteq (LG)_{\sigma}.\]
In particular, $(LG)_{0} = G$ and $(LG)_{\mathrm{int} \mathfrak{A}} = T$.

\subsection{Positive Energy Representation}
Let $S^{1}_{\mathrm{rot}}$ be the rotation group on $S^{1}$ and $\partial$ its infinitesimal generator.
Consider a unitary representation of $S^{1}_{\mathrm{rot}}\ltimes \widehat{LG}$ on a Hilbert space $V$, on which the central circle acts by scalar multiplication.

\begin{definition}
We say that $V$ is a \emph{positive energy representation} if  the  operator $\partial$ is self-adjoint with spectrum bounded below. Moreover, we say that $V$ \emph{has level k} if the central circle of $\widehat{LG}$ acts with weight $k$.
\end{definition}

The positive energy representations of loop groups behave quite analogously to the representation theory of compact Lie groups. For example, every irreducible positive energy representation is uniquely determined by the highest weight. To be more precise, let $T$  be a maximal torus of $G$ and $\Lambda^{*}$ the weight lattice. We take $S^{1}_{\mathrm{rot}} \times T \times S^{1}$ as the maximal torus of $S^{1}_{\mathrm{rot}}\ltimes \widehat{LG} $, where the second $S^{1}$ factor comes from the central extension. The \emph{affine weights} of $LG$ are in the forms of $(m, \lambda, k)$, where $m \in \Z$ is the \emph{energy}, $\lambda \in \Lambda^{*}$ is the weight of $G$, and $k$ is the \emph{level}.

The \emph{affine Weyl group}  $W_{\mathrm{aff}} = W \ltimes \Lambda$ acts on affine weights as follow: the Weyl group $W$ acts as usual on $\Lambda^{*}$ and the action of $z \in \Lambda$ is given by
\[
z \cdot (m,\lambda, k) = (m + \langle \lambda, z \rangle + \frac{k}{2} \cdot \|z\|^{2}, \lambda+ k \cdot z, k).
\]
The level $k$ is fixed by the affine Weyl group action and the energy is shifted so as to preserve the inner product:
\begin{equation}
\label{weight product}
(m_{1}, \lambda_{1}, k_{1}) \cdot (m_{2}, \lambda_{2}, k_{2})= \langle \lambda_{1}, \lambda_{2} \rangle - m_{1}k_{2} - m_{2}k_{1}.
\end{equation}
For a fixed level $k$, every irreducible positive energy representation $V$ of $LG$ is uniquely determined by the dominant weight $\lambda$ at the minimum energy $m$. We call $\boldsymbol{\lambda}  = (m, \lambda, k)$ the \emph{highest weight} of $V$.

Let $V(n)$ be a subspace of the Hilbert space $V$, on which the energy operator acts on $V(n)$ with weight $n$. The positive energy condition asserts that there is an integer $n_{\mathrm{min}}$ so that $V(n) = 0$ for all $n < n_{\mathrm{min}}$. The algebraic direct sum
\[
V^{\mathrm{fin}}:= \bigoplus_n V(n)
\]
consists of vectors of \emph{finite energy}. It is a dense subspace of $V$. In addition, we can always normalize so that the lowest energy level $n_{\mathrm{min}}$ equals zero.

It is well-known by the Borel-Weil theorem that  all the irreducible $G$-representations are parameterized by the dominant weights $P_{G,+} = \Lambda^{*} \cap \mathfrak{t}^{*}_{+}$. Similarly there is a 1-1 correspondence between irreducible positive energy representation at level $k$ and weight in
\[
P_{k,+} = k \mathfrak{A} \cap \Lambda^{*} =  \{ \lambda \in \Lambda^{*} \big| \frac{\lambda}{k} \in \mathfrak{A} \}.
\]
The abelian group $R_{k}(LG)$ generated by irreducible positive energy representations at level $k$  has a finite basis and a ring structure known as \emph{fusion product}. But we won't discuss it in this paper.

\section{Dirac operators in the algebraic setting}
The \emph{cubic Dirac operator} is an algebraically defined operator introduced by Kostant \cite{kostant99} for finite-dimensional Lie algebras. It has now been generalized to infinite-dimensional case and plays an important role in the theory of loop groups: its application in representation theory was first demonstrated in the lecture notes of Wassermann \cite{Wassermann-notes}. Later Landweber and Posthuma generalize it to different homogeneous settings \cite{landweber01, Posthuma11}. A family version of the cubic Dirac operator was  used by Freed-Hopkins-Teleman \cite{Freed13} to construct the isomorphism between the twisted K-theory and fusion ring of loop groups. In addition Meinrenken \cite{MR2894443} discusses its application in general Kac-Moody algebra.

\subsection{Finite dimensional case}
Let $G$ be a compact Lie group and $\mathfrak{g}$ its Lie algebra equipped with an ad-invariant inner product $\langle \ , \ \rangle_\mathfrak{g}$. Let $\mathrm{Cliff}(\mathfrak{g})$ be the $\Z_2$-graded complex Clifford algebra of $\mathfrak{g}$ and  $S_{\mathfrak{g}}$ an irreducible $\Z_{2}$-graded $\mathrm{Cliff}(\mathfrak{g})$-module.

Fix an orthonormal basis
\[
X_{a}, \hspace{5mm} a = 1, \dots, \mathrm{dim}\mathfrak{g}.
\]
We define a map $\mathrm{ad}^{\mathfrak{g}}: \mathfrak{g} \to \mathrm{Cliff}(\mathfrak{g})$ by the formula
\[
\mathrm{ad}^{\mathfrak{g}}(X) := \frac{1}{4}\cdot \sum_{a=1}^{\mathrm{dim}\mathfrak{g}} [X, X_{a}] \cdot X_{a}, \hspace{5mm} X \in \mathfrak{g}.
\]

Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. We consider the \emph{non-commutative Weil algebra}
\[
U(\mathfrak{g}) \otimes \mathrm{Cliff}(\mathfrak{g})
\]
introduced by Alekseev-Meinrenken\cite{Meinrenken00}.
\begin{definition}
\label{cubic operator}
The \emph{cubic Dirac operator} $D_\mathfrak{g}$ is an element of the
algebra $U(\mathfrak{g}) \otimes \mathrm{Cliff}(\mathfrak{g})$ defined as
\[
D_{\mathfrak{g}}= \sum_{a=1}^{\mathrm{dim}\mathfrak{g}} \big( X_{a} \otimes X_{a} +  \frac{1}{3} \otimes  \mathrm{ad}^{\mathfrak{g}}(X_{a}) \cdot X_{a} \big)
\]
\end{definition}
The key property of the cubic Dirac operator is that its square is simple and nice.

More generally, let $H \subset G$ be a closed subgroup of the equal rank. Using the inner product we write
\[
\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{p}, \hspace{5mm} \mathfrak{p} = \mathfrak{h}^\perp.
\]
This decomposition induces isomorphisms:
\[
\mathrm{Cliff}({\mathfrak{g}}) \cong \mathrm{Cliff}({\mathfrak{h}}) \otimes \mathrm{Cliff}({\mathfrak{p}}), \hspace{5mm}
S_{\mathfrak{g}} \cong S_{\mathfrak{h}} \otimes S_{\mathfrak{p}},
\]
where $S_{\mathfrak{h}}, S_{\mathfrak{p}}$ are spinor modules of $\mathrm{Cliff}({\mathfrak{h}})$ and $\mathrm{Cliff}({\mathfrak{p}})$ respectively.

\begin{definition}
We define the \emph{relative cubic Dirac operator}
\[
D_{\mathfrak{g},\mathfrak{h}} \in U(\mathfrak{g}) \otimes \mathrm{Cliff}(\mathfrak{p})
\]
by
\begin{equation}
\label{relative cubic operator}
D_{\mathfrak{g}, \mathfrak{h}} =\sum^{(\mathfrak{p})}\big(X_{a} \otimes X_{a} + \frac{1}{3} \otimes   \mathrm{ad}^{\mathfrak{p}}(X_{a}) \cdot X_{a} \big).
\end{equation}
Here $\sum^{(\mathfrak{p})}$ indicates the summation over basis of $\mathfrak{p}$.

\end{definition}

As it stands, the element $D_{\mathfrak{g}, \mathfrak{h}}$ gives us an operator on  $W \otimes S_{\mathfrak{p}}$ for any $\mathfrak{g}$-representation $W$. To exhibit the structure of $D_{\mathfrak{g},\mathfrak{h}}$, we decompose $W\otimes S_{\mathfrak{p}}$ with respect to the $\mathfrak{h}$-action and denote by $M(\nu)$ the isotypic $\mathfrak{h}$-summand with highest weight $\nu$.

\begin{theorem}[\cite{kostant99}]
\label{relative cubic}
Suppose that $W_{\lambda}$ is an irreducible $\mathfrak{g}$-representation with highest weight $\lambda$. The following formula holds
\[
D_{\mathfrak{g},\mathfrak{h}}^{2}\big|_{M(\nu)} = \|\lambda +\rho_{G} \|^{2}- \|\nu + \rho_{H}\|^{2}.
\]
\end{theorem}

\subsection{Dirac operators on Homogeneous spaces}

Suppose now that $M = G/H$ is an orbit of the coadjoint $G$-action on $\mathfrak{g}^*$. It is known that $M$ has a $G$-invariant complex structure, which determines a $H$-equivariant splitting: $\mathfrak{p} = \mathfrak{p}^+ \oplus \mathfrak{p}^-$. One can check that the spinor bundle associated to the complex structure on $M$ is given by
\begin{equation}\label{spinor for lg}
S_{M} = G \times_H (\wedge \mathfrak{p}^-) \cong G \times_{H} (S_{\mathfrak{p}}^{*} \otimes \C_{\rho_G - \rho_H}),
\end{equation}
and the canonical line bundle
\begin{equation}\label{canonical for lg}
K_M = G \times_H\C_{2 \cdot(\rho_G - \rho_H)}.
\end{equation}
Hence, the Hilbert space $L^{2}(M, S_{M})$ can be identified with
\begin{equation}\label{G dec}
 \big[ L^{2}(G) \otimes S_{\mathfrak{p}}^{*} \otimes  \C_{\rho_G - \rho_H}\big]^{H}  \cong \bigoplus_{\lambda \in P_{G,+}} W_{\lambda} \otimes [W_{\lambda}^{*} \otimes S_{\mathfrak{p}}^{*} \otimes  \C_{\rho_G - \rho_H}]^{H},
\end{equation}
where the isomorphism comes from the Peter-Weyl theorem.

We  define Dirac operators on $M$ in two different ways. First of all, the Levi-Civita connection $\nabla^{TM}$ lifts to a Hermitian connection $\nabla^{S_{M}}$ on $S_{M}$. In particular, the connection $\nabla^{S_{M}}$ is compatible with the Clifford action in the sense that
\[
[X, \nabla^{S_{M}}_Y] = \nabla^{TM}_Y X, \hspace{5mm} X, Y \in TM.
\]
We define a geometric Spin$^{c}$-Dirac operator by
\[
D_{\mathrm{geo}} = \sum_{a=1}^{\mathrm{dim}M} X_{a} \cdot \nabla^{S_{M}}_{X_{a}},
\]
where $\{X_{a}\}_{i=1}^{\mathrm{dim}M}$ is an orthonormal basis of $TM$.

