Character formulæ and GKRS multiplets in equivariant K-theory

Abstract

Let $G$ be a compact Lie group, $H$ a closed subgroup of maximal rank and $X$ a topological $G$-space. We obtain a variety of results concerning the structure of the $H$-equivariant K-ring $K_H^*(X)$ viewed as a module over the $G$-equivariant K-ring $K_G^*(X)$. One result is that the module has a nonsingular bilinear pairing; another is that the module contains multiplets which are analogous to the Gross–Kostant–Ramond–Sternberg multiplets of representation theory.

Appendices

Appendix: twisted K-theory

In this appendix we summarize the necessary facts from twisted K-theory in a form suited to our purpose. Like the original treatment by Donovan and Karoubi [11], we cover only K-theory twisted by torsion classes. We denote by $X$ a compact topological space and by $G$ and $H$ two (not necessarily connected) compact Lie groups. Contrary to our convention elsewhere in the paper, we do not assume $H$ to be a subgroup of $G$.

1.1 Twists and twisted vector bundles

A twist of $X$ is a pair $\tau =(P,H^{(\tau )})$, where ${\text{ pr}}:P\rightarrow X$ is a principal $H$-bundle over $X$ and $H^{(\tau )}$ is a central extension of $H$ by $\mathbf{U}(1)$. We regard $P$ as an $H^{(\tau )}$-space on which the central circle $\mathbf{U}(1)$ acts trivially. A $\tau$ -twisted vector bundle over $X$ is an $H^{(\tau )}$-equivariant complex vector bundle over $P$ which is of level $1$, in the sense that the central circle of $H^{(\tau )}$ acts on $E$ by scalar multiplication on the fibres.

A morphism between $\tau$-twisted vector bundles $E_1$ and $E_2$ is an $H^{(\tau )}$-equivariant vector bundle homomorphism $E_1\rightarrow E_2$. We denote by $\mathfrak Vec (X,\tau )$ the category of $\tau$-twisted vector bundles. This is an additive category, in which there is an obvious notion of an exact sequence. Thus, we can form the Grothendieck group of $\mathfrak Vec (X,\tau )$, which we will denote by $K(X,\tau )$.

A twist $\tau$ is a simple example of a gerbe with band $\mathbf{U}(1)$ over $X$, and $K(X,\tau )$ is called the $\tau$ -twisted K-group of $X$, or the K-group of $X$ with coefficients in  $\tau$. See [9, § 2] or [45, § 2.5] for more general notions of a gerbe and a comparison with other versions of twisted K-theory.

We can multiply a twisted bundle $F$ by an ordinary vector bundle $E$ on $X$ by the rule $E\cdot F={\text{ pr}}^*E\otimes F$. This rule turns $K(X,\tau )$ into a $K(X)$-module. More generally, let $\tau _1=(P_1,H_1^{(\tau _1)})$ and $\tau _2=(P_2,H_2^{(\tau _2)})$ be two twists of $X$. The sum of $\tau _1$ and $\tau _2$ is the twist $\tau =(P,H^{(\tau )})$, where $P$ is the fibred product $P_1\times _XP_2$, viewed as a principal bundle over $X$ with structure group $H=H_1\times H_2$, and the central extension $H^{(\tau )}$ is the quotient of $H_1^{(\tau _1)}\times H_2^{(\tau _2)}$ by the anti-diagonal copy of $\mathbf{U}(1)$. Any $\tau _1$- and $\tau _2$-twisted bundles on $X$ can be lifted to $P$ and the tensor product of the lifts is a $\tau _1+\tau _2$-twisted bundle. This defines a multiplication law

$$\begin{aligned} K(X,\tau _1)\times K(X,\tau _2)\longrightarrow K(X,\tau _1+\tau _2). \end{aligned}$$

A twist $\tau$ pulls back under a continuous map $f:Y\rightarrow X$ in an evident way, and we have an induced homomorphism

$$\begin{aligned} f^*:K(X,\tau )\longrightarrow K(Y,f^*\tau ). \end{aligned}$$

The twist $\tau$ is (Morita) trivial if there exists a trivialization, that is, a principal $H^{(\tau )}$-bundle $Q$ over $X$ such that $P$ is the quotient of $Q$ by the central circle of $H^{(\tau )}$. (If the extension $H^{(\tau )}$ of $H$ is trivial, then the twist $\tau$ is Morita trivial, but the converse is false.) If $\tau$ is Morita trivial, then

$$\begin{aligned} K(X,\tau )\cong K_{H^{(\tau )}}(Q)\cong K\bigl (Q{/}H^{(\tau )}\bigr )\cong K(X). \end{aligned}$$

(6.1)

This isomorphism depends on the choice of the trivialization $Q$.

Example 6.1.1

Let $V$ be an oriented real vector bundle of rank $m$ over $X$ provided with a Riemannian metric. The orientation twist or orientation system associated with $V$ is the pair $\omega _V=(\mathbf{SO}(V),\mathbf{Spin}^c(m))$. Here $\mathbf{SO}(V)$ is the oriented orthogonal frame bundle of $V$, which has structure group $\mathbf{SO}(m)$, and $\mathbf{Spin}^c(m)$ is the $\mathbf{Spin}^c$-group of the Euclidean space $\mathbf{R}^m$. The orientation system is Morita trivial precisely when $V$ possesses a $\mathbf{Spin}^c$-structure, tht is, an orientation in K-theory. If $X$ is an oriented Riemannian manifold, the orientation twist $\omega _X$ of $X$ is defined to be the orientation twist of the tangent bundle of $X$.

Example 6.1.2

Let $\tau$ be a central extension of a compact Lie group $H$ by $\mathbf{U}(1)$. By letting $H$ act on itself by right multiplication we can view $H$ as a principal bundle over the one-point space $X=\mathrm{pt}$. From this point of view $\tau$ is nothing but a twist of a point. As such it is Morita trivial, and therefore it follows from (6.1) that $K(\mathrm{pt},\tau )\cong \mathbf{Z}$.

