Natural Equivariant Dirac Operators

NA TURAL EQUIV ARIANT DIRAC OPERA TORS IGOR PROKHORENKO V AND KEN RICHARDSON Abstract. W e int ro duce a new cla ss of natural, explicitly defined, transversally elliptic dif- ferential op erators o ver manifolds with co mpact gro up actions. Under certain assumptions, the symbols of these op erators ge nerate all the po ssible v alues of the equiv ariant index. W e also show that the comp onen ts of the representation-v alued e quiv a rian t index coincide with those o f an elliptic op erator constructed from the or iginal da ta. Contents 1. In tro duction 1 2. Restrictions of Clifford structures 2 3. T ransv erse Dirac Op erators for Distributions 4 4. Equiv a r ian t op erators o n the fra me bundle 8 4.1. Equiv arian t structure of the orthonormal frame bundle 8 4.2. Dirac-type op erators on the frame bundle 10 5. T op ological prop erties o f the lifted Dirac o perators 11 5.1. Equiv arian t Multiplicativ e Prop erties of K -theory 11 5.2. Index o f Lifted Dira c op erators 13 6. Example 13 6.1. A t r a ns v ersal Dira c op erator on the sphere 13 6.2. Calculation of ke r D σ n S 2 16 6.3. The op erator o n F O  G 17 References 19 1. Intr oduction The represen tation-v alued equiv ariant index of a transv ersally elliptic op erator is an im- p ortan t in v ariant in K -theory (see [1]). There are few kno wn nontrivial examples in the literature where this in v arian t is explicitly computed. P art of the motiv a tion of this pap er is to provide an interes ting a nd sufficien tly general class of examples of transv ersally elliptic differen tial op erators for whic h suc h computations are p ossible. It is we ll-know n that eac h compactly supp orted K -theory class o f the cotangent bundle o v er an ev en-dimensional spin c manifold is represen ted b y the sym b ol of a Dirac-type o p- erator. This implies tha t Dira c- t yp e sym b ols map on to the imag e of the K -theory index homomorphism. In this pap er, w e will generalize the second fact to the case of transv er- sally elliptic op erators o v er a compact manifold endo w ed with a compact Lie group actio n. Date : May , 200 8. 2000 Mathematics Subje ct Classific ation. 58J20 ; 58J70; 19 L 4 7. Key wor ds and phr ases. Equiv ariant index, g roup action, Dirac o perator . 1 2 IGOR PROKHORENKO V AND KEN RICHARD SON The role of the D irac-t yp e op erators will b e play ed b y a new class of transve rsally elliptic differen tial op erators intro duc ed in this pap er. T o construct these op erators, we lift the group action to a principal bundle so that all orbits in the principal bundle ha v e the same dimension. There is a na t ural tra nsve rsal Dirac op erator asso ciated to this action. This op erator induces a transv ersally elliptic differen tial op erator on the base manifold with the desired prop erties. In the case when a ll orbits hav e the same dimension, the or bits on the base manifold fo rm a R iem annian fo liation, and our construction pro duces a transv ersal Dirac op erator a s studied by [11], [12], [8], [10], [13], [14], a nd others. This new op erator will generate all p ossible v alues of the represen tation-v alued equiv ariant index. F urther, w e sho w that the decomp osition of this equiv ar ia n t index represen tatio n in to irreducible com- p onen ts ma y b e computed b y means of equiv a rian t indices of elliptic op erators. Th us, the tec hniques of A tiy ah and Segal [2] for elliptic op erators b ecome applicable to transv ersally elliptic op erators as well. No w we describ e the con ten t of the pap er. Let M b e a closed Riemannian manifold. Let Q ⊂ T M b e a smo oth distribution ov er M ; we do no t assume that Q or its normal bundle a re in v olutiv e. Section 2 con tains preliminary results a bout connections asso ciated to restrictions of Clifford structures. In Section 3, w e assume only that E → M is a C l ( Q )-bundle with corresp onding compatible Clifford connection. Such a connection alwa ys exists if E is in addition a C l ( T M )-bundle; see Section 2. Using this C l ( Q ) connection, we construct an op erator D Q whose principal sym b ol σ ( D Q ) ( ξ x ) is inv ertible f or all ξ x ∈ Q x \ { 0 } and pro v e that it is essen tially self-adjoin t. In the case o f a Riemannian fo liation with normal bundle Q , this construction pro duces the we ll-know n self-adj o in t v ersion of t he transv ersal Dirac op erator (see [11], [1 3 ], etc.). In Section 4, w e assume that there is an isometric action of a compact Lie group G on M . The action of G lifts to t he orthonormal f r a me bundle F O of M . Giv en an equiv ariant transv ersal Dirac op erator on F O and irreducible represen tation of the or t ho gonal group, we sho w how to construct a transv ersally elliptic op erator on M . Similarly , using the tra ns v ersal Dirac op erator a nd a n irreducible represen tation of G , w e