On the other hand, since the cubic Dirac operator $D_{\mathfrak{g},\mathfrak{h}}$ is $H$-equivariant, it restricts to an operator on
\[
[W_{\lambda}^{*} \otimes S_{\mathfrak{p}}^{*}  \otimes  \C_{\rho_G - \rho_H}]^{H}.
\]
Tensoring  the identity operator on each $W_{\lambda}$, and summing over $W_{\lambda}$, one obtains an  operator $D_{\mathrm{alg}}$ on (\ref{G dec}).

\begin{lemma}
\label{geo alg}
The difference between $D_{\mathrm{alg}}$ and $D_{\mathrm{geo}}$ on (\ref{G dec}) is a bounded operator.
\begin{proof}
We rewrite the geometric connection $\nabla^{S_M}$  as
\[
\nabla^{S_M}_X = X + \frac{1}{2}\mathrm{ad}^{\mathfrak{p}}(X).
\]
It follows that
\[
D_{\mathrm{geo}} =\sum_{i=1}^{\mathrm{dim}\mathfrak{p}} X_{a} \otimes \big(X_{a} + \frac{1}{2}\mathrm{ad}^{\mathfrak{p}}(X_{a})\big).
\]
We deduce the lemma by comparing the above with (\ref{relative cubic operator}).  An alternative proof can be found in \cite[Chapter 9]{Meinrenken-book}.
\end{proof}
\end{lemma}

\subsection{Infinite dimensional case}
The definitions of spin representation and cubic Dirac operator can be extended to the infinite-dimensional loop algebra $L\mathfrak{g}$.

Let now $G$ be a compact, simple and simple connected Lie group with Lie algebra $\mathfrak{g}$. The loop algebra $L\mathfrak{g}$ carries an inner product defined by
\begin{equation}
\label{inner product}
B(X, Y) = \frac{1}{2\pi}\int_{0}^{2\pi} \langle X(\theta), Y(\theta) \rangle_{\mathfrak{g}} d\theta, \hspace{5mm} X, Y \in L\mathfrak{g}.
\end{equation}
As in the finite dimensional case, we can define the Clifford algebra $\mathrm{Cliff}(L\mathfrak{g})$, and its spin representation $S_{L\mathfrak{g}}$. Here $S_{L\mathfrak{g}}$ is a $\Z_2$-graded complex Hilbert space, and also a positive energy $LG$-representation with highest weight
\[
\boldsymbol{\rho}_G= (0, \rho_G, h^{\vee}).
\]
The explicit construction of $S_{L\mathfrak{g}}$ is given in \cite{PS86}. For the general theory of Clifford algebras and  representations for infinite dimensional Hilbert spaces we refer to \cite{Plymen94}.

Let us fix an orthonormal basis $\{X_{a}\}$  of $\mathfrak{g}$. For $n \in \Z$, we write $X_{a}^{n}$ for the loop
\[
s \mapsto e^{i n s}\cdot X_{a}, \hspace{5mm} s \in \mathbb{R},
\]
and $\mathfrak{g}(n)$ the vector space spanned by $\{X^{n}_{a}\}_{a=1}^{\mathrm{dim}\mathfrak{g}}$.
The algebraic direct sum
\[
L\mathfrak{g}^{\mathrm{fin}}:= \bigoplus_{n \in \Z} \mathfrak{g}(n)
\]
is dense in $L\mathfrak{g}_{\C}$. The dense subspace of the spin representation $S_{L\mathfrak{g}}$ which consists of vectors with  finite energy may be realized as
\[
S_{L\mathfrak{g}}^{\mathrm{fin}}= S_{\mathfrak{g}} \otimes \bigotimes_{k>0} \Lambda^*\big(\mathfrak{g}_\C z^k\big).
\]

\begin{definition}
We define the \emph{cubic Dirac operator}
\[
D_{L\mathfrak{g}} \in U(L\mathfrak{g}) \otimes \mathrm{Cliff}(L\mathfrak{g})
\]
by
\begin{equation}\label{d_lg}
D_{L\mathfrak{g}} = \sum_{n\in \Z} \sum_{a=1}^{\mathrm{dim}\mathfrak{g}} \big( X_{a}^{n}\otimes X_{a}^{-n}+ \frac{1}{3}\otimes \mathrm{ad}^{L\mathfrak{g}}(X_{a}^{n}) \cdot X_{a}^{-n} \big),
\end{equation}
\end{definition}
\begin{remark}
The map
\begin{equation}\label{ad_lg}
\mathrm{ad}^{L\mathfrak{g}}(X) := \frac{1}{4} \sum_{n\in \Z} \sum_{a=1}^{\mathrm{dim}\mathfrak{g}} [X, X_{a}^n] \cdot X_{a}^{-n} \in \mathrm{Cliff}(L\mathfrak{g}), \hspace{5mm} X \in L\mathfrak{g}^{\mathrm{fin}}.
\end{equation}
is defined only on vectors with finite energy. To justify the infinite summation in (\ref{d_lg}) and (\ref{ad_lg}), we refer to \cite{MR2894443, Wassermann-notes}. For any positive energy $LG$-representation $V$, $D_{L\mathfrak{g}}$ gives us an unbounded operator on $\big(V \otimes S_{L\mathfrak{g}}\big)^{\mathrm{fin}}$ which is a  dense subspace of $V \otimes S_{L\mathfrak{g}}$.
\end{remark}

Let $\mathfrak{h}$ be an isotropy Lie algebra $\mathfrak{h}$ of the coadjoint $LG$-action. We decompose the Lie algebra $L\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{h}$, where $\mathfrak{p} = \mathfrak{h}^{\perp}$. By the multiplicative property of spin representation
\[
S_{L\mathfrak{g}} = S_{\mathfrak{p}} \otimes S_{\mathfrak{h}},
\]
where $S_{\mathfrak{p}}$  is a $\Z_2$-graded irreducible representation of the Clifford algebra $\mathrm{Cliff}(\mathfrak{p})$.

\begin{definition}
We define  $D_{L\mathfrak{g}, \mathfrak{h}}\in U(L\mathfrak{g}) \otimes \mathrm{Cliff}(\mathfrak{p})$ by the formula
\[
D_{L\mathfrak{g}, \mathfrak{h}} = \sum_{a, n}^{(\mathfrak{p})}\big( X_a^n \otimes X_a^{-n} + \frac{1}{3} \otimes \mathrm{ad}^\mathfrak{p}(X_{a}^n)\cdot X_a^{-n} \big),
\]
where the summation ranges over a basis of $\mathfrak{p}$.
\end{definition}

We denote by $\widehat{H}$ the central extension of $H$ induced by the inclusion $H \hookrightarrow LG$ and the central extension $\widehat{LG}$. We decompose $V \otimes S_{\mathfrak{p}}$ with respect to the $S^{1}_{\mathrm{rot}} \times \widehat{H}$-action and denote by $M(\boldsymbol{\nu})$ the isotypic  component labeled by
\[
\boldsymbol{\nu} = (n, \nu, k+h^\vee) \in \Z \times \Lambda^{*} \times \Z.
\]
We have an analog of Theorem \ref{relative cubic} for the infinite-dimensional case.
\begin{theorem}
\label{square operator-4}
Suppose that $V_{\lambda}$ is an irreducible positive energy representation with highest weight $\boldsymbol{\lambda}  = (0, \lambda, k)$. If we restrict to the isotypic component $M(\boldsymbol{\nu})$ of $V_{\lambda} \otimes S_{\mathfrak{p}}$, we have that
\[
D_{L\mathfrak{g},\mathfrak{h}}^{2}\big|_{M(\boldsymbol{\nu})}= \|\boldsymbol{\lambda} + \boldsymbol{\rho}_{G}\|^{2} - \|\boldsymbol{\nu} + \boldsymbol{\rho}_{H}\|^{2},
\]
where $\boldsymbol{\rho}_{G} = (0, \rho_{G}, h^{\vee})$ and $ \boldsymbol{\rho}_{H} = (0, \rho_{H}, 0)$.
\begin{proof}
\cite[Theorem 7.5]{MR2894443}.
\end{proof}
\end{theorem}

\section{Hamiltonian $LG$-spaces and q-Hamiltonian $G$-spaces}
The theory of q-Hamiltonian $G$-spaces was developed in \cite{AMM98}. It provides a finite-dimensional model for Hamiltonian $LG$-spaces. In this section, we begin by reviewing the basic definitions, and then discuss their cross-section theorems. We assume that $G$ is a compact, simple and simple connected Lie group.

\subsection{Basic definitions}
Recall that a \emph{Hamiltonian $G$-space} is a triple $(M, \omega, \mu)$, with $\omega$ the $G$-equivariant symplectic 2-form, and $\mu: M \to \mathfrak{g}^{*}$ the \emph{moment map} satisfying that
\[
\iota_{\xi_{M}} \omega =d \langle \mu, \xi \rangle, \hspace{5mm} \xi \in \mathfrak{g},
\]
where $\xi^M$ is the vector field on $M$ induced by the infinitesimal action of $\xi$.

The above definition can be extended to the loop group setting. Let $\mathcal{M}$ be an infinite-dimensional Banach manifold. We say that it is \emph{weakly symplectic} if it is equipped with a closed 2-form $\omega \in \Omega^{2}(\mathcal{M})$ so that the induced map
\[
\omega^{\flat}: T_{m}\mathcal{M} \to T_{m}^{*}\mathcal{M}
\]
is injective.

\begin{definition}
A \emph{Hamiltonian $LG$-space} is a weakly symplectic Banach manifold $(\mathcal{M}, \omega)$ together with a $LG$-action and
a $LG$-equivariant map $\mu: \mathcal{M} \to L\mathfrak{g}^{*}$ so that $\iota_{\xi_{\mathcal{M}}} \omega =d \langle \mu, \xi \rangle$ for all $\xi \in L\mathfrak{g}$.
\end{definition}
For example, the coadjoint $LG$-orbit is a Hamiltonian $LG$-space, with moment map the inclusion.

Let $(\mathcal{M}, \omega, \mu)$ be a Hamiltonian $LG$-space. Since the based loop group $\Omega G$ acts freely on $L\mathfrak{g}^{*}$, it acts freely on $\mathcal{M}$ as well by the equivariance of $\mu$. We thus obtain a commuting square
\begin{diagram}
 \mathcal{M} & \rTo^{\mu} &   L\mathfrak{g}^{*}\\
\dTo &  &\dTo^{\mathrm{Hol}}\\
M & \rTo^{\phi} & G\\,
\end{diagram}
where the quotient $M = \mathcal{M}/\Omega G$ is a finite-dimensional compact smooth manifold provided that $\mu$ is proper.

Alekseev-Malkin-Meinrenken \cite{AMM98} give a set of conditions a $G$-space $M$ must satisfy in order to arise from a Hamiltonian $LG$-space by such a construction.

Choose an invariant inner product $\langle \cdot, \cdot \rangle_{\mathfrak{g}}$ on $\mathfrak{g}$ and denote by $\theta^{L}, \theta^{R} \in \Omega^{1}(G,\mathfrak{g})$ the left and right invariant Maurer-Cartan forms on $G$ and  the \emph{Cartan 3-form}
\[
\chi = \frac{1}{12} \langle \theta^{L}, [\theta^{L}, \theta^{L}] \rangle_{\mathfrak{g}} = \frac{1}{12} \langle \theta^{R}, [\theta^{R}, \theta^{R}]\rangle_{\mathfrak{g}} \in \Omega^{3}(G).
\]

\begin{definition}[\cite{AMM98}]
\label{q-Hamiltonian}
A \emph{q-Hamiltonian} $G$-space is a compact $G$-manifold $M$, together with an equivariant 2-form $\omega$, and an equivariant  map $\phi: M \to G$ satisfying the following properties:
\begin{enumerate}
\item
$d \omega = \phi^{*}\chi;$
\item
$\iota_{\xi_{M}} \omega = \frac{1}{2}\langle \phi^{*}(\theta^{L} + \theta^{R}), \xi \rangle_{\mathfrak{g}}$ for all  $\xi \in \mathfrak{g};$
\item
$\mathrm{ker}(\omega) \cap \mathrm{ker}(d\phi) = 0. $
\end{enumerate}
We call $\phi$ the \emph{group-valued moment map}.
\end{definition}
According to \cite[Theorem 8.3]{AMM98}, there is a 1-1 correspondence between Hamiltonian $LG$-spaces with proper moment map and q-Hamiltonian $G$-spaces. One can always choose to work with infinite-dimensional Hamiltonian $LG$-spaces with more conventional definitions or to use finite dimensional q-Hamiltonian $G$-spaces. The counterparts of coadjoint orbits for q-Hamiltonian $G$-spaces are conjugacy classes $\mathcal{C}$ in $G$ with group-valued moment map the embedding $\mathcal{C} \hookrightarrow G$.