Suppose the compact Lie group $G$ acts continuously on $X$. A twist $\tau =(P,H^{(\tau )})$ of $X$ is $G$ -equivariant if the principal bundle $P$ is $G$-equivariant, that is, equipped with a $G$-action by bundle maps which lifts the $G$-action on the base $X$ and which commutes with the action of the structure group $H$. A $G$ -equivariant $\tau$ -twisted vector bundle over $X$ is a $G\times H^{(\tau )}$-equivariant complex vector bundle over $P$ which is of level $1$ with respect to $H^{(\tau )}$. Such twisted bundles are the objects of an exact category $\mathfrak Vec _G(X,\tau )$, whose K-group $K_G(X,\tau )$ is the equivariant K-group of $X$ with coefficients in $\tau$.

Example 6.1.3

The orientation twist of an oriented Riemannian vector bundle $V$ is equivariant with respect to any compact Lie group which acts on $V$ by orientation-preserving isometric bundle maps. In particular, the orientation twist of an oriented Riemannian manifold $X$ is equivariant with respect to any compact Lie group which acts on $X$ by orientation-preserving isometries.

Example 6.1.4

Let $X=\mathrm{pt}$ and let $\tau$ be as in Example 6.1.2. We turn $\tau$ into an $H$-equivariant twist by letting $H$ act on $H$ by left multiplication. Every $H\times H^{(\tau )}$-equivariant vector bundle $E$ over $H$ trivializes equivariantly to a product bundle $E\cong H\times U$. On this product bundle $H$ acts on the base $H$ by left multiplication and trivially on the vector space $U$, and $H^{(\tau )}$ acts on the base by right multiplication and linearly on the fibre. Thus, the bundle $E$ is of level $1$ if and only if the $H^{(\tau )}$-module $U$ is of level $1$. It follows that the category $\mathfrak Vec _H(\mathrm{pt},\tau )$ is equivalent to the category of level $1$ $H^{(\tau )}$-modules. We conclude that $K_H(\mathrm{pt},\tau )\cong R(H,\tau )$, the twisted representation module of $H$.

Example 6.1.5

Generalizing Example 6.1.4, we let $X$ be a topological $H$-space and $p:X\rightarrow \mathrm{pt}$ the constant map. The $H$-equivariant twist $\tau$ pulls back to the $H$-equivariant twist $p^*\tau$ on $X$. We view $X$ as an $H^{(\tau )}$-space on which the central circle acts trivially. As in Example  6.1.4 one shows that the category $\mathfrak Vec (X,p^*\tau )$ is equivalent to the category of $H^{(\tau )}$-equivariant level $1$ vector bundles on $X$. Thus $K_H(X,p^*\tau )$ is the Grothendieck group of $H^{(\tau )}$-equivariant level $1$ vector bundles on $X$. To simplify the notation we will often write this group as $K_H(X,\tau )$.

Example 6.1.6

(induced twists) Continuing Example 6.1.5, we suppose that $H$ is a subgroup of $G$ and let $i:H\rightarrow G$ the inclusion map. We view the product $G\times X$ as a $G$-equivariant principal $H$-bundle over the associated bundle $G\times ^HX$. Hence the pair

$$\begin{aligned} i_*\tau =\bigl (G\times X,H^{(\tau )}\bigr ) \end{aligned}$$

is a $G$-equivariant twist of $G\times ^HX$, called the twist induced by $\tau$. If $E$ is a $\tau$-twisted vector bundle over $X$, then $G\times E$ is a $i_*\tau$-twisted vector bundle over $G\times ^HX$. The map $E\mapsto G\times E$ defines an equivalence of categories between $\mathfrak Vec (X,\tau )$ and $\mathfrak Vec (G\times ^HX,i_*\tau )$. Thus we have a natural isomorphism

$$\begin{aligned} K_G(G\times ^HX,i_*\tau )\cong K_H(X,\tau ). \end{aligned}$$

For $X=\mathrm{pt}$ this specializes to $K_G(M,i_*\tau )\cong R(H,\tau )$, where $M=G{/}H$. For simplicity we will often write $K_G(G\times ^HX,\tau )$ instead of $K_G(G\times ^HX,i_*\tau )$.

1.2 Relative twisted K-theory

Relative twisted K-classes are presented by complexes of twisted vector bundles. Let $X$ be a compact $G$-space and let $\tau =(P,H^{(\tau )})$ be a $G$-equivariant twist of $X$. Consider the category $\mathfrak Vec _G^*(X,\tau )$ of bounded complexes associated with the additive category $\mathfrak Vec _G(X,\tau )$. Thus an object of $\mathfrak Vec _G^*(X,\tau )$ is a $\mathbf{Z}$-graded $G$-equivariant $\tau$-twisted bundle $E^*$ such that $E^j=0$ for almost all $j$, furnished with a differential of degree $1$. Let $Y$ be a closed $G$-invariant subspace of $X$. We denote by $\mathfrak Vec _G^*(X,Y,\tau )$ the full subcategory of $\mathfrak Vec _G^*(X,\tau )$ comprising all objects $E^*$ with the property that the restriction of $E^*$ to the subspace ${\text{ pr}}^{-1}(Y)$ of $P$ is an exact complex. The set $L_G(X,Y,\tau )$ of isomorphism classes of $\mathfrak Vec _G^*(X,Y,\tau )$ is an abelian monoid. A quotient of $L_G(X,Y,\tau )$ by an appropriate submonoid (which is defined in the same way as in ordinary K-theory; see [34, § 3]) is the relative twisted K-group $K_G(X,Y,\tau )$ of $X$. The Euler characteristic map $L_G(X,Y,\tau )\rightarrow K_G(X,\tau )$ defined by $[E^*]\mapsto \sum _j(-1)^j[E^j]$ induces a homomorphism

$$\begin{aligned} K_G(X,Y,\tau )\rightarrow K_G(X,\tau ), \end{aligned}$$

which is an isomorphism if $Y$ is empty.