construct an elliptic op erator on F O  G . The precise relationship b et w een the eigenspaces of these tw o op erators is stated in Prop osition 4.4. In Section 5, w e study the equiv ariant index of the K - theory class of op erators constructed in Section 4. T o do this, w e first deriv e a m ultiplicativ e prop ert y o f the equiv ariant index on a ss o ciated fib er bundles with compact fib ers. This prop ert y , stated in Theorem 5.1, is a generalization of the m ultiplicativ e pro perty of the index for sphere bundles shown by Atiy ah and Singer in [3]. The main result of Section 5 is Theorem 5.2, in whic h we sho w that the sym b ols of the lifted tra ns v ersal Dirac op erators generate all the p ossible equiv ar ia n t indices, if the F O is G -t r a ns v ersally spin c . In Section 6, w e demonstrate our constructions o f the lifted t ransv ersal Dira c op erator and verify our results by explicit calculations on the tw o- sphere. The reader ma y consult [1] fo r the ba s ic prop erties o f tra nsve rsally elliptic equiv ar ian t op erators and their equiv aria nt indices. In teresting and relev ant results also app ear in [2], [4], [5], [6], [7], [8], and [1 7 ]. 2. Restrictions of Cliff ord structures Let M b e a closed R iemannian ma nif o ld with metric h · , · i , and let E b e a Clifford bundle o v er M . Recall that a Clifford bundle E is a complex Hermitian v ector bundle endo w ed with NA TURAL EQU IV ARIANT DIRAC OPERA TORS 3 a Clifford action c : T M ⊗ C → End ( E ) and a connection ∇ E compatible with this action and the metric. Let Q be a subbundle of T M , and let L b e the orthogonal complemen t of Q in T M . The Levi-Civita connection ∇ M induces a connection ∇ Q on Q b y the following form ula. Giv en a n y section Y ∈ Γ Q and any v ector field X ∈ Γ ( T M ), define ∇ Q X Y = π ∇ M X Y , (2.1) where π : T M → Q is the orthogonal bundle pro jection. It is elemen tary to c hec k that F orm ula (2 .1) yields a metric connection on Q (with the restricted metric). Remark 2.1. Note that this c o n ne ction i s not gen er al ly Q -torsion-fr e e, b e c a use the torsion - fr e e pr op erty is e quivalent to the inte gr ability of Q (i.e. [Γ Q, Γ Q ] ⊂ Γ Q ). W e now modify the connection ∇ E so that it ha s the desired compatibilit y with ∇ Q . Eve ry metric connection f ∇ E on E satisfies g ∇ E X = ∇ E X + B X , where B X is a sk ew-Hermitian endomorphism of E that is C ∞ ( M )-linear in X . In order that f ∇ E is a C l ( Q )-connection compatible with ∇ Q , we m ust ha v e that if X ∈ Γ ( T M ) , Y ∈ Γ Q , s ∈ Γ E , g ∇ E X c ( Y ) s = c  ∇ Q X Y  s + c ( Y ) g ∇ E X s . By (2.1 ) w e see that B X c ( Y ) s = − c  (1 − π ) ∇ M X Y  s + c ( Y ) B X s, or c  (1 − π ) ∇ M X Y  s = [ c ( Y ) , B X ] s. (2.2) Computing with a lo cal orthonor ma l frame { e 1 , ..., e p } for L , w e hav e (1 − π ) ∇ M X Y = p X m =1  ∇ M X Y , e m  e m = − p X m =1  Y , ∇ M X e m  e m = − p X m =1  Y , π ∇ M X e m  e m = 1 2 p X m =1  c ( Y ) c  π ∇ M X e m  + c  π ∇ M X e m  c ( Y )  e m . Then (2 .2 ) implies that 1 2 p X m =1  c ( Y ) c  π ∇ M X e m  c ( e m ) − c  π ∇ M X e m  c ( e m ) c ( Y )  = ( c ( Y ) B X − B X c ( Y )) , or B X − 1 2 p X m =1 c  π ∇ M X e m  c ( e m ) ! c ( Y ) = c ( Y ) B X − 1 2 p X m =1 c  π ∇ M X e m  c ( e m ) ! . W e conclude that the requiremen t that f ∇ E is a C l ( Q )-connection defines B X (and thus g ∇ E X ) up to a sk ew-adjoin t endomorphism of E that comm utes with Cliffor d multiplication by 4 IGOR PROKHORENKO V AND KEN RICHARD SON v ectors in Q . W e ma y a lw ay s take B X = 1 2 p X m =1 c  π ∇ M X e m  c ( e m ) . This c hoice of B X (and th us g ∇ E X ) is well-define d and canonical, since the form ula is inde- p enden t o f the lo cal orthonormal frame for L . W e no w sho w how t o express B X in terms of any lo cal orthonormal fr a me f 1 , ..., f q for Q . B X = 1 2 p X m =1 c  π ∇ M X e m  c ( e m ) = 1 2 p X m =1 q X j =1  ∇ M X e m , f j  c ( f j ) c ( e m ) = − 1 2 p X m =1 q X j =1  e m , ∇ M X f j  c ( f j ) c ( e m ) = 1 4 p X m =1 q X j =1  c ( e m ) c  ∇ M X f j  + c  ∇ M X f j  c ( e m )  c ( f j ) c ( e m ) = 1 4 q X j =1 p X m =1  − c ( e m ) c  ∇ M X f j  c ( e m ) c ( f j ) + c  ∇ M X f j  c ( f j )  = 1 4 q X j =1  − pc  π ∇ M X f j  c ( f j ) − ( p − 2) c  (1 − π ) ∇ M X f j  c ( f j ) + pc  ∇ M X f j  c ( f j )  = 1 2 q X j =1 c  (1 − π ) ∇ M X f j  c ( f j ) . Observ e that this expression for B X is t he same as the original expression for B X with Q replaced by L . W e hav e show n the following. Prop osition 2.2. L et M b e a close d Riemannian manifold, a nd let  E , ∇ E , c  b e a Her- mitian Cliffo r d bund le over M . L et Q b e a subbund le o f T M , an d let ∇ Q denote the m etric c onne ction on Q define d b y ∇ Q X Y = π ∇ M X Y , wher e π : T M → Q is the ortho gonal bund le pr oje ction. Then the c onne ction f ∇ E define d by g ∇ E X = ∇ E X