\subsection{Cross-section theorems}
The Hamiltonian $LG$-spaces and their equivalent finite-dimensional models behave in many respects like the usual Hamiltonian $G$-spaces. This is due to the existence of the cross-section theorem we shall now describe.

 Let us first introduce a partial order of open faces of $\mathfrak{A}$ by setting $\tau \preceq\sigma$ if $\tau \subseteq \overline{\sigma}$. The isotropy group $(LG)_{\xi}$ of the coadjoint $LG$-action on $L\mathfrak{g}^{*}$ depends only on the open face $\sigma$ of $\mathfrak{A}$ containing $\xi$ and will be denoted by $(LG)_{\sigma}$ (note however $(LG)_{\sigma}$ will generally contain non-constant loops). One has that
\[
\sigma \preceq \tau \Rightarrow (LG)_{\tau} \subseteq (LG)_{\sigma}.
\]
In particular, $(LG)_{0} = G$ and $(LG)_{\mathrm{int} \mathfrak{A}} = T$.

We define a $(LG)_{\sigma}$-invariant open subset of $(L\mathfrak{g})_{\sigma}^{*}$ by
\[
\mathcal{A}_{\sigma} = (LG)_{\sigma} \cdot \bigcup_{\sigma \preceq\tau} \tau.
\]
Note that $\mathcal{A}_{\sigma}$ is a slice for all $\xi \in \sigma$ for the action of $LG$ in the sense that
\[
LG \times_{(LG)_{\sigma}} \mathcal{A}_{\sigma} \to LG \cdot \mathcal{A}_{\sigma}
\]
is a diffeomorphism of Banach manifolds.
\begin{theorem}
\label{cross section LG}
Let $(\mathcal{M}, \omega, \mu)$ be a Hamiltonian $LG$-space with proper moment map. For every open face $\sigma$ of $\mathfrak{A}$, the cross-section
\[
\mathcal{V}_{\sigma} = \mu^{-1}(\mathcal{A}_{\sigma})
\]
is a finite-dimensional symplectic submanifold with Hamiltonian $(LG)_{\sigma}$-action. The restriction of $\mu|_{\mathcal{V}_{\sigma}}$ is a moment map of the $\widehat{(LG)}_{\sigma}$-action (the central circle acts trivially on $\mathcal{V}_{\sigma}$).
\begin{proof}
\cite[Theorem 4.8]{MW98}.
\end{proof}
\end{theorem}

The symplectic cross-section theorem  carries over to q-Hamiltonian $G$-spaces. The centralizer $G_{\mathrm{exp}(\xi)}$ with $\xi \in \mathfrak{A}$ is isomorphic to $(LG)_{\xi}$ and it depends only on the open face $\sigma$ of $\mathfrak{A}$ containing $\xi$. We denoted it by $G_{\sigma}$. The subset
\[
 A_{\sigma} = \mathrm{Ad}(G_{\sigma}) \cdot \mathrm{exp}(\bigcup_{\sigma \preceq\tau}\tau) \subset G_{\sigma} \subset G
\]
is smooth and is a slice for the $\mathrm{Ad}$(G)-action at points in $\sigma$.
\begin{theorem}
 Let $(M, \omega, \phi)$ be a q-Hamiltonian $G$-space. The cross-section
\begin{equation}
\label{slice}
V_{\sigma} = \phi^{-1}(A_{\sigma})
\end{equation}
 is a smooth $G_{\sigma}$-invariant submanifold and
\[
G \times_{G_{\sigma}} V_{\sigma} \cong G \cdot V_{\sigma}
\]
is a $G$-invariant open subset of $M$. Moreover, $V_{\sigma}$
 is a q-Hamiltonian $G_{\sigma}$-space with the restriction of $\phi$ as the group-valued moment map.
\begin{proof}
\cite[Proposition 7.1]{AMM98}.
\end{proof}
\end{theorem}

\begin{remark}
\label{identify}
It is important to point out that if we identify $G_{\sigma}\cong (LG)_{\sigma}$, the two cross-sections
\[
 \mathcal{V}_{\sigma}\subset \mathcal{M}, \hspace{5mm} V_{\sigma} \subset M
\]
are equivariantly diffeomorphic. In particular, every $V_\sigma$ is also a Hamiltonian $G_\sigma$-space.
\end{remark}

\begin{comment}
\subsection{Equivalence theorem}
Every Hamiltonian $LG$-space with proper moment map determines a q-Hamiltonian $G$-space and vice versa. Since the based loop group $\Omega G$ acts on $L\mathfrak{g}^{*}$ freely, its action on $\mathcal{M}$ is free as well. We can thus form the quotient
\[
M = \mathrm{Hol}(\mathcal{M}) := \mathcal{M} / \Omega G.
\]
By the assumption that $\phi$ is proper, the holonomy manifold $M$ is a smooth finite-dimensional manifold. In fact, we have the following theorem:

\begin{theorem}[Equivalence theorem]
Let $(\mathcal{M}, \sigma, \mu)$ be a Hamiltonian $LG$-space with proper moment map $\mu$. Then the holonomy manifold $M = \mathrm{Hol}(\mathcal{M})$ is a q-Hamiltonian $G$-space with group-valued moment map $\phi = \mathrm{Hol}(\mu) : M \to G$.
\end{theorem}

Suppose conversely that we are given a q-Hamiltonian $G$-space $(M, \omega, \phi)$. The Hamiltonian $LG$-space can be reconstructed from
\[
\mathcal{M} = \{ (\xi, m) \in L\mathfrak{g}^{*} \times M| \phi(m) = \mathrm{Hol}(\xi)\},
\]
with moment map $\mu(\xi, m) = \xi \in L\mathfrak{g}^{*}$.

\end{comment}

\section{Twisted spinor bundle and twisted pre-quantum bundle}
In this section, we construct twisted spinor bundle and twisted pre-quantum bundles on q-Hamiltonian $G$-spaces.

\subsection{Construction of the twisted spinor bundle}
Let $G$ be a compact, simple, and simply connected Lie group. Let $(M, \omega)$ be a q-Hamiltonian $G$-space and $\mathcal{M}$ its corresponding Hamiltonian $LG$-space.

We first replace the cross-sections $V_\sigma$ in (\ref{slice}) with smaller open subsets. To be more precise, for every vertex $\sigma$ of $\mathfrak{A}$, let  $Y_{\sigma}$ be a  $G_{\sigma}$-invariant,  open subset of $V_\sigma$ so that $\overline{Y_\sigma} \subset V_\sigma$, and
\[
M/G \subseteq \bigcup_{\sigma, \mathrm{dim}\sigma =0}  Y_{\sigma}.
\]
Then we form an open cover of $M$ by
\[
\{ U_{\sigma} = G \times_{G_{\sigma}}Y_{\sigma}\}_{\sigma, \mathrm{dim}\sigma = 0}.
\]
For all open faces $\tau$ of $\mathfrak{A}$ with $\mathrm{dim}\tau > 0$, we define
\[
Y_{\tau} = \bigcap_{\sigma \preceq\tau, \mathrm{dim}\sigma = 0} Y_{\sigma},  \hspace{5mm} U_\tau = G \times_{G_\tau} Y_{\tau}.
\]
Remark at this point that each $Y_{\tau}$ is a Hamiltonian
$G_\tau$-space and admits $G_\tau$-invariant almost complex structures.

\begin{lemma}
\label{homotopy}
There exists a collection of $G_{\sigma}$-invariant almost complex structures on the collection of $Y_{\sigma}$ such that the embedding
\[
Y_{\tau} \hookrightarrow Y_{\sigma},\hspace{5mm} \sigma \prec \tau
\]
is almost complex. In addition, any two almost complex structures with the required properties are homotopic. We denote by $S_{Y_{\sigma}}$ the spinor bundle on $Y_{\sigma}$ associated to the almost complex structures.
\begin{proof}
\cite[Lemma 3.2]{Meinrenken-W-2001}.
\end{proof}
\end{lemma}

Let $\pi : U_\sigma \to G/G_\sigma$ be the projection. The tangent bundle decomposes $G$-equivariantly
\[
TU_\sigma \cong \pi^*T(G/G_\sigma) \oplus G \times_{G_\sigma} TY_\sigma.
\]
The base manifold $G/G_\sigma$ is a conjugacy class and might not have a $G$-equivariant Spin$^c$-structure in general. Thus the total space $U_\sigma$ doesn't have to be Spin$^c$ either.

On the other hand, the coadjoint $LG$-orbit $\mathcal{O} =LG/(LG)_\sigma$ is a complex manifold. By the discussion in  \cite{Meinrenken-W-2001}, the weight
\[
2 \cdot (\rho_{G} -\rho_{\sigma}, h^{\vee})
\]
is fixed by $\widehat{(LG)}_\sigma$ and the tensor product
\[
S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})}
\]
is a $(LG)_\sigma$-space. The associated spinor bundle and canonical line bundle on $\mathcal{O}$ are given  by
\[
S_\mathcal{O} = \widehat{LG} \times_{\widehat{(LG)}_\sigma} (S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})})
\]
and
\[
K_\mathcal{O} = \widehat{LG} \times_{\widehat{(LG)}_\sigma}  \C_{2 \cdot (\rho_{G} -\rho_{\sigma}, h^{\vee})}).
\]
 One can compare them with (\ref{spinor for lg}) and (\ref{canonical for lg}).

Motivated by the above, we define a bundle of Hilbert space on $U_\sigma$ by
\begin{equation}
\label{spinor-2}
S_{U_{\sigma}}^{\mathrm{spin}} = G \times_{G_{\sigma}} \big(S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})} \otimes S_{Y_{\sigma}}\big),
\end{equation}
where $G_{\sigma}$ acts on
\[
S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})}
\]
factoring through the identification $G_{\sigma} \cong (LG)_{\sigma}$. In addition, we equip $S_{U_{\sigma}}^{\mathrm{spin}}$ with a $\Z_{2}$-grading induced by that on $S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}}$ and $S_{Y_\sigma}$. To sum up, we obtain a collection of $G$-equivariant bundles of $\Z_{2}$-graded Hilbert space $\{ S_{U_{\sigma}}^{\mathrm{spin}}\}$.  We next show that such a collection of bundles can be glued together.
\begin{lemma}
For any $\sigma \prec \tau$, the normal bundle $\nu_{\tau}^{\sigma}$ of $Y_{\tau} \hookrightarrow Y_{\sigma}$ has a $G_{\tau}$-equivariant almost complex structure with spinor bundle isomorphic to
\begin{equation}
\label{normal} S_{(L\mathfrak{g})_{\sigma}/(L\mathfrak{g})_{\tau}}^{*} \otimes \C_{(\rho_{\sigma} - \rho_{\tau})},
\end{equation}
where $\rho_{\tau}, \rho_{\sigma}$ are the half-sums of positive roots for $G_\tau, G_\sigma$ respectively.
\begin{proof}
By the cross-section theorem, the normal bundle $\nu_{\tau}^{\sigma}$ is  isomorphic to the trivial bundle
\[
\mathfrak{g}_{\sigma}/\mathfrak{g}_{\tau} \cong (L\mathfrak{g})_{\sigma}/(L\mathfrak{g})_{\tau}
\]
with equivariant almost complex structure. In fact,  let  $\mathfrak{R}_{\sigma}, \mathfrak{R}_{\tau}$ be compatible sets of positive roots for $G_{\sigma}$ and $G_{\tau}$. Then
\[
\mathfrak{g}_{\sigma} /\mathfrak{g}_{\tau} = \bigoplus_{\alpha \in \mathfrak{R}_{\sigma}\setminus \mathfrak{R}_{\tau}} \C_{\alpha}
\]
and
\[
\mathrm{det}_{\C}(\nu_{\tau}^{\sigma}) = \bigotimes_{\alpha \in \mathfrak{R}_{\sigma}\setminus \mathfrak{R}_{\tau}} \C_{\alpha} = \C_{2(\rho_{\sigma} - \rho_{\tau})}.
\]
\end{proof}
\end{lemma}