The group $K_G^0(X,Y,\tau )=K_G(X,Y,\tau )$ is the degree $0$ part of the $\mathbf{Z}{/}2\mathbf{Z}$-graded K-group $K_G^*(X,Y,\tau )$. The degree $1$ part is defined by

$$\begin{aligned} K_G^1(X,Y,\tau ) =K_G\bigl (X\times [0,1],(Y\times [0,1])\cup (X\times \{0,1\}),\tau \bigr ). \end{aligned}$$

1.3 The Thom isomorphism

Let $X$ be a compact $G$-space, and let $\tau$ be a $G$-equivariant twist of $X$. Let $\pi :V\rightarrow X$ be a $G$-equivariant oriented real vector bundle of even rank $m=2l$ equipped with an invariant Riemannian metric, and let $\omega _V$ be the orientation twist of $V$. We denote the unit ball bundle of $V$ by BV, the unit sphere bundle by SV and the zero section by $\zeta :X\rightarrow V$. Let $P=\mathbf{SO}(V)$. The spinor module $S=S^0\oplus S^1$ of the Clifford algebra $\mathbf{Cl}(\mathbf{R}^{2l})$ is a level $1$ $\mathbf{Spin}^c(2l)$-module, so the product bundle $E=\pi ^*(P)\times S$ is a $\mathbf{Z}{/}2\mathbf{Z}$-graded $\pi ^*(\omega _V)$-twisted vector bundle over (the total space of) $V$. Consider the two-term $G\times \mathbf{Spin}^c(2l)$-equivariant complex of vector bundles

$$\begin{aligned} E^0\overset{\text{ cliff}}{\longrightarrow }E^1 \end{aligned}$$

(6.2)

defined by placing the term $E^j$ in degree $j$ and for each $v\in V$ letting $\text{ cliff}(v):E_v^0\rightarrow E_v^1$ be Clifford multiplication by $v$. Since $\text{ cliff}(v)$ is an isomorphism for $v\ne 0$, the complex (6.2) defines a class in $L_G(BV,SV,\pi ^*\omega _V)$, and hence a class

$$\begin{aligned} \mathbf{th}(V)\in K_G^0(BV,SV,\pi ^*\omega _V), \end{aligned}$$

which is called the Thom class of $V$. The Thom map is the map

$$\begin{aligned} \zeta _*:K_G^*(X,\tau )\rightarrow K_G^*\bigl (BV,SV,\pi ^*(\tau +\omega _V)\bigr ) \end{aligned}$$

defined by $\zeta _*(a)=\pi ^*(a)\mathbf{th}(V)$. The following result is [18, Theorem IV.6.21]. See also [10, § 3.2] for a discussion closer to our treatment.

Theorem 6.3.1

The Thom map is an isomorphism of graded $K_G^*(X)$-modules.

Appendix: Central extensions

In this appendix we gather a few elementary facts regarding central extensions. The notation is as stated at the beginning of Sect. 2. In particular $G$ denotes a compact connected Lie group. In addition $\check{G}$ denotes a connected central extension of $G$ by a compact abelian Lie group $C$,

$$\begin{aligned} 1\longrightarrow C\longrightarrow \check{G}\longrightarrow G\longrightarrow 1. \end{aligned}$$

A complex $\check{G}$-module $V$ has central character  $\chi \in \mathcal{X }(C)$ if the subgroup $C$ acts on $V$ by $c\cdot v=\chi (c)v$. Let $\mathfrak Rep ^\chi (\check{G})$ be the category of finite-dimensional complex $\check{G}$-modules of central character $\chi$. We call the Grothendieck group $R^\chi (\check{G})$ of $\mathfrak Rep ^\chi (\check{G})$ the $\chi$ -twisted representation module of $G$. The category $\mathfrak Rep ^0(\check{G})$ (where $\chi =0$ is the trivial character) is equivalent to $\mathfrak Rep (G)$, so the groups $R^0(\check{G})$ and $R(G)$ are isomorphic. The tensor product functor

$$\begin{aligned} \mathfrak Rep ^{\chi _1}(\check{G})\times \mathfrak Rep ^{\chi _2}(\check{G})\longrightarrow \mathfrak Rep ^{\chi _1+\chi _2}(\check{G}) \end{aligned}$$

induces a bi-additive map

$$\begin{aligned} R^{\chi _1}(\check{G})\times R^{\chi _2}(\check{G})\longrightarrow R^{\chi _1+\chi _2}(\check{G}) \end{aligned}$$

In particular $R^\chi (\check{G})$ is an $R(G)$-module for all $\chi$.

Lemma 7.1

• (i) The $R(G)$-module $R(\check{G})$ is the direct sum of the submodules $R^\chi (\check{G})$ over all $\chi \in \mathcal{X }(C)$. Each summand $R^\chi (\check{G})$ is nonzero.

• (ii) Let $\check{T}\subset \check{G}$ be the inverse image of $T$. Then $\check{T}$ is a maximal torus of $\check{G}$; for each character $\chi$ of $C$ the submodule $R^\chi (\check{T})$ of $R(\check{T})$ is preserved by the $W_G$-action; and the restriction homomorphism $R^\chi (\check{G})\rightarrow R^\chi (\check{T})$ is an isomorphism onto $R^\chi (\check{T})^{W_G}$.

Proof

Every $\check{G}$-module $V$ decomposes under the action of $C$ into a direct sum $\bigoplus _{\chi \in \mathcal{X }(C)}V^\chi$ of isotypical submodules $V^\chi$. This decomposition is functorial and defines an equivalence of categories

$$\begin{aligned} \mathfrak Rep (\check{G})\overset{\sim }{\longrightarrow }\bigoplus _{\chi \in \mathcal{X }(C)}\mathfrak Rep ^\chi (\check{G}). \end{aligned}$$

Passing to Grothendieck groups we obtain the direct sum decomposition in (i). Each of the submodules $R^\chi (\check{G})$ is nonzero, because there exists an irreducible representation of $\check{G}$ with central character $\chi$, for instance an appropriate subrepresentation of the formally induced representation ${\text{ ind}}_C^{\check{G}}(\mathbf{C}_\chi )$. Since $C$ is central in $\check{G}$, the group $\check{T}$ is a maximal torus of $\check{G}$, and the homomorphism $\check{G}\rightarrow G$ induces isomorphisms of root systems $\mathcal R _{\check{G}}\cong \mathcal R _G$ and of Weyl groups $W_{\check{G}}\cong W_G$. The homomorphism $\check{T}\rightarrow T$ is $W_G$-equivariant. The $W_G$-action on $\check{T}$ fixes $C$ and therefore the submodule $R^\chi (\check{T})$ of $R(\check{T})$ is $W_G$-stable for each $\chi \in \mathcal{X }(C)$. The isomorphism $R^\chi (\check{G})\cong R^\chi (\check{T})^{W_G}$ now follows from $R(\check{G})\cong R(\check{T})^{W_G}$. $\square$

