Harmonic Spinors on Reductive Homogeneous Spaces

Abstract

An integral intertwining operator is given from certain principal series representations into spaces of harmonic spinors for Kostant’s cubic Dirac operator. This provides an integral representation for harmonic spinors on a large family of reductive homogeneous spaces.

1 Appendix: Geometric vs. Algebraic Dirac Operators

For the convenience of the reader we provide here a proof of Proposition 8. Recall that V denotes a smooth admissible representation of G, V _K the space of K-finite vectors in V, $V _{K}^{\star }$ the K-finite dual of V _K and $S_{\mathfrak{s}}$ is the spin representation for $\mathfrak{k}$. Let E be a finite-dimensional representation of $\mathfrak{k}$ such that the tensor product $E \otimes S_{\mathfrak{s}}$ lifts to a representation of the group K, and denote by E ^⋆ the dual of E. The K-representation $E \otimes S_{\mathfrak{s}}$ induces a homogeneous bundle $\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E}\longrightarrow G/K$ over G∕K whose space of smooth sections, on which G acts by left translations, is denoted by $C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E})$. The map

$$\displaystyle\begin{array}{rcl} \varPsi:\mathrm{ Hom}_{G}(V,C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E}))\longrightarrow \mathrm{Hom}_{ K}(E^{\star },S_{\mathfrak{s}} \otimes V _{ K}^{\star })& & {}\\ \end{array}$$

defined by $\varPsi (T)(e^{\star })(v) = 1 \otimes e^{\star }T(v)(1)$, is an isomorphism, where 1 denotes the identity G.

Next, as in Sect. 2, consider the (cubic) Dirac operators

$$\displaystyle\begin{array}{rcl} & & \mathcal{D}_{G/K}(\mathcal{E}): C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E})\longrightarrow C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E})\text{ and } {}\\ & & D_{V _{K}^{\star }}: S_{\mathfrak{s}} \otimes V _{K}^{\star }\longrightarrow S_{\mathfrak{s}} \otimes V _{ K}^{\star }, {}\\ \end{array}$$

and define the maps

$$\displaystyle\begin{array}{rcl} \mathcal{D}_{{\ast}}:\mathrm{ Hom}_{G}(V,C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E}))\longrightarrow \mathrm{Hom}_{ G}(V,C^{\infty }(G/K,\mathcal{S}_{\mathfrak{s}} \otimes \mathcal{E}))& & {}\\ \end{array}$$

and

$$\displaystyle\begin{array}{rcl} D_{{\ast}}:\mathrm{ Hom}_{K}(E^{\star },S_{\mathfrak{s}} \otimes V _{ K}^{\star })\longrightarrow \mathrm{Hom}_{ K}(E^{\star },S_{\mathfrak{s}} \otimes V _{ K}^{\star })& & {}\\ \end{array}$$

by

$$\displaystyle\begin{array}{rcl} & & (D_{{\ast}}(T))(v) = \mathcal{D}_{G/K}(\mathcal{E})(T(v)), {}\\ & & (D_{{\ast}}(A))(e^{\star }) = D_{ V _{K}^{\star }}(A(e^{\star })). {}\\ \end{array}$$

We claim that the following diagram is commutative:

317171_1_En_6_Figa_HTML.gif

Indeed one has

$$\displaystyle\begin{array}{rcl} \varPsi (D_{{\ast}}(T))(e^{\star })(v) = (1 \otimes e^{\star })\big((D_{ {\ast}}(T))(v)(1)\big) = (1 \otimes e^{\star })\big(\mathcal{D}_{ G/K}(T(v))(1)\big)& & {}\\ \end{array}$$

and

$$\displaystyle\begin{array}{rcl} & & \mathcal{D}_{G/K}(T(v))(1) {}\\ & & \qquad \qquad =\sum _{i} \frac{d} {dt}\vert _{t=0}(\gamma (X_{i}) \otimes 1)(T(v)(\exp (tX_{i})(1)) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes 1)(T(v)(1)) {}\\ & & \qquad \qquad =\sum _{i} \frac{d} {dt}\mid _{t=0}(\gamma (X_{i}) \otimes 1)(T(\exp (-tX_{i})v)(1)) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes 1)(T(v)(1)) {}\\ & & \qquad \qquad = -\sum _{i}(\gamma (X_{i}) \otimes 1)(T(X_{i}v)(1)) - (1 \otimes \gamma (\mathbf{c_{\mathfrak{s}}}))(T(v)(1)) {}\\ \end{array}$$

which means that

$$\displaystyle\begin{array}{rcl} \varPsi (D_{{\ast}}(T))(e^{\star })(v) = -\sum _{ i}(\gamma (X_{i}) \otimes e^{\star })(T(X_{ i}v)(1)) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes e^{\star })(T(v)(1)).& & {}\\ \end{array}$$

On the other hand, one has

$$\displaystyle\begin{array}{rcl} & & \Big((D_{{\ast}}(\varPsi (T))(e^{\star })\Big)(v) =\Big (D_{ V _{K}^{\star }}(\varPsi (T)(e^{\star }))\Big)(v) {}\\ & & \qquad \quad = -\sum _{i}\big((\gamma (X_{i}) \otimes X_{i})(\varPsi (T)(e^{\star })\big)(v) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes 1)(\varPsi (T)(e^{\star })(v)) {}\\ & & \qquad \quad = -\sum _{i}(\gamma (X_{i}) \otimes 1)(\varPsi (T)(e^{\star })(X_{ i}v)) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes 1)(\varPsi (T)(e^{\star })(v)) {}\\ & & \qquad \quad = -\sum _{i}(\gamma (X_{i}) \otimes e^{\star })(T(X_{ i}v)(1)) - (\gamma (\mathbf{c_{\mathfrak{s}}}) \otimes e^{\star })(T(v)(1)). {}\\ \end{array}$$

We deduce the following isomorphism relating algebraic and geometric harmonic spinors:

$$\displaystyle\begin{array}{rcl} \varPsi:\mathrm{ Hom}_{G}(V,\mathrm{ker}(\mathcal{D}_{G/K}(\mathcal{E}))) \simeq \mathrm{ Hom}_{K}(E^{\star },\mathrm{ker}(D_{ V _{K}^{\star }})),& & {}\\ \end{array}$$

therefore proving Proposition 8.