\begin{lemma}
There are canonical isomorphisms
\[
\Psi_{\tau, \sigma}: S_{U_{\tau}}^{\mathrm{spin}}  \cong S_{U_{\sigma}}^{\mathrm{spin}} |_{U_{\tau}}, \hspace{5mm} \sigma \prec \tau
\]
and they automatically satisfy the cocycle condition.
\begin{proof}
By the above lemma, we have that
\[
S_{Y_{\sigma}}|_{Y_{\tau}} \cong S_{(L\mathfrak{g})_{\sigma}/(L\mathfrak{g})_{\tau}}^{*} \otimes \C_{(\rho_{\sigma} - \rho_{\tau})}  \otimes S_{Y_{\tau}}.
\]
The claim follows form the construction.
\end{proof}
\end{lemma}

\begin{definition}
We define the \emph{twisted spinor bundle} $S^{\mathrm{spin}}$ to be the $G$-equivariant bundle of  $\Z_{2}$-graded Hilbert space over $M$ with the property that
\[
S^{\mathrm{spin}}|_{U_{\sigma}} \cong S_{U_{\sigma}}^{\mathrm{spin}},
\]
for all vertexes $\sigma$ of $\mathfrak{A}$.
\end{definition}
The twisted spinor bundle is determined by the choice of almost complex structures on subsets $\{Y_{\sigma}\}$. By Lemma \ref{homotopy}, all the choices are homotopic. Hence, the twisted spinor bundle $S^{\mathrm{spin}}$ is unique up to homotopy.

\subsection{Construction of the twisted pre-quantum bundle}
Let us recall that the pre-quantum line bundle of a symplectic manifold is traditionally defined to be a line bundle whose first Chern class is an integral lift of the symplectic 2-form.
\begin{definition}\label{pre-quantum LG}
We say that a Hamiltonian $LG$-space $\mathcal{M}$ is \emph{pre-quantizable at level $k$} $(k >0)$ if there exists a $\widehat{LG}$-equivariant line bundle
\[
\mathcal{L} \to \mathcal{M}
\]
such that the central circle acts with weight $k$ and the first Chern class $c_{1}(\mathcal{L})$ equals to the symplectic 2-form on $\mathcal{M}$.
\end{definition}
Because the pre-quantum line bundle $\mathcal{L}$ is $\widehat{LG}$-equivariant instead of $LG$-equivariant, it might not descend to an actual line bundle on its corresponding q-Hamiltonian $G$-space $M$.
\begin{remark}\label{pre-quantum q}
The 2-form $\omega$ for a q-Hamiltonian $G$-space $M$ is not closed in general. Instead the condition $d\omega = \phi^{*}\chi$ and the fact that$\chi$ is a closed 3-form imply that the pair $(\omega, \chi)$ defines a cocyle for the relative de Rham theory (see \cite[Appendix B]{MR2989614} for a reference). We denote by $[(\omega, \chi)] \in H^{3}(\phi, \mathbb{R})$ its cohomology class. We say that a q-Hamiltonian $G$-space $(M, \omega, \phi)$ is \emph{pre-quantizable at level $k$} if $k \cdot [(\omega, \chi)]$ is integral. By the 1-1 correspondence between q-Hamiltonian $G$-spaces and Hamiltonian $LG$-spaces, their pre-quantum conditions are equivalent.
\end{remark}

Let $Y_\sigma$ be the cross-section defined before. We identify it as a subset in $\mathcal{M}$.  If $\mathcal{M}$ has a pre-quantum line bundle $\mathcal{L}$ at level $k$, then there exists a $\widehat{(LG)}_{\sigma}$-equivariant line bundle obtained by restriction
\[
L_{Y_{\sigma}} = \mathcal{L}|_{Y_{\sigma}} \to Y_\sigma,
\]
on which the central circle acts with weight $k$. The collection of line bundles $\{L_{Y_{\sigma}}\}$ satisfy a compatibility condition in the sense that
\begin{equation}
\label{normal-2}
(LG)_{\sigma} \times_{(LG)_{\tau}}L_{Y_{\tau}} \cong L_{Y_{\sigma}}|_{Y_{\tau}^{\sigma}}, \hspace{5mm} \sigma \prec \tau.
\end{equation}
where
\[
Y_{\tau}^{\sigma} =(LG)_{\sigma} \times_{(LG)_{\tau}} Y_{\tau}
\]
is a $(LG)_{\sigma}$-invariant open subset of $Y_{\sigma}$.

For any irreducible positive energy representation $V_{\lambda}$ at level $k$, we denote $V_{\lambda}^*$ its dual. Comparing to (\ref{spinor-2}), we define a bundle of Hilbert space on
\[
U_\sigma = G\times_{G_\sigma} Y_\sigma
\]
by
\begin{equation}
\label{pre-q-2}
S^{\mathrm{pre}}_{\lambda,U_{\sigma}} = G \times_{G_{\sigma}} (V_{\lambda}^* \otimes L_{Y_{\sigma}}).
\end{equation}
Here  the central circle of $\widehat{(LG)}_\sigma$  acts trivially on the tensor product $V_{\lambda}^* \otimes L_{Y_{\sigma}}$, and $G_\sigma$ acts factoring through $G_{\sigma} \cong (LG)_\sigma$.

By the compatibility condition (\ref{normal-2}), there are canonical isomorphisms
\[
\Psi_{\tau, \sigma}: S^{\mathrm{pre}}_{\lambda,U_{\tau}}  \cong S^{\mathrm{pre}}_{\lambda,U_{\sigma}} |_{U_{\tau}}, \hspace{5mm} \sigma \prec \tau,
\]
satisfying the cocycle condition. Thus, they can be glued together. We define $S^{\mathrm{pre}}_{\lambda}$  the unique $G$-equivariant bundle of Hilbert space over $M$
with the property that
\[
S^{\mathrm{pre}}_{\lambda}|_{U_{\sigma}} =S^{\mathrm{pre}}_{\lambda,U_{\sigma}}.
\]

\begin{definition}
We define the \emph{twisted pre-quantum bundle} by
\begin{equation}
\label{twisted pre-quantum}
S^{\mathrm{pre}}:= \bigoplus_{\lambda \in P_{k,+}} V_{\boldsymbol{\lambda}} \otimes S^{\mathrm{pre}}_{\lambda}.
\end{equation}
It is a $\widehat{LG} \times G$-equivariant bundle of Hilbert space over the q-Hamiltonian $G$-space $M$.
\end{definition}

\begin{remark}
There is a global construction of the twisted pre-quantum bundle.  Let us introduce  a $\widehat{LG} \times \widehat{LG}$-space:
\[
H_{\mathrm{wzw}, k} = \bigoplus_{\lambda \in P_{k,+}} V_{\lambda} \otimes V_{\lambda}^{*}.
\]
This is the so-called Hilbert space of the Wess-Zumino-Witten model (see \cite{Gawedzki00} for a reference). The aim of this paper is not to justify this choice of the Hilbert space, but  morally one can consider it as the analog of $L^2(G)$ for loop groups, in the spirit of the Peter-Weyl decomposition of $L^2(G)$. The tensor product
\[
(\mathcal{M} \times H_{\mathrm{wzw}, k}) \otimes \mathcal{L} = \bigoplus_{\lambda \in P_{k,+}} V_{\lambda} \otimes (V_{\lambda}^{*} \otimes \mathcal{L}) \to \mathcal{M}
\]
is a $\widehat{LG} \times LG$ equivariant bundle. We obtain a $\widehat{LG}\times G$-equivariant bundle of Hilbert space by taking its $\Omega G$-invariant part
\[
S^{\mathrm{pre}} = V_{\lambda} \otimes [ V^{*}_{\boldsymbol{\lambda}} \otimes \mathcal{L}]^{\Omega G}  \to M = \mathcal{M}/ \Omega G.
\]
\end{remark}

\section{Dirac operators on q-Hamiltonian $G$-spaces}
With the twisted spinor bundle and twisted pre-quantum bundle defined in last section, we now proceed to construct Hilbert spaces and Dirac operators. We keep the same notations as in last section.  Let $G$ be a compact, simple and simply connected Lie group, and $M$ a pre-quantizable q-Hamiltonian $G$-space at level $k$.

\subsection{Dirac operators on cross-sections}
The idea of constructing Dirac operators is that we first define Dirac operators on local cross-sections, and then patch them together using partition of unity.

Let $\{ Y_\tau \}$ be the collection of cross-sections defined in last section and open subsets $U_\tau = G \times_{G_\tau} Y_\tau$.
Fixing an irreducible positive energy $LG$-representation $V_\lambda$, we define
\[
\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G}
\]
the space of $G$-invariant, smooth sections of $S^{\mathrm{spin}} \otimes S^{\mathrm{pre}}_{\lambda}$ with compact support in $U_\tau$, with norm given by
\[
\|s\|^2 := \int_{U_\tau} \langle s(m), s(m) \rangle dm.
\]
\begin{lemma}
\label{cross-1}
We have that
\begin{equation}
\begin{aligned}
&\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G}\\
 \cong & \big[V_{\lambda}^{*} \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\tau}} \otimes \C_{(\rho_{G} -\rho_\tau, h^\vee)} \otimes  \Gamma_{c}^\infty(Y_\tau, S_{Y_\tau} \otimes L_{Y_\tau})\big]^{G_{\tau}}.
\end{aligned}
\end{equation}
\begin{proof}
The assertion follows immediately from the isomorphisms (\ref{spinor-2}) and (\ref{pre-q-2}).
\end{proof}
\end{lemma}

\begin{comment}
\begin{remark}
Although the tensor products
Every factor on the right-hand side of equation (\ref{e-1}) might not be a $G_{\sigma}$-space individually but $\mathfrak{g}_{\sigma}$-space. However, the tensor product
\[
S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})}
\]
as well as
\[
V_{\lambda}^{*} \otimes  \Gamma_{L^{2}}(\mathcal{Y}_{\sigma}, S_{\mathcal{Y}_{\sigma}} \otimes L_{\mathcal{Y}_{\sigma}})
\]
are actual $G_{\sigma}$-representations.
\end{remark}
\end{comment}

Let
\[
D_{\mathrm{alg}} \in U(L\mathfrak{g}) \otimes \mathrm{Cliff}(L\mathfrak{g}/(L\mathfrak{g})_\tau)
\]
be the cubic Dirac operator acting on $V_{\lambda}^{*} \otimes S_{L\mathfrak{g}/(L\mathfrak{g})_\tau}^{*}$ and $D_{\mathrm{geo}}$ the equivariant  geometric Spin$^{c}$-Dirac operator on  $\Gamma_{c}^\infty(Y_\tau, S_{Y_\tau} \otimes L_{Y_\tau})$. Here we choose $D_{\mathrm{geo}}$ so that it is symmetric. Since the sum
\[
D_{\mathrm{alg}} \otimes 1 + 1\otimes D_{\mathrm{geo}}
\]
is equivariant, it descends to the $G_\tau$-invariant part. That is, we obtain a collection of operators
\[
\big[D_{ \mathrm{alg}} \otimes 1 + 1\otimes D_{\mathrm{geo}}\big]^{G_\tau}
\]
on
\[
 \big[V_{\lambda}^{*} \otimes S^{*}_{\mathfrak{p}} \otimes \C_{(\rho_{G} -\rho_\tau, h^\vee)}  \otimes  \Gamma_{c}^\infty(Y_\tau, S_{Y_\tau} \otimes L_{Y_\tau})\big]^{G_\tau},
\]
and thus an operator on
\[
\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G}.
\]
We denote it by $D_{U_\tau}$. By definition, every $D_{U_\tau}$ is an unbounded operator.
Since $D_{\mathrm{alg}}$ is defined only on vectors with finite energy,  the domain of $D_{U_\tau}$ is  given by
\[
\big[(V_{\lambda}^{*} \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\tau}})^{\mathrm{fin}} \otimes \C_{(\rho_{G} -\rho_\tau, h^\vee)} \otimes  \Gamma_{c}^\infty(Y_\tau, S_{Y_\tau} \otimes L_{Y_\tau})\big]^{G_{\tau}},
\]
which is a dense subspace.