Let $V$ be an irreducible $\check{G}$-module. By Schur’s lemma, the central subgroup $C$ acts on $V$ by a character $\chi _V$. Define

$$\begin{aligned} c\cdot [V]=\chi _V(c)[V] \end{aligned}$$

(7.1)

for $c\in C$. By linear extension, this formula defines a $C$-action on the complexified representation ring $R(\check{G})_\mathbf{C}$ by ring automorphisms. The ring of $C$-invariants is

$$\begin{aligned} (R(\check{G})_\mathbf{C})^C\cong R(G)_\mathbf{C}. \end{aligned}$$

(7.2)

Recall that the rank of a module $A$ over a domain $R$ is the dimension of the vector space $F\otimes _RA$, where $F$ is the fraction field of $R$, and is denoted by ${\text{ rank}}_R(A)$.

Lemma 7.2

${\text{ rank}}_{R(G)}(R^\chi (\check{G}))=1$ for every $\chi \in \mathcal{X }(C)$.

Proof

Assume first that $C$ is finite. Since $G$ and $\check{G}$ are connected, the rings $R(G)$ and $R(\check{G})$ have no zero divisors, so we can form the fraction fields $\mathbf{K}$ of $R(G)_\mathbf{C}$ and $\check{\mathbf{K}}=R(\check{G})_\mathbf{C}\otimes _{R(G)_\mathbf{C}}\mathbf{K}$ of $R(\check{G})_\mathbf{C}$. It follows from (7.2) that $\check{\mathbf{K}}$ is a Galois extension of $\mathbf{K}$ with Galois group $C$, which implies $\dim _\mathbf{K}(\check{\mathbf{K}})=|C|$. For $\chi \in \mathcal{X }(C)$ let

$$\begin{aligned} \check{\mathbf{K}}^\chi =R^\chi (\check{G})_\mathbf{C}\otimes _{R(G)_\mathbf{C}}\mathbf{K}, \end{aligned}$$

which is a $\mathbf{K}$-linear subspace of $\check{\mathbf{K}}$. It follows from Lemma 7.1(i) that as a vector space over $\mathbf{K}$

$$\begin{aligned} \check{\mathbf{K}}=\bigoplus _{\chi \in \mathcal X (C)}\check{\mathbf{K}}^\chi , \end{aligned}$$

where each of the summands is nonzero. The number of summands is $|C|$ because of the fact that $\mathcal{X }(C)\cong C$, and therefore $\dim _\mathbf{K}(\check{\mathbf{K}}^\chi )=1$ for all $\chi$. Hence

$$\begin{aligned} {\text{ rank}}_{R(G)}\bigl (R^\chi (\check{G})\bigr ) =\dim _\mathbf{K}\bigl (\check{\mathbf{K}}^\chi \bigr )=1. \end{aligned}$$

For general $C$ we make the basic observation, which appears to go back to Shapiro [36], that the compact central extension $\check{G}$ is the pushout of a finite central extension

$$\begin{aligned} 1\longrightarrow Z\longrightarrow \tilde{G}\longrightarrow G\longrightarrow 1. \end{aligned}$$

One produces $\tilde{G}$ by choosing a Lie algebra homomorphism $\kappa :\mathfrak{g }\rightarrow \check{\mathfrak{g }}$ which splits the exact sequence

$$\begin{aligned} 0\rightarrow \mathfrak c \rightarrow \check{\mathfrak{g }}\rightarrow \mathfrak{g }\rightarrow 0. \end{aligned}$$

One can choose $\kappa$ to be defined over $\mathbf{Q}$; this ensures that it exponentiates to a Lie group homomorphism $\kappa :\tilde{G}\rightarrow \check{G}$, where $\tilde{G}$ is a finite connected covering group of $G$. Let $Z$ be the kernel of the covering $\tilde{G}\rightarrow G$; then $\kappa (Z)$ is contained in $C$. Let $\chi$ be a character of $C$ and $\kappa ^*(\chi )$ its pullback to $Z$. Then the $R(G)$-module $R^\chi (\check{G})$ is isomorphic to $R^{\kappa ^*(\chi )}(\tilde{G})$, and we already know that the latter is of rank $1$. $\square$

Example 7.3

Let $\check{G}_1=\check{G}_2=\mathbf{SU}(2)$ and

$$\begin{aligned} \check{G}=\check{G}_1\times \check{G}_2\cong \mathbf{Spin}(4),\qquad C=\{\pm (I,I)\},\qquad G=\check{G}/C\cong \mathbf{SO}(4). \end{aligned}$$

Then $R(\check{G})\cong R(\check{G}_1)\otimes R(\check{G}_2)\cong \mathbf{Z}[x_1,x_2]$. Identify $\mathcal{X }(C)$ with $\mathbf{Z}{/}2\mathbf{Z}=\{0,1\}$; then $C$ acts on the monomial $x_1^{r_1}x_2^{r_2}\in R(\check{G})$ with weight $(r_1+r_2)\text{ mod}\,2$. Hence $R(\check{G})$ is the direct sum of the submodules

$$\begin{aligned} R(G)\cong R^0(\check{G})=\bigoplus _{r_1+r_2\equiv 0}\mathbf{Z}\cdot x_1^{r_1}x_2^{r_2},\qquad R^1(\check{G})=\bigoplus _{r_1+r_2\equiv 1}\mathbf{Z}\cdot x_1^{r_1}x_2^{r_2}, \end{aligned}$$

where the congruences are modulo $2$. As a ring,

$$\begin{aligned} R(G)\cong \mathbf{Z}[y_1,y_2,y_3]{/}(y_1y_2-y_3^2), \end{aligned}$$

where the inclusion $R(G)\rightarrow R(\check{G})$ is given by

$$\begin{aligned} y_1\longmapsto x_1^2,\qquad y_2\longmapsto x_2^2,\qquad y_3\longmapsto x_1x_2. \end{aligned}$$

The twisted module $R^1(\check{G})$ is generated by $x_1$ and $x_2$, which are subject to the single relation $y_3x_1-y_1x_2=0$. Over the quotient field of $R(G)$ the two generators are multiples of each other, $x_2=(y_2y_3^{-1})x_1$ and $x_1=(y_1y_3^{-1})x_2$, so $R^1(\check{G})$ is of rank $1$, as predicted by Lemma  7.2. However, $R^1(\check{G})$ is not generated by any single element and is therefore not free.

Appendix: Shifted anti-invariants

This appendix is devoted to the proof of the statement below (Proposition 3.2.5 in the main text). We use the notation defined in Sect. 3. In particular, $G$ is a compact connected Lie group with maximal torus $T$, $\omega _{G{/}T}$ is the orientation system of the flag variety $G{/}T$ defined in (3.2), $J_G$ is the anti-symmetrizer (3.5), and $\mathbf{d}_G$ is the Weyl denominator (3.7). Recall that $A^{-W_G}$ denotes the set of anti-invariant elements of a $W_G$-module $A$. If $G$ is semisimple and simply connected, then $\rho _G\in \mathcal{X }(T)$ and therefore the $R(T)$-module $R(T,\omega _{G{/}T})$ is $W_G$-equivariantly isomorphic to $R(T)$ by Lemma  3.2.1(iii). The proposition is then a standard fact; see e.g. [8, § VI.3, Proposition 2]. We will deduce the general case from this special case.

Proposition.

• (i) The $W_G$-action on $R(T^{(\omega _{G{/}T})})$ preserves $R(T,\omega _{G{/}T})$.

• (ii) The set of anti-invariants $R(T,\omega _{G{/}T})^{-W_G}$ is a free $R(G)$-module of rank $1$ generated by $\mathbf{d}_G$.

• (iii) The elements $J_G(e^\lambda )$, with $\lambda \in \rho _G+\mathcal{X }(T)$ strictly dominant, form a basis of the $\mathbf{Z}$-module $R(T,\omega _{G{/}T})^{-W_G}$.

Proof

Put

$$\begin{aligned} W=W_G,\qquad J=J_G,\qquad \omega =\omega _{G{/}T},\qquad \mathbf{d}=\mathbf{d}_G,\qquad \rho =\rho _G. \end{aligned}$$

We identify $R(T)$ with $\mathbf{Z}[\mathcal{X }(T)]$ and $R(T,\omega )$ with $R(T)\,e^\rho =\mathbf{Z}[\rho +\mathcal{X }(T)]$ as in Remark 3.2.2. The fact that $\rho -w(\rho )$ is in the root lattice for all $w\in W$ implies that the $W$-action on $\mathcal{X }(T)_\mathbf{Q}$ preserves the affine lattice $\rho +\mathcal{X }(T)$. This proves (i). Let $\phi :\tilde{G}\rightarrow G$ be a compact connected covering group which is the product $\tilde{G}=C\times \bar{G}$ of a torus $C$ and a simply connected group $\bar{G}$. Let $\tilde{T}$ be the maximal torus $\phi ^{-1}(T)$ of $\tilde{G}$ and identify $\mathcal{X }(T)$ with its image $\phi ^*(\mathcal{X }(T))$. Let $\bar{T}$ be the maximal torus $\bar{G}\cap \tilde{T}$ of $\bar{G}$. Since $\bar{G}$ is simply connected, $\rho \in \mathcal{X }(\bar{T})$ and therefore $\mathbf{d}\in R(\bar{T})$. We have

$$\begin{aligned} R(\tilde{T})^W=R(C\times \bar{T})^W=\bigl (R(C)\otimes _\mathbf{Z}R(\bar{T})\bigr )^W=R(C)\otimes _\mathbf{Z}R(\bar{T})^W, \end{aligned}$$

because $W$ acts trivially on $R(C)$ and $R(C)$ is a free abelian group. Since $\bar{G}$ is simply connected, it follows from [8, § VI.3, Proposition 2] that

$$\begin{aligned} R(\bar{T})^{-W}=R(\bar{T})^W\cdot \mathbf{d}. \end{aligned}$$

Therefore

$$ R(\tilde{T})^{-W}=R(C\times \bar{T})^{-W}=\bigl (R(C)\otimes _\mathbf{Z}R(\bar{T})\bigr )^{-W}=R(C)\otimes _\mathbf{Z}R(\bar{T})^{-W}\\ =R(C)\otimes _\mathbf{Z}\bigl (R(\bar{T})^W\cdot \mathbf{d}\bigr )=\bigl (R(C)\otimes _\mathbf{Z}R(\bar{T})^W\bigr )\cdot \mathbf{d}=R(\tilde{T})^W\cdot \mathbf{d}. $$

(8.1)

Now let $a\in R(T,\omega )^{-W}\subset R(\tilde{T})^{-W}$. It follows from (8.1) that $a=b\mathbf{d}$ for some $b\in R(\tilde{T})^W$. We need to argue that $b\in R(T)$. Let $K\subset \tilde{T}$ be the kernel of the covering homomorphism $\phi :\tilde{G}\rightarrow G$. This group acts on the complexified representation ring $R(\tilde{T})_\mathbf{C}$ as in (7.1), and the ring of $K$-invariants $(R(\tilde{T})_\mathbf{C})^K$ is isomorphic to $R(T)_\mathbf{C}$. Since $a$ and $\mathbf{d}$ are in $R(T,\omega )=R(T)e^\rho$, we have $k\cdot a=\rho (k)a$ and $k\cdot \mathbf{d}=\rho (k)\mathbf{d}$ for all $k\in K$ and hence

$$\begin{aligned} b\mathbf{d}=a=\rho (k)^{-1}k\cdot a=\rho (k)^{-1}k\cdot (b\mathbf{d}) =\rho (k)^{-1}(k\cdot b)(k\cdot \mathbf{d})=(k\cdot b)\mathbf{d}. \end{aligned}$$

It follows that $k\cdot b=b$ for all $k\in K$, that is, $b\in R(T)$. This proves (ii). (iii) is proved in exactly the same way as [8, § VI.3, Proposition 1]. $\square$

Appendix: Homogeneous $\mathbf{Spin}^c$-structures

In this appendix we review the classification of invariant $\mathbf{Spin}^c$-structures on equal-rank homogeneous spaces, which is surely well known but for which we could not find a reference. (But see [16, §2.6] and also Example  9.5 below for remarks on the $\mathbf{Spin}$ case.) See [28, Example 4.6] and [25] for examples of maximal-rank homogeneous spaces that do not carry invariant $\mathbf{Spin}^c$-structures.