Suppose that $\sigma_1, \sigma_2$ are two vertexes of $\mathfrak{A}$ and $\tau$ an open face of $\mathfrak{A}$ so that $ \sigma_1 \preceq \tau$ and $ \sigma_2 \preceq \tau$. In particular, one has that
\[
U_\tau \subseteq U_{\sigma_1}, \hspace{5mm} U_\tau \subseteq U_{\sigma_2}.
\]

\begin{proposition}
\label{overlap}
If we restrict to
\[
\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G},
\]
the difference between $D_{U_{\sigma_1}}$ and $D_{U_{\sigma_2}}$ is a bounded operator.
\begin{proof}
Since $U_\tau = G \times_{G_\tau} Y_\tau$, the Lemma \ref{cross-1} gives us an isomorphism:
\begin{equation} \label{iso-2}
\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G} \cong  \big[  V_\lambda^*\otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\tau}} \otimes \C_{(\rho_{G} -\rho_{\tau}, h^{\vee})} \otimes  \Gamma_{c}^\infty(Y_{\tau}, S_{Y_{\tau}} \otimes L_{Y_{\tau}})\big]^{G_{\tau}}.
\end{equation}
The Dirac operator $D_{U_\tau}$ is defined as the combination of a cubic Dirac operator on
\[
V_\lambda^*\otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\tau}} \cong  \cong V_\lambda^* \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma_1}} \otimes S^*_{\mathfrak{g}_{\sigma_1}/\mathfrak{g}_\tau}
\]
and a Spin$^c$-Dirac operator on
\[
  \Gamma_{c}^\infty(Y_{\tau}, S_{Y_{\tau}} \otimes L_{Y_{\tau}}).
\]

On the other hand, we have that
\[
U_{\tau} \cong G \times_{G_{\sigma_1}} Y_{\tau}^{\sigma_1}, \hspace{5mm} Y_{\tau}^{\sigma_1}= G_{\sigma_1} \times_{G_{\tau}} Y_{\tau}.
\]
Applying Lemma \ref{cross-1} again,
\begin{equation}\label{iso-1}
\big[\Gamma_{c}^\infty(U_{\tau}, S^{\mathrm{spin}}  \otimes S^{\mathrm{pre}}_{\lambda}) \big]^{G}
\cong  \big[ V_\lambda^* \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma_1}} \otimes \C_{(\rho -\rho_{\sigma_1}, h^{\vee})} \otimes  \Gamma_{c}^\infty(Y_{\tau}^{\sigma_1},S_{Y_{\sigma_1}} \otimes L_{Y_{\sigma_1}} )\big]^{G_{\sigma_1}}.
\end{equation}
Under the isomorphism (\ref{iso-1}), the operator $D_{U_{\sigma_1}}$ decomposes into the sum of a cubic Dirac operator on
\[
V_\lambda^* \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma_1}}
\]
and a Spin$^c$-Dirac operator on
\[
 \Gamma_{c}^\infty(Y_{\tau}^{\sigma_1},S_{Y_{\sigma_1}} \otimes L_{Y_{\sigma_1}} ).
\]
It follows immediately that the two operators  $D_{U_\tau}$ and $D_{U_{\sigma_1}}$ are identical on
\[
V_\lambda^* \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma_1}}.
\]

Note that the spinor bundle
\[
S_{Y_{\sigma_1}}|_{Y_{\tau}^{\sigma_1}} \cong  G_{\sigma_1} \times_{G_{\tau}} (S_{Y_{\tau}}  \otimes S^{*}_{\mathfrak{g}_{\sigma_1}/ \mathfrak{g}_{\tau}} \otimes \C_{\rho_{\sigma_1}-\rho_{\tau}}),
\]
and the pre-quantum line bundle
\[
L_{Y_{\sigma_1}}|_{Y_{\tau}^{\sigma_1}} \cong  G_{\sigma_1} \times_{G_{\tau}} L_\tau.
\]
Therefore,  the  space
\[
\Gamma_{c}^\infty(Y_{\tau}^{\sigma_1},S_{Y_{\sigma_1}} \otimes L_{Y_{\sigma_1}} )
\]
decomposes into:
\[
\big[ \big(C^\infty(G_{\sigma_1}) \otimes S^{*}_{\mathfrak{g}_{\sigma_1}/ \mathfrak{g}_{\tau}}\big) \otimes \C_{\rho_{\sigma_1}-\rho_{\tau}} \otimes \Gamma_{c}^\infty(Y_{\tau}, S_{Y_{\tau}} \otimes  L_{Y_{\tau}} )\big]^{G_{\tau}}.
\]
By definition, both $D_{U_\tau}$ and $D_{U_{\sigma_1}}$ act as Spin$^{c}$-Dirac operators on
\[
\Gamma_{c}^\infty(Y_{\tau}, S_{Y_{\tau}} \otimes  L_{Y_{\tau}} );
\]
while on the factor
\[
C^\infty(G_{\sigma_1}) \otimes S^{*}_{\mathfrak{g}_{\sigma_1}/ \mathfrak{g}_{\tau}},
\]
$D_{U_\tau}$ acts as the cubic Dirac operator, and $D_{U_{\sigma_1}}$ acts as the Spin$^{c}$-Dirac operator.  By Lemma \ref{geo alg}, their difference is bounded.

We just show that the difference between $D_{U_{\sigma_1}}$ and $D_{U_{\tau}}$ is a bounded operator on the overlap $U_{\tau}$. Similarly one can show that the difference between $D_{U_{\sigma_2}}$ and $D_{U_{\tau}}$ is bounded as well. This completes the proof.
\end{proof}
\end{proposition}

\subsection{Construction of Dirac operator and main theorem}
For a fixed irreducible positive energy representation $V_\lambda$ at level $k$, we define
\[
\mathcal{H}_{\lambda} =  \big[L^2(M, S^{\mathrm{spin}}\otimes S^{\mathrm{pre}}_\lambda)\big]^{G}.
\]
The $\Z_2$-grading on the twisted spinor bundle $S^{\mathrm{spin}}$ equips $\mathcal{H}_\lambda$ with a $\Z_2$-grading. That is
\[
\mathcal{H}_\lambda = \mathcal{H}_\lambda^+ \oplus \mathcal{H}_\lambda^-.
\]

Select a $G$-invariant, smooth partition of unity $\{f_{\sigma}^{2}\}$ which is subordinate to the cover
\[
\{U_\sigma = G \times_{G_\sigma} Y_\sigma \}_{\sigma, \mathrm{dim}\sigma = 0}.
\]
We define a Dirac operator $\mathcal{D}$ on $\mathcal{H}$ by the formula:
\[
\mathcal{D}= \sum_{\sigma, \mathrm{dim}\sigma = 0} f_{\sigma} \cdot  D_{U_\sigma} \cdot f_{\sigma}.
\]

\begin{proposition} \label{same up to bounded}
The Dirac operator $\mathcal{D}$ doesn't depend on the choice of $\{f_\sigma^2\}$ up to homotopy.
\begin{proof}
Let $\{f_{\sigma}'^{2}\}$ be another partition of unity.
We define a Dirac operator $\mathcal{D}'$ on $\mathcal{H}_\lambda$ by
\[
\mathcal{D}'= \sum_{\sigma, \mathrm{dim}\sigma = 0} f'_{\sigma} \cdot  D_{U_\sigma} \cdot f'_{\sigma}.
\]
It suffices to show that
\[
\mathcal{D}' - \mathcal{D} \in \mathbb{B}(\mathcal{H}_{\lambda}).
\]

We compute that
\begin{equation}\label{step-1}
\begin{aligned}
\mathcal{D}&= \sum_{\sigma} f_{\sigma} \cdot  D_{U_\sigma} \cdot f_{\sigma} = \sum_{\sigma, \tau} f_{\sigma} \cdot  D_{U_\sigma}\cdot f'^2_\tau \cdot f_{\sigma}\\
& =  \sum_{\sigma, \tau} \big( f'_\tau \cdot f_{\sigma} \cdot  D_{U_\sigma} \cdot f_{\sigma} \cdot f'_\tau + f_{\sigma} \cdot  [D_{U_\sigma}, f'_\tau] \cdot f'_\tau \cdot f_{\sigma}\big)\\
& =  \sum_{\sigma, \tau} \big( f'_\tau  \cdot  D_{U_\sigma} \cdot f^2_{\sigma} \cdot f'_\tau  + f_{\sigma} \cdot  [D_{U_\sigma}, f'_\tau] \cdot f'_\tau \cdot f_{\sigma} + f'_{\tau} \cdot  [D_{U_\sigma}, f_\sigma] \cdot f'_\tau \cdot f_{\sigma}\big).\\
\end{aligned}
\end{equation}
Since the functions $f_\sigma$ are $G$-invariant, they commute with the cubic Dirac operator $D_{\mathrm{alg}}$. Their commutators with Spin$^c$-Dirac operators:
\[
c(df_\sigma) = [f_\sigma, D_{\mathrm{geo}}]
\]
are all bounded. It shows that
\begin{equation}\label{step-2}
 [D_{U_\sigma}, f'_\tau],\hspace{5mm}   [D_{U_\sigma}, f_\sigma]  \in \mathbb{B}(\mathcal{H}_{\lambda}).
\end{equation}

In addition, by Proposition \ref{overlap},
\[
 f'_\tau  \cdot ( D_{U_\sigma} - D_{U_\tau}) \cdot f^2_{\sigma} \cdot f'_\tau
\]
are bounded for all $\sigma, \tau$. Therefore,
\begin{equation}\label{step-3}
\begin{aligned}
&\sum_{\sigma, \tau} f'_\tau  \cdot  D_{U_\sigma} \cdot f^2_{\sigma} \cdot f'_\tau\\
&= \sum_{\tau, \sigma} f'_\tau  \cdot  D_{U_\tau}  \cdot f^2_\sigma \cdot f'_\tau + \sum_{\tau, \sigma}f'_\tau  \cdot ( D_{U_\sigma} - D_{U_\tau}) \cdot f^2_{\sigma} \cdot f'_\tau \\
& = \mathcal{D}' + \sum_{\tau, \sigma}f'_\tau  \cdot ( D_{U_\sigma} - D_{U_\tau}) \cdot f^2_{\sigma} \cdot f'_\tau
\end{aligned}
\end{equation}
The proposition follows from (\ref{step-1}),  (\ref{step-2}) and  (\ref{step-3}).
\end{proof}
\end{proposition}

\begin{proposition}\label{self-adjoint}
The Dirac operator $\mathcal{D}$ is essentially self-adjoint.
\begin{proof}
It is enough to show that each $f_\sigma \cdot D_{U_\sigma} \cdot f_\sigma$ is essentially self-adjoint.
Let us write
\[
D_{U_\sigma} = \big[D_{ \mathrm{alg}} \otimes 1 + 1\otimes D_{\mathrm{geo}}\big]^{G_\sigma}
\]
The algebraic part $D_{ \mathrm{alg}}$ is certainly self-adjoint, and the geometric part, $D_{\mathrm{geo}}$ is a symmetric Spin$^c$-Dirac operator on $Y_\sigma$. Recall that every symmetric Dirac operator on a complete manifold is essentially self-adjoint. But it doesn't apply directly to our case since $Y_\sigma$ is not complete. Nevertheless, we can get around this by the following trick.