The notation and the assumptions are as explained at the beginning of Sects. 2 and 3. Recall that $G$ denotes a compact connected Lie group, $H$ a closed and connected subgroup of maximal rank and $T$ a common maximal torus of $G$ and $H$. Recall also that $\mathfrak{m }=T_{\bar{1}}M$ denotes the tangent space at the identity coset of the homogeneous space $M=G{/}H$, and $\eta :H\rightarrow \mathbf{SO}(\mathfrak{m })$ denotes the tangent representation. We denote the set of equivalence classes of $G$-invariant $\mathbf{Spin}^c$-structures on $M$ by $\mathfrak Spin _G^c(M)$.

Definition 9.1

The orthogonal representation $\eta$ is c-spinorial if it lifts to a homomorphism $H\rightarrow \mathbf{Spin}^c(\mathfrak{m })$. The subgroup $H$ is c-spinorial if $\eta$ is c-spinorial.

Note that $\eta$ is c-spinorial if and only if $M$ possesses a $G$-invariant $\mathbf{Spin}^c$-structure, that is, if and only if $\mathfrak Spin _G^c(M)$ is nonempty. Moreover, liftings of $\eta$ to $\mathbf{Spin}^c(\mathfrak{m })$ correspond bijectively to elements of $\mathfrak Spin _G^c(M)$.

A lifting of $\eta$ to $\mathbf{Spin}^c(\mathfrak{m })$ determines a trivialization $s$ of the central extension

defined in (3.2). Conversely, given a section $s$ of $\phi$ we can compose it with the canonical homomorphism $\hat{\eta }:H^{(\omega )}\rightarrow \mathbf{Spin}^c(\mathfrak{m })$ to obtain a lifting of $\eta$,

$$\begin{aligned} H\overset{s}{\longrightarrow }H^{(\omega )}\overset{\hat{\eta }}{\longrightarrow }\mathbf{Spin}^c(\mathfrak{m }). \end{aligned}$$

Thus, we have a natural one-to-one correspondence between $\mathfrak Spin _G^c(M)$ and the set of trivializations of the orientation system $\omega$.

The determinant character is the character $\det :H^{(\omega )}\rightarrow \mathbf{U}(1)$ obtained by pulling back the determinant character of $\mathbf{Spin}^c(\mathfrak{m })$, which is defined by $\det ([z,a])=z^2$. The homomorphism

$$\begin{aligned} \psi =\det \times \phi :H^{(\omega )}\longrightarrow \mathbf{U}(1)\times H \end{aligned}$$

is a double covering map. Let $s$ be a section of $\phi$, and let $\gamma$ be the character $\det \circ s$ of $H$. Then $s$ is a lifting homomorphism of $\gamma \times {\text{ id}}$, as in the commutative diagram

and so $s=s_\gamma$ is uniquely determined by $\gamma$.

Definition 9.2

A character $\gamma \in \mathcal{X }(H)$ is c-spinorial (relative to the orthogonal $H$-module $\mathfrak{m }$) if the lifting $s_\gamma$ of $\gamma \times {\text{ id}}$ exists. We denote the set of $c$-spinorial characters by $\mathcal{X }(H)^c$.

The conclusion is that we have natural bijections between the set $\mathfrak Spin _G^c(M)$, the set of trivializations of $\omega$ and the set $\mathcal{X }(H)^c$. This proves the first part of the following proposition. The map $\nu :\mathcal{X }(H)\rightarrow \frac{1}{2}\mathcal{X }(T)$ in the second part is defined by

$$\begin{aligned} \nu (\gamma ) =\frac{1}{2}j_H^*(\gamma )-\frac{1}{2}\sum _{\alpha \in \mathcal R _M^+}\alpha =\frac{1}{2}j_H^*(\gamma )-\rho _M. \end{aligned}$$

Proposition 9.3

• (i) For a $c$-spinorial character $\gamma$, define a $G$-invariant principal $\mathbf{Spin}^c(\mathfrak{m })$-bundle over $M$ by

  $$\begin{aligned} \mathbf{P}_\gamma =G\times ^H\mathbf{Spin}^c(\mathfrak{m }), \end{aligned}$$

  where the right-hand side denotes the quotient of $G\times \mathbf{Spin}^c(\mathfrak{m })$ by the $H$-action $h\cdot (g,k)=(gh^{-1},\hat{\eta }(s_\gamma (h))k)$. The map

  $$\begin{aligned} f:\mathcal{X }(H)^c\longrightarrow \mathfrak Spin _G^c(M) \end{aligned}$$

  which sends $\gamma$ to the equivalence class of $\mathbf{P}_\gamma$ is bijective. In particular $\mathfrak Spin _G^c(M)$ is nonempty if and only if $\mathcal{X }(H)^c$ is nonempty.

• (ii) A character $\gamma$ of $H$ is $c$-spinorial if and only if $\nu (\gamma )\in \mathcal{X }(T)$. Hence $\mathcal{X }(H)^c=\nu ^{-1}(\mathcal{X }(T))$.

Proof

It remains to prove (ii). By covering space theory and our standing assumption that $H$ is connected, a lifting $s$ of $\gamma \times {\text{ id}}$ exists if and only if it exists on the level of fundamental groups, as in the diagram

(9.1)

Let $\mathcal{Y }(T)={\text{ Hom}}(\mathbf{U}(1),T)$ denote the cocharacter group of $T$. The fundamental group $\pi _1(H)$ is naturally isomorphic to $\mathcal{Y }(T){/}Q(\mathcal R _H^\vee )$, where $Q(\mathcal R _H^\vee )$ is the coroot lattice in $\mathcal{Y }(T)$. (See e.g. [8, § IX.4.6].) Since $\psi$ is a covering map, $\psi _*$ maps the coroot system of $H^{(\omega )}$ bijectively to that of $\mathbf{U}(1)\times H$. Therefore, the lifting problem (9.1) is equivalent to the lifting problem

Since $\mathcal{X }(T)$ and $\mathcal{Y }(T)$ are dual abelian groups, this lifting problem is equivalent to the dual extension problem

(9.2)

By Lemma 3.2.1(ii), $\mathcal{X }(T^{(\omega )})$ is the direct sum of $\phi ^*(\mathcal{X }(T))$ and $\mathbf{Z}\cdot (\varepsilon _0-\rho _M)$. Let $\mathcal{X }$ be the image of $\psi ^*$, which is a sublattice of $\mathcal{X }(T^{(\omega )})$ of index $2$. Since $2\rho _M$ is in $\mathcal{X }(T)$ and $\psi ^*$ maps the generator $\varepsilon _0$ of $\mathcal{X }(\mathbf{U}(1))$ to the determinant character $\det =2\varepsilon _0$, the element $\varepsilon _0-\rho _M\in \mathcal{X }(T^{(\omega )})$ is a representative of the nontrivial coset in $\mathcal{X }(T^{(\omega )}){/}\mathcal{X }$. Because $2(\varepsilon _0-\rho _M)$ is equal to $\psi ^*(\varepsilon _0-2\rho _M)$, the extension problem (9.2) is soluble if and only if

$$\begin{aligned} 2\nu (\gamma )=j_H^*(\gamma )-2\rho _M=\gamma ^*(\varepsilon _0)-2\rho _M\in \mathcal{X }(T) \end{aligned}$$

is divisible by $2$ in $\mathcal{X }(T)$. If this is the case, the extension $s^*=s_\gamma ^*$ is uniquely determined by the formula

$$\begin{aligned} s_\gamma ^*(\varepsilon _0-\rho _M)=\nu (\gamma )\in \mathcal{X }(T). \end{aligned}$$

(9.3)

This proves (ii). $\square$

We call $\gamma \in \mathcal{X }(H)^c$ the character of the $\mathbf{Spin}^c$-structure $\mathbf{P}_\gamma$. Let $\gamma \in \mathcal{X }(H)^c$ and $\chi \in \mathcal{X }(H)$. Then

$$\begin{aligned} \nu (\gamma +2\chi )=\nu (\gamma )+j_H^*(\chi )\in \mathcal{X }(T), \end{aligned}$$

so $\gamma +2\chi \in \mathcal{X }(H)^c$. Thus we have an action of the abelian group $\mathcal{X }(H)$ on the set of $c$-spinorial characters defined by $\chi \cdot \gamma =\gamma +2\chi$. Let $\mathbf{L}_\chi =G\times ^H\mathbf{C}_\chi$ be the homogeneous complex line bundle on $M$ defined by $\chi \in \mathcal{X }(H)$. Recall the natural isomorphisms

$$\begin{aligned} \mathcal{X }(H)\overset{\cong }{\longrightarrow }{\text{ Pic}}_G(M)\overset{\cong }{\longrightarrow }H_G^2(M,\mathbf{Z}), \end{aligned}$$

(9.4)

where ${\text{ Pic}}_G(M)$ denotes the (topological) Picard group of isomorphism classes of homogeneous complex line bundles on $M$. The first map sends a character $\chi$ to the class of the bundle $\mathbf{L}_\chi$, and the second map sends the class of a bundle $L$ to its equivariant Chern class $c_G^1(L)$. (See e.g. [14, Theorem C.47].) A natural action of ${\text{ Pic}}_G(M)$ on $\mathfrak Spin _G^c(M)$ is defined by

$$\begin{aligned}{}[L]\cdot [\mathbf{P}]=\bigl [\mathbf{U}(L)\times ^{\mathbf{U}(1)_M}\mathbf{P}\bigr ], \end{aligned}$$

where $\mathbf{U}(L)$ denotes the circle bundle associated with $L$ and we identify the circle $\mathbf{U}(1)$ with the kernel of $\psi :\mathbf{Spin}^c(\mathfrak{m })\rightarrow \mathbf{SO}(\mathfrak{m })$. (The quotient in the right-hand side is taken in the category of manifolds over $M$, and $\mathbf{U}(1)_M=M\times \mathbf{U}(1)$ denotes the constant groupoid over $M$ with fibre $\mathbf{U}(1)$.)

Proposition 9.4

The notation is as in Proposition 9.3.

• (i) The bijection $f:\mathcal{X }(H)^c\rightarrow \mathfrak Spin _G^c(M)$ is $\mathcal{X }(H)$-equivariant with respect to the group isomorphism (9.4).

• (ii) $\mathfrak Spin _G^c(M)$ is a principal homogeneous space for the abelian group $\mathcal{X }(H)$. In particular $\mathfrak Spin _G^c(M)$ consists of at most one element if $H$ is semisimple.

• (iii) Let $\gamma \in \mathcal{X }(H)^c$. The determinant $\text{ line}$ bundle of $\mathbf{P}_\gamma$ is $\mathbf{L}_\gamma =G\times ^H\mathbf{C}_\gamma$. Its equivariant Chern class $c_1(\mathbf{L}_\gamma )\in H_G^2(M,\mathbf{Z})\cong \mathcal{X }(H)$ is equal to $\gamma$.

• (iv) Let $\gamma \in \mathcal{X }(H)^c$ and let be the $\mathbf{Spin}^c$ Dirac operator associated with $\mathbf{P}_\gamma$. The equivariant Euler class of is , where is the twisted $\mathbf{Spin}^c$ Dirac operator of $M$.