Since the function $f_\sigma$ has compact support in $Y_\sigma$, we can find a smaller  subset $\widetilde{Y}_{\sigma}$ so that
\[
\mathrm{Supp} f_\sigma \subset \widetilde{Y}_{\sigma} \subset Y_\sigma
\]
and the closure of $\widetilde{Y}_{\sigma}$ is contained in $Y_\sigma$. We denote by $g$ the metric on $Y_\sigma$ and $\chi$ a positive function on $Y_\sigma$ so that $\chi|_{\widetilde{Y}_{\sigma}} \equiv 1$ and $\chi(m)$ tends to infinity as $m$ tends to the boundary of $Y_\sigma$.  Under the rescaled metric
\[
g^\chi(\cdot, \cdot) = \chi^2 \cdot g(\cdot, \cdot),
\]
the manifold $Y_\sigma$ becomes complete.

Let $\widetilde D_{\mathrm{geo}}$ be an essentially self-adjoint Spin$^c$-Dirac operator  on the complete manifold $Y_\sigma$. Consider a new Dirac operator
\[
\widetilde D_{U_\sigma} := [D_{\mathrm{alg}} \otimes 1 + 1\otimes \widetilde D_{\mathrm{geo}}]^{G_{\sigma}},
\]
which is essentially self-adjoint. Since the metric remains the same within $\widetilde{Y}_\sigma$, we have that
\[
\widetilde D_{U_\sigma} s = D_{U_\sigma} s
\]
for all $s$ with $\mathrm{Supp} s \subseteq \widetilde{Y}_\sigma$. Thus,
\[
f_\sigma \cdot \widetilde D_{U_\sigma} \cdot f_\sigma
\]
can be viewed as an operator on $\mathcal{H}_\lambda$, and
\[
f_\sigma \cdot D_{U_\sigma} \cdot f_\sigma = f_\sigma \cdot \widetilde D_{U_\sigma} \cdot f_\sigma
\]
is essentially self-adjoint.

\end{proof}
\end{proposition}

We define an operator by functional calculus,
\[
\mathcal{F} = \frac{\mathcal{D}}{\sqrt{1 + \mathcal{D}^2}}.
\]
Because $\mathcal{D}$ has a dense domain in $\mathcal{H}_\lambda$, $\mathcal{F}$ extends to a self-adjoint bounded operator on $\mathcal{H}_\lambda$, which anti-commutes with the $\Z_2$-grading on $\mathcal{H}_\lambda$.

\begin{theorem}\label{main theorem}
The bounded operator $\mathcal{F}$ is Fredholm on $\mathcal{H}_{\lambda}$. We define its Fredholm index by
\[
\mathrm{Ind}(\mathcal{D})_{\lambda}:=  \mathrm{ker}(\mathcal{F}) \cap \mathcal{H}^{+}_{\lambda} - \mathrm{ker}(\mathcal{F}) \cap \mathcal{H}^{-}_{\lambda} \in \Z.
\]
\end{theorem}
By Proposition \ref{same up to bounded},  the index is independent of the choice of partition of unity. Let us define a Hilbert space
\[
\mathcal{H} = \big[L^2(M, S^{\mathrm{spin}} \otimes S^{\mathrm{pre}})\big]^{G} \cong  \bigoplus_{\lambda \in P_{k,+}} V_{\lambda} \otimes \mathcal{H}_{\lambda},
\]
and a Dirac operator
\[
\mathcal{D}_M = \bigoplus_{\lambda \in P_{k,+}} 1 \otimes \mathcal{D}|_{\mathcal{H}_\lambda}.
\]
\begin{definition}
Let $(M, \omega)$ be a q-Hamiltonian $G$-space. If it is pre-quantizable at level $k$, we define its \emph{quantization}
\[
Q(M) =\mathrm{Ind}(\mathcal{D}_M) =  \bigoplus_{\lambda \in P_{k,+}} \mathrm{Ind}(\mathcal{D})_{\lambda}  \cdot  V_{\lambda} \in R_{k}(LG).
\]
\end{definition}
This generalizes Bott's Spin$^c$-quantization for Hamiltonian $G$-spaces to q-Hamiltonian $G$-spaces.

\begin{remark}
In \cite{MR2989614} Meinrenken develops a quantization from pre-quantized q-Hamiltonian $G$-spaces to the equivariant twisted $K$-homology of $G$ using push-forward maps.  By the work of Freed-Hopkins-Teleman \cite{Freed13} the equivariant twisted $K$-homology of $G$ at level $k$ is isomorphic to fusion ring of loop group $R_{k}(LG)$.
\end{remark}

\begin{example}
Let
\[
\mathcal{C} = G \cdot \exp(\xi) \cong G/H, \hspace{5mm} \xi \in \mathfrak{A}
\]
be a conjugacy class. We assume that it is pre-quantizable at level $k$. By Remark \ref{pre-quantum q},
\[
(\eta, k) = (k \cdot \xi, k) \in \widehat{\mathfrak{t}}^*_+
\]
is an integral weight. By definition, the twisted spinor bundle and twisted pre-quantum bundle are given by
\[
S^{\mathrm{spin}} = G \times_{H} (S_{L\mathfrak{g}, \mathfrak{h}}^* \otimes \C_{(\rho_{G} - \rho_{H}, h^{\vee})}), \hspace{5mm} S^{\mathrm{pre}} = G\times_{H}(H_{\mathrm{wzw}, k} \otimes \C_{(\eta, k)}).
\]
The Hilbert space
\[
\big[L^2(\mathcal{C}, S^{\mathrm{spin}}\otimes S^{\mathrm{pre}})\big]^G  \cong
\bigoplus_{\lambda \in P_{k,+}} V_{\boldsymbol{\lambda}} \otimes \big(V_{\boldsymbol{\lambda}}^{*} \otimes S_{L\mathfrak{g}/\mathfrak{h}}^{*} \otimes \C_{(\rho_{G}-\rho_{H} +\eta, k +h^{\vee})} \big)^{H}.
\]
By Theorem \ref{square operator-4}, one can calculate that
\[
Q(\mathcal{C})= \mathrm{ind}(\mathcal{D}_M) = V_{\eta} \in R_{k}(LG).
\]
This is an algebraic version  of the Borel-Weil construction for loop groups \cite{PS86}.
\end{example}

\subsection{Proof of Theorem \ref{main theorem}}
We will prove the main theorem in this subsection. Let us begin with a lemma.
\begin{lemma}
\label{bounded perturb}
Let $\mathcal{D},\mathcal{D}'$ be two self-adjoint unbounded operators on a Hilbert space $\mathcal{H}$ such that $\mathcal{D} - \mathcal{D}' \in  \mathbb{B}(\mathcal{H})$. For $\alpha = 0,1$, if
\[
\mathcal{D}^\alpha \cdot (1+ \mathcal{D}^2)^{-1} \in \mathbb{K}(\mathcal{H}),
\]
then
\[
\mathcal{D}'^\alpha \cdot (1+ \mathcal{D}'^2)^{-1} \in \mathbb{K}(\mathcal{H}).
\]
\begin{proof}
Suppose that $\mathcal{D}' = \mathcal{D} + B$ with  $B \in  \mathbb{B}(\mathcal{H})$. By straightforward calculation,
\begin{equation}\label{op-2}
\begin{aligned}
& (1+ \mathcal{D}^2)^{-1} -  (1+ \mathcal{D}'^2)^{-1} \\
& =(1+ \mathcal{D}^2)^{-1} \cdot(\mathcal{D}'^2 - \mathcal{D}^2) \cdot (1+ \mathcal{D}'^2)^{-1}\\
& = (1+ \mathcal{D}^2)^{-1} \cdot(\mathcal{D} \cdot B + B \cdot \mathcal{D}') \cdot (1+ \mathcal{D}'^2)^{-1}
\end{aligned}
\end{equation}
By the fact that the product of bounded operator and compact operator is again a compact operator and the assumption, we deduce that
\[
(1+ \mathcal{D}'^2)^{-1}\in \mathbb{K}(\mathcal{H}).
\]

We next consider
\begin{equation}\label{op-2}
\begin{aligned}
& \mathcal{D} \cdot (1+ \mathcal{D}^2)^{-1} -  \mathcal{D} \cdot  (1+ \mathcal{D}'^2)^{-1} \\
& = \mathcal{D} \cdot (1+ \mathcal{D}^2)^{-1} \cdot(\mathcal{D}'^2 - \mathcal{D}^2) \cdot (1+ \mathcal{D}'^2)^{-1}\\
& = \mathcal{D} \cdot (1+ \mathcal{D}^2)^{-1} \cdot(\mathcal{D} \cdot B + B \cdot \mathcal{D}') \cdot (1+ \mathcal{D}'^2)^{-1}
\end{aligned}
\end{equation}
For the same reason as above, it follows that
\[
\mathcal{D} \cdot (1+ \mathcal{D}'^2)^{-1}\in \mathbb{K}(\mathcal{H}).
\]
Because $\mathcal{D}' -\mathcal{D}$ is bounded, we conclude that
\[
\mathcal{D}' \cdot (1+ \mathcal{D}'^2)^{-1}\in \mathbb{K}(\mathcal{H}).
\]
\end{proof}
\end{lemma}

Fix a vertex $\sigma$ of $\mathfrak{A}$. Let us write
\[
D_{U_\sigma} = \big[D_{ \mathrm{alg}} \otimes 1 + 1\otimes D_{\mathrm{geo}}\big]^{G_\sigma},
\]
acting on
\[
 \big[V_{\lambda}^{*} \otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_\sigma, h^\vee)} \otimes  \Gamma_{c}^\infty(Y_\sigma, S_{Y_\sigma} \otimes L_{Y_\sigma})\big]^{G_{\sigma}}.
\]
By the trick used in Proposition \ref{self-adjoint}, we can assume that $Y_\sigma$ is complete and $D_{\mathrm{geo}}$ is an essentially self-adjoint Spin$^c$-Dirac operator on $Y_\sigma$.

 \begin{lemma}\label{compact op}
For $\alpha = 0,1$, one has that
\[
f_\sigma \cdot  D^{\alpha}_{U_\sigma} \cdot (1+ D^{2}_{U_\sigma})^{-1}  \cdot f_\sigma  \in \mathbb{K}(\mathcal{H}_\lambda).
\]
\begin{proof}
We decompose
\[
L^2(Y_\sigma, S_{Y_\sigma} \otimes L_{Y_\sigma})
\]
with respect to the $S^{1}_{\mathrm{rot}} \times \widehat{G}_\sigma$-action and denote by $M(\boldsymbol{\nu})$ the isotypic  component labeled by
\[
\boldsymbol{\nu} = (n, \nu, k) \in \Z \times \Lambda^{*} \times \Z.
\]
Since $f_{\sigma}$ has compact support, it follows from the Rellich's lemma that
\[ f_\sigma \cdot D_{\mathrm{geo}}^{\alpha} \cdot  (1 + D_{\mathrm{geo}}^{2} )^{-1} \cdot f_{\sigma}
\]
is a compact operator on
\[
L^2(Y_\sigma, S_{Y_\sigma} \otimes L_{Y_\sigma}).
\]
It implies that the norm of the restriction of
\[
f_\sigma \cdot  D^{\alpha}_{U_\sigma} \cdot (1+ D^{2}_{U_\sigma})^{-1}  \cdot f_\sigma
\]
to
\[
\big[ V_{\lambda}^*\otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})} \otimes  M(\boldsymbol{\nu})\big]^{G_{\sigma}}.
\]
tends to 0 as $\|\boldsymbol{\nu}\|$ tends to infinity. Thus, it suffices to show that the operator
\[
f_\sigma \cdot  D^{\alpha}_{U_\sigma} \cdot (1+ D^{2}_{U_\sigma})^{-1}  \cdot f_\sigma
\]
is compact after restricted to a fixed component
\[
\big[ V_{\lambda}^*\otimes S^{*}_{L\mathfrak{g}/(L\mathfrak{g})_{\sigma}} \otimes \C_{(\rho_{G} -\rho_{\sigma}, h^{\vee})} \otimes  M(\boldsymbol{\nu})\big]^{G_{\sigma}}.
\]