Proof

Let $\gamma \in \mathcal{X }(H)^c$ and $\chi \in \mathcal{X }(H)$. The fibre of $\mathbf{U}(\mathbf{L}_\chi )\times ^{\mathbf{U}(1)_M}\mathbf{P}_\gamma$ over the identity coset $\bar{1}\in M$ is

$$\begin{aligned} \mathbf{U}(\mathbf{C}_\chi )\times ^{\mathbf{U}(1)}\mathbf{Spin}^c(\mathfrak{m }), \end{aligned}$$

where $\mathbf{U}(\mathbf{C}_\chi )$ denotes the unit circle in $\mathbf{C}_\chi$. The map

$$\begin{aligned} \mathbf{U}(\mathbf{C}_\chi )\times ^{\mathbf{U}(1)}\mathbf{Spin}^c(\mathfrak{m })\longrightarrow \mathbf{Spin}^c(\mathfrak{m }) \end{aligned}$$

defined by $(z,k)\mapsto zk$ is a diffeomorphism and is equivariant with respect to the $H$-actions defined on the left-hand side by

$$\begin{aligned} h\cdot [z,k]=[\chi (h)^{-1}z,\hat{\eta }(s_\gamma (h))k] \end{aligned}$$

and on the right-hand side by $h\cdot k=\hat{\eta }(s_{\gamma +2\chi }(h))k$. It follows that $[L_\chi ]\cdot [\mathbf{P}_\gamma ]=[\mathbf{P}_{\gamma +2\chi }]$, which proves (i). It is easy to verify that the $\mathcal{X }(H)$-action on $\mathcal{X }(H)^c$ is free and transitive. Thus (ii) follows from (i). The determinant line bundle of $\mathbf{P}_\gamma$ is

$$\begin{aligned} \mathbf{L}_\gamma =\mathbf{P}_\gamma \times ^{\mathbf{Spin}^c(\mathfrak{m })}\mathbf{C}_{\det } =\bigl (G\times ^H\mathbf{Spin}^c(\mathfrak{m })\bigr )^{\mathbf{Spin}^c(\mathfrak{m })}\mathbf{C}_{\det } =G\times ^H\mathbf{C}_\gamma . \end{aligned}$$

The isomorphism $\mathcal{X }(H)\rightarrow H_G^2(M,\mathbf{Z})$ defined in (9.4) maps $\gamma$ to $c_G^1(\mathbf{L}_\gamma )$, which proves (iii). By (2.10) and (3.4), the Euler class of is

where $S$ is the spinor module of $\mathbf{Cl}(\mathfrak{m })$. Using Lemma 3.2.1(iv) and (9.3) we obtain

which establishes (iv). $\square$

Example 9.5

($\mathbf{Spin}$-structures) Let us call the orthogonal representation $\eta :H\rightarrow \mathbf{SO}(\mathfrak{m })$, or the subgroup $H$, spinorial if $\eta$ lifts to a homomorphism $\tilde{\eta }:H\rightarrow \mathbf{Spin}(\mathfrak{m })$. Thus, $H$ is spinorial if and only if $M$ has a $G$-invariant $\mathbf{Spin}$-structure, and such a structure is unique up to equivalence, because $H$ is connected. A $\mathbf{Spin}$-structure determines a $\mathbf{Spin}^c$-structure by means of the homomorphism $\kappa \circ \tilde{\eta }:H\rightarrow \mathbf{Spin}^c(\mathfrak{m })$, where $\kappa :\mathbf{Spin}(\mathfrak{m })\rightarrow \mathbf{Spin}^c(\mathfrak{m })$ is the inclusion map. The corresponding $c$-spinorial character of $H$ is $\gamma =\det \circ \kappa \circ \tilde{\eta }=0$, because $\mathbf{Spin}(\mathfrak{m })$ is the kernel of the determinant character. Hence, by Proposition  9.3(ii), $M$ is $G$-invariantly $\mathbf{Spin}$ if and only if $\rho _M\in \mathcal{X }(T)$. The Euler class of the $\mathbf{Spin}$ Dirac operator is equal to that of the twisted $\mathbf{Spin}^c$ Dirac operator .

Remark 9.6

If $H$ is c-spinorial and semisimple, then a lifting $H\rightarrow \mathbf{Spin}^c(\mathfrak{m })$ takes values in the commutator subgroup $\mathbf{Spin}(\mathfrak{m })$, so $H$ is spinorial.

Example 9.7

(almost complex structures) Up to homotopy, $G$-invariant almost complex structures on $M$ correspond bijectively to $H$-invariant orthogonal complex structures on $\mathfrak{m }$. Let $J$ be such a structure; then the tangent representation $\eta :H\rightarrow \mathbf{SO}(\mathfrak{m })$ takes values in $\mathbf{U}(\mathfrak{m },J)$, the group of $J$-holomorphic orthogonal maps. We equip $M$ with the $\mathbf{Spin}^c$-structure defined by composing this homomorphism with the canonical injection $\iota$ (see e.g. [14, § D.3.1]),

$$\begin{aligned} H\overset{\eta }{\longrightarrow }\mathbf{U}(\mathfrak{m },J)\overset{\iota }{\hookrightarrow }\mathbf{Spin}^c(\mathfrak{m }). \end{aligned}$$

The associated $c$-spinorial character $\gamma$ is then $\gamma =\det \circ \iota \circ \eta$. Since $\det \circ \iota$ is equal to the usual complex determinant character of $\mathbf{U}(\mathfrak{m },J)$, we have $j_H^*(\gamma )=\sum _{k=1}^l\beta _k$, where the $\beta _k$ are the weights of the $T$-action on $\mathfrak{m }$ with respect to $J$. To compute this, let $\mathcal R _M^+=\{\alpha _1,\alpha _2, \ldots , \alpha _l\}$ and let $J_0$ be the $T$-invariant complex structure on $\mathfrak{m }$ given by the decomposition $\mathfrak{m }=\bigoplus _{k=1}^l\mathfrak{m }_k$, where $\mathfrak{m }_k\cong \mathfrak{g }_\mathbf{C}^{\alpha _k}$. On $\mathfrak{m }_k$ we have $J=c_kJ_0$, where $c_k=\pm 1$. Hence $\beta _k=c_k\alpha _k$ and $j_H^*(\gamma )=\sum _{k=1}^lc_k\alpha _k$.

Example 9.8

(complex structures) As a special case of Example 9.7, let us take $H$ to be the centralizer of a subtorus of $T$. This is the case precisely when $M$ has an integrable invariant almost complex structure, as one sees from the well-known identification $M\cong G_\mathbf{C}{/}P$ ([8, § IX.4, Exercise 8]), where $P$ is the parabolic subgroup of $G_\mathbf{C}$ generated by $H$ and the negative root spaces. The corresponding orthogonal almost complex structure $J$ on $\mathfrak{m }$ is positive with respect to the inner product on $\mathfrak{m }$, so $c_k=1$ for all $k$ and hence

The Kähler structure on $M$ and the associated $\mathbf{Spin}^c$-structure depend on the choice of the basis of $\mathcal R _G$.