By Theorem \ref{square operator-4},  the operator $D^2_{\mathrm{alg}} = D^2_{L\mathfrak{g},(L\mathfrak{g})_\sigma}$ acts on the above space by
\[
 \|\boldsymbol{\lambda} + \boldsymbol{\rho}_{G}\|^{2} - \|\boldsymbol{\nu} + \boldsymbol{\rho}_{G}\|^{2}.
\]
By the formula in (\ref{weight product}),
\[
 \|\boldsymbol{\lambda} + \boldsymbol{\rho}_{G}\|^{2} - \|\boldsymbol{\nu} + \boldsymbol{\rho}_{G}\|^{2} =  \|\boldsymbol{\lambda} + \boldsymbol{\rho}_{G}\|^{2} + 2 \cdot n \cdot (k+ h^\vee) - \|\nu + \rho_G\|^2.
\]
Since $ \|\boldsymbol{\lambda} + \boldsymbol{\rho}_{G}\|^{2}$ and $ \|\nu + \rho_g\|^2$ are fixed constants, the operator $D^2_{\mathrm{alg}}$ tends to infinity as the energy level $n$ goes to infinity. This proves the assertion.
\end{proof}
\end{lemma}

\begin{lemma}\label{key lemma}
One has that
\[
f_\sigma \cdot (1+\mathcal{D}^{2})^{-1}  \cdot f_\sigma, \hspace{5mm} [f_\sigma,  (1+\mathcal{D}^{2})^{-1}] \cdot f_\sigma \in \mathbb{K}(\mathcal{H}_{\lambda})
\]
\begin{proof}
As shown in Proposition \ref{same up to bounded}, there exists a bounded operator $B$ so that
\[
\mathcal{D} \cdot s = D_{U_\sigma}\cdot s + B \cdot s.
\]
for any $s \in \mathcal{H}_\lambda$ with $\mathrm{Supp} s \subset U_\sigma$. By the choice of function $f_\sigma$,  the support of $f_\sigma \cdot s$ is automatically contained in $U_\sigma$ for all $s \in \mathcal{H}_\lambda$. One can verify that
\[
f_\sigma \cdot \mathcal{D}^\alpha \cdot  (1+\mathcal{D}^{2})^{-1}  \cdot f_\sigma = f_\sigma \cdot (D_{U_\sigma} + B)^\alpha \cdot \big(1+(D_{U_\sigma} + B)^{2} \big)^{-1}  \cdot f_\sigma.
\]
By Lemma \ref{bounded perturb} and \ref{compact op}, we conclude that for $\alpha=0,1$,
\[
f_\sigma \cdot \mathcal{D}^\alpha \cdot  (1+\mathcal{D}^{2})^{-1}  \cdot f_\sigma \in \mathbb{K}(\mathcal{H}_\lambda).
\]

For the second half, we calculate that
\begin{equation}
\begin{aligned}
&\big[f_\sigma,  (1+\mathcal{D}^{2})^{-1}\big]\cdot f_\sigma \\
= &(1+\mathcal{D}^{2})^{-1} \cdot [f_\sigma, \mathcal{D}^2] \cdot(1+\mathcal{D}^{2})^{-1}\cdot f_\sigma\\
=& (1+\mathcal{D}^{2})^{-1} \cdot c(df_\sigma)\cdot \mathcal{D}   \cdot(1+\mathcal{D}^{2})^{-1}\cdot f_\sigma\\
+& (1+\mathcal{D}^{2})^{-1} \cdot \mathcal{D} \cdot c(df_\sigma) \cdot(1+\mathcal{D}^{2})^{-1}\cdot f_\sigma.
\end{aligned}
\end{equation}
We point out here that $c(df_\sigma)$ is a bounded operator with support  in $U_\sigma$. As in Lemma \ref{compact op}, one can similarly prove that
\[
 c(df_\sigma)\cdot \mathcal{D}^\alpha   \cdot(1+\mathcal{D}^{2})^{-1}\cdot f_\sigma \in \mathbb{K}(\mathcal{H}_\lambda).
\]
This completes the proof.
\end{proof}
\end{lemma}

Because the operator
\[
\mathcal{F} = \frac{\mathcal{D}}{\sqrt{1 + \mathcal{D}^2}} \in \mathbb{B}(\mathcal{H}_\lambda),
\]
we have that
\begin{equation}
\begin{aligned}
 &1 - \mathcal{F}^2 = (1+\mathcal{D}^{2})^{-1}\\
 = &\sum_{\mathrm{dim}\sigma = 0}  (1+\mathcal{D}^{2})^{-1} \cdot f^2_\sigma\\
 =&  \sum_{\mathrm{dim}\sigma = 0} \big( f_\sigma \cdot (1+\mathcal{D}^{2})^{-1} \cdot f_\sigma +  [ f_\sigma,  (1+\mathcal{D}^{2})^{-1}] \cdot f_\sigma \big).
\end{aligned}
\end{equation}
By Lemma \ref{key lemma},
\[
 1 - \mathcal{F}^2 \in \mathbb{K}(\mathcal{H}_{\lambda}),
\]
which implies that $\mathcal{F}$ is a Fredholm operator on $\mathcal{H}_{\lambda}$.

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


text:
Liberals Outraged by 'Nazi' Judge's Ruling in Kyle Rittenhouse Case
A judge's decision to let Kyle Rittenhouse's attorneys refer to the men their client shot as "rioters" and "looters" has liberals melting down in rage.
The "fiery but mostly peaceful" narrative won't die.
Schroeder sided with the defense during a pre-trial hearing Monday in Kenosha, Wisconsin, after previously ruling prosecutors cannot refer to the three men shot by Rittenhouse as "victims" in his upcoming murder trial.
During the Rittenhouse motions today the State was ridiculed by the judge for saying "all we are talking about here is arson" to which the judge responded "I can't believe some of the things you're saying.. come on!" pic.twitter.com/3fVqKxlBUD
— John Curtis (@Johnmcurtis) October 25, 2021
Schroeder was responding Monday to the state's request that Rittenhouse's attorneys be barred from using "pejorative" terms to describe the men, Joseph Rosenbaum and Anthony Huber, who were killed, and Gaige Grosskreutz, who was wounded.
If prosecutors will be able to argue Rittenhouse "is a cold-blooded killer, then why couldn't [an attorney] call someone a rioter?" the judge asked the prosecutors Monday, per Kenosha News.
Schroeder also scoffed at Kenosha County Assistant District Attorney Tom Binger's attempts to downplay Rosenbaum's actions on the night of Aug. 25, 2020, when he was killed amid destructive racial justice protests in Kenosha.
"All we're talking about is arson. We're talking about being loud and disorderly," Binger told the judge, who replied: "I can't believe some of the things you're saying."
Reaffirming his ruling against calling the men "victims," Schroeder explained it is "a loaded, loaded word."
THE REACTION
On Twitter, the Daily Wire's Matt Walsh and other conservatives, among whom Rittenhouse's case has become a cause célèbre, said he should be acquitted and more.
It will be an outrage if Kyle Rittenhouse just walks away from this a free man. He should walk away a free and rich man after suing for malicious prosecution.
— Matt Walsh (@MattWalshBlog) October 27, 2021
But left-leaning journalists and activists condemned Schroeder's decision as proof the U.S. criminal justice system is "rigged" and America is irredeemably "racist."
"Our white supremacist court system will bend in any direction possible to make life easier for white supremacist murderer posterboy Kyle Rittenhouse," tweeted Cameron Kasky, a Parkland shooting survivor and gun control activist.
The popular Occupy Democrats account suggested Schroeder is a Nazi sympathizer and called for him to be "REMOVED FROM THE CASE."
Will Bunch, a Philadelphia Inquirer columnist, commiserated with his fellow "seething" liberals and listed Schroeder's supposed transgressions on the bench, such as his reputation for doling out stiff sentences and the time he urged AIDS tests for sex workers.
1. So you're probably seething about the judge in the Kyle Rittenhouse case saying lawyers can't call the 3 people he shot "victims," while there's a chance they can be called "rioters" or "looters"
Did you wonder what's up with this guy? I did.
— Will Bunch (@Will_Bunch) October 27, 2021
Rittenhouse, now 18, has been charged with two counts of homicide and one count of attempted homicide for shooting the men.
He has pleaded not guilty of all the charges and argued the shootings were in self-defense.
The trial is scheduled to begin Nov. 1.

redpajama_set_name: RedPajamaCommonCrawl


text:
Difference between revisions of "General Resources"
From AccessGA Wiki
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1 Newsletters
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The ATWiki
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SC ATAC
The South Carolina Assistive Technology Advisory Committee (ATAC) is a technology architecture committee with a primary focus on information technology accessibility. ATAC is composed of representatives from many of South Carolina's state agencies.
G3ict
The Global Initiative for Inclusive Information and Communication Technologies is an advocacy initiative launched in December 2006 by the United Nations Global Alliance for ICT and Development, in cooperation with the Secretariat for the Convention on the Rights of Persons with Disabilities at UN DESA. Its mission is to facilitate and support the implementation of the dispositions of the Convention on the Rights of Persons with Disabilities on the accessibility of Information Communication Technologies (ICTs) and assistive technologies.
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The American Foundation for the Blind removes barriers, creates solutions, and expands possibilities so people with vision loss can achieve their full potential. Among several resources and services, AFB provides valuable information on the creation of accessible websites at the following link: Creating Accessible Websites
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Flowers are the natural blossom, a God's gift to the nature that we live in. There is a wide variety of flowers available across the country. Flowers are perfect for any occasion and there are different flowers for different occasions. Once the flowers are being plucked from the garden it starts losing its life. Although the life span of different flowers is different and can stay fresh for a long time even after it has been plucked. Maintaining the flower after it has been plucked is a tough job but a bouquet of dead flowers does not look good.
A flower bouquet should always be fresh with the fresh natural aroma of the flower which should spread the surrounding with its fragrance. Fresh cut flowers always please the receiver and make them happy. Flower decorators and shop keepers pluck the bloomed flowers in advance before offering them to the customers.
It is important for them to maintain the freshness of the flower and its natural aroma for a long time. Shop keepers earlier used to sprinkle some scented water on the fresh flowers to keep them fresh but that did not help and reduced the life span of the flowers because of excessive water on them.
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We have a range of verity floral baskets, floral containers and floral vases that would definitely add to the beauty of your surroundings.
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Floral Foam is widely used in designing flower arrangements. Floral Foam is wettable foam used for a variety of fresh floral design projects. We have floral foam in a variety of densities which is pH controlled to prevent stem burning. And our floral color spray is delicate enough to color and preserve fresh flowers.
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'Germany is a country of migration': Stats
DPA/The Local - [email protected] • 14 Nov, 2014 Updated Fri 14 Nov 2014 16:12 CEST
New data released from the 2013 Microcensus showed that one in five Germans are from a migration background, Destatis announced on Friday.
More than 16 million people in Germany have roots outside of the country, Destatis said.
"The phrase 'Germany is a country of migration' is supported by these new numbers from the Federal Statistics Office," said Social Democrat (SPD) politician and government migration commissioner Aydan Özoguz. "Our country is as diverse as ever."
In the year 2013, the data showed that the migrant population in Germany grew by 3.8 percent, making it "the largest increase over the previous years" since the survey began in 2005, the Federal Statistics Office noted.
Destatis officer Joscha Dick said that the number crunchers counted anyone who had migrated to Germany since 1950 and their descendants and all foreigners. Of that group, 60 percent hold a German passport and around a third were born in Germany.
The survey also showed that there was a difference between the former East and West countries.
"Even 25 years after the fall of the wall, the former countries show very different numbers," said Dick.
Almost all, 97 percent, are found in the former West and in Berlin. In Thuringia, only four percent of the population have a migration background.
Professor Ludger Preis Foundation for Integration and Migration said that those low numbers could have dire consequences for states like Thuringia.
"Immigration has become the most important factor for compensating the population decline," he said.
Preis said that the idea that migrants will drain German social services is an old prejudice that Germans must get away from.
"Since the 1990s, there are more and more migrants from central and eastern Europe who area well educated," he said.
Stephan Sievert from the Berlin Institute for Population and Development said that current debates around "migration poverty" are distorting the big picture of migrants coming to Germany.
"Or course there are challenges," he said. "But there is also a lot being thrown into the pot with a lot of potential... We need immigration."
Most migrants in Germany stem from Turkey (13 percent) followed by Poland, Russia, Kazakhstan and then Romania to round out the top five.
Ozuguz said that while the federal government has done plenty to integrate migrants, "There remains much to do. We have yet to succeed in enabling children to have educational success regardless of their social background - whether in school or vocational training."
#Immigration #Destatis


To achieve its objectives, MISTRA has brought together experts and researchers in various fields as full-time and part time researchers, research fellows and as partners of various kinds.
A MISTRA Fellowship system has been established to bring into the MISTRA fold experts in a variety of fields as researchers on specific themes or high-level partners in the content work of MISTRA.
is an NRF-rated Humanities Research Professor at the University of Johannesburg. The Iowa-born scholar relocated to South Africa in 2004, joining Wits University where he was Professor of Philosophy and founding Director of the Centre for Ethics. He has worked on an eclectic range of topics and is recognised internationally for his rigorous research into value theory, including the sub-Saharan ethic of ubuntu. Professor Metz argues that ubuntu, in the African milieu, is the most defensible moral theory and be should be developed further for its contrasting potential with dominant Western/Eastern theories and belief systems such as utilitarianism and Kantianism. Included in his research repertoire is Kant's notion of justice, grounded on the concept of human dignity to which he has applied a variety of contemporary social issues of particular concern to South Africa and the world. He is the project leader of MISTRA's priority research project titled The Philosophy of Chinese Civilisation: The Rise, Decline and Rise of Civilisations. Part of his contribution to this project is to compare the value systems of ubuntu and Confucianism and their role in a country's economy, technology, innovation, politics, and society.
is a graduate of Wits University. She joined the National Centre for Occupational Health (NCOH) which later became the National Institute for Occupational Health (NIOH). Currently she is the head of the Toxicology Section at the institute while holding an honorary professorial post in the Haematology and Molecular Medicine Department, University of the Witwatersrand. In her university post she is involved in post graduate training and presents short courses in Toxicology and Health Risk assessment.
Amongst her many associations, Professor Gulumian represents South Africa in the OECD Working Party on Manufactured Nanomaterial's as well as serving on a related national expert technical committee. She is a member of the Society for Free Radical Biology and Medicine (SFRBM), Society of Toxicology (SOT) and Society of Risk Analysis (SRA).
Professor Gulumian's research interests include hazard identification and communication as well as elucidation of mechanisms of toxicity of micro and nano particles, toxic metal ions, and pesticides. She leads numerous nationally funded research projects, has authored and co-authored several scientific publications and has been invited to present at conferences across the globe. She has provided expert consultation to industry and government departments on the toxicity of chemicals in working and ambient environments. MISTRA is privileged to have Professor Gulumian providing an advisory role on its priority research project Beyond the Imagination which is focusing on nano technology.
currently leads the division of mathematics education at the School of Education, University of the Witwatersrand. She further leads the Marang mathematical literacy programme. Professor Venkatkrishnan completed a PhD in mathematics education in 2004 at Kings College - her doctoral thesis was awarded the British Educational Research Association award in 2005 with the distinction of being the best educational thesis submitted for 2004. Her research has focused on the impact of policy initiatives on mathematics teaching and learning in schools in the UK and South Africa. Her recent publications focus on understanding the implementation of Mathematical Literacy in the FET phase in South Africa. In addition to this ongoing focus, she is currently working on the development of a primary mathematics specialism within mathematics education teaching and research, spanning the work of the school and Marang. MISTRA is honoured that she has agreed to contribute to the priority research project, The Pedagogy of Mathematics.
has worked in a wide range of sectors — businesses, non-governmental organisations, international development organisations and academia, in China, South Africa and The Netherlands.
He gained his PhD in International Development Studies at the University of Groningen, Netherlands. His research covered social, institutional and political economic dimensions of natural resource governance in China and Africa, and China's engagement with Africa on agricultural and sustainable development issues. Professor Zhao has produced three books on land reform, and his most recent book China's Disappearing Countryside: Towards Sustainable Land Governance for the Poor was published by Ashgate in 2013. He has also been widely published in journals and these cover his work collaborations on research projects in Africa, the sub-continent, South East Asia, China, South America and Europe.
Professor Zhao spent many years working in China and more recently joined the (UK) Department for International Development (DFID) in Beijing as an advisor. He has since occupied a variety of posts at the University of Groningen. Currently he is Assistant Professor in Globalisation Studies and Humanitarian Action at the university, and MISTRA is honoured to be associated with him as he contributes to the priority research project, The philosophy of Chinese civilisation: the rise and decline and rise of civilisations.
Prof Pitika Ntuliis a scholar with wide ranging expertise. He is an artist, sculptor, poet and a respected arts activist. Deeply connected to South Africa's struggle for liberation, Professor Ntuli actively pursues through his art, a campaign to preserve memory and forge a path of healing in our country.
Professor Ntuli has chaired the arts and culture ministerial task team as well as the ministerial advisory committee on indigenous advisory systems and a range of other related issues over the past twenty years. In addition, he has contributed to a wealth of committees including the National Film and Video Foundation, World Congress of Sociologists, Womad (World of music, art and drama), the 1995 Venice Biennale and the SA Art and Cultural Association, to name a few.
Professor Ntuli has held various posts at universities, (UKZN and UDW, Wits) institutes and art schools, and was the Artistic Director at the Jenako Arts Community Arts Centre in Brixton, London.
Professor Ntuli has published and exhibited widely. He has been an artist in residence, a curator of exhibitions and has made a rich contribution to film and television. MISTRA is honoured to have him lead the priority research project Arts and national development: an evolving South African challenge.
Dr Edmund Thamsanqa (Thami) Mazwais executive chairman of Mtiya Dynamics (Pty) Ltd a company specialising in the development of enterprise and supply. Formerly a journalist and editor, he changed direction towards small business development because of his interest in entrepreneurship. Dr Mazwai started the business section of the Sowetan and later ran his own business magazine, Enterprise, which focused on black entrepreneurs. He was formerly director of the Centre for Small Business Development at the University of Johannesburg and is currently Resident Executive at the Wits Business School, advising on small business support programmes.
He is Adjunct Professor of Entrepreneurship at the University of Fort Hare. In January 2013 Mazwai was in Russia, part of a panel of experts looking at methodologies to integrate small business into the value chains of the Brazil, Russia, India, China and South Africa (BRICS) countries. He is a member of the Skills System Review Technical task team of the Human Resource Development Council of South Africa and his brief is to help integrate small business into the national skills development agenda.
Dr Mazwai has an MBA from the UK-based Henley Management College and a D.Comm from the University of Pretoria. He has designed, and is currently working on, an enterprise development strategy for rural areas pioneered by the Services Seta, and has designed a strategy for the Department of Trade and Industry (dti) that seeks to identify robust small businesses that can grow into formidable entities. The small business strategy will be launched later this year by the dti and will be implemented by the Small Enterprise Development Agency.
Dr Mazwai has published books and papers and he contributes regularly in the media on issues pertaining to Black Economic Empowerment and SME development. He was the founder chairperson of the Black Editors Forum and founder chairperson of the South African National Editors Forum. He is a past chairperson of the Wholesale and Retail Seta (WRSeta).
Prof Gauhar Razais based at the New Delhi based Academy of Scientific and Innovative Research, and chief scientist and Head of the Science Communication through Multimedia Division, at the National Institute of Science Communication and Information Resources. He was a member of the core team that conceived and planned a national science communication incentive Bharat Jan Vigyan Jatha in India in the 1990s. Professor Raza is an electrical engineer by qualification, a scientist by designation, a communicator and a poet by passion. He initiated research on public understanding of science in India and has conducted large-scale public understanding surveys. The national project conceived and directed by him has probed attitudinal shifts in public understanding among the Indian population for the past 25 years. His published work includes books, research papers, and monographs on public understanding of science, science communication, cultural studies and indigenous knowledge systems. He has produced a number of documentary films on various topics of science and technology, and produced the first science serial for TV in Hindi, and anchored a number of popular science serials for television channels, including Kasauti (the Touch Stone), Scientifically Yours and Eureka.
Professor Raza is a member of the MISTRA research team on The philosophy of China: rise, decline and rise of civilisations.
Dr van Heerden's work experience over the last 2 decades has been in strategic support to the ANC in the late 1990's, the South African Qualifications Authority in 2000, and independent management consultancies. He will be publishing his thesis in due course, and is currently writing chapters for the Institute for Global Dialogue (on development diplomacy) and the CSIR (on South African foreign relations), respectively.
For the past year, Dr van Heerden has worked for the City of Johannesburg. He is currently in the Executive Mayor's office, responsible for the strategic orientation of both the domestic and foreign stakeholders of the metropole. Dr van Heerden remains an active member of the Thabo Mbeki Foundation's 'Concerned Africans Forum' and the Mapungubwe Institute of Strategic reflections (MISTRA).


This extension point allows a plug-in to register compare/merge viewers for structural content types. The viewer is expected to extend org.eclipse.jface.viewers.Viewer. However, since viewers don't have a default constructor, the extension point must implement the factory interface for viewers org.eclipse.compare.IViewerCreator.
extensions - a comma separated list of file extensions e.g. "zip, jar"
A contentTypeBinding binds a structure merge viewer to a content type.
The Compare UI plugin defines a structure compare viewer for zip archives.


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Home Blog Chicago's e-scooter pilot program ends Tuesday
Chicago's e-scooter pilot program ends Tuesday
CHICAGO (October 15, 2019) – Chicago's experiment with electric scooters officially ends Tuesday. The test began with high expectations and concern over safety, but now the city has to decide whether to make scooters permanent.
The city's analysis of the program will include studying where and when scooters were used and examining hospital injury reports.
The city reports the 2,500 scooters were taken out for nearly 800,000 rides since mid-June – about 7,000 trips a day. The most popular spots for scooter use tended to be in the eastern part of the pilot area, in trendier, wealthier neighborhoods. The Near West Side, for example, saw 622 scooter departures and 465 scooter destinations on July 24, while only 23 trips started in the West Side neighborhood of West Garfield Park on the same day.
The program has drawn some mixed reviews.
Critics of the program have said "scooter litter" was one of their biggest complaints, or riders zipping on sidewalks instead of streets. The Active Transportation Alliance published a report Monday with their recommendations on how to move this program forward. One of the group's recommendations was to institute docking or public post stations.
The ATA's report also concluded that scooters offer fewer health and environmental benefits than walking, biking and transit.
Injury reports from area hospitals have not yet been released publicly. Hospitals have been asked to report scooter injuries to the Chicago Department of Public Health, which will make the data available after the program ends. But according to the Chicago Tribune, among the most seriously injured was a man who was riding his bike home from work when he was hit by a scooter rider who was going the wrong way and took off after the crash.
Despite the problems, the scooters have been popular and they gained in acceptance the more people tried them. Those who support the program say not only are they a fun way to get around, but they are also are a good way of connecting commuters to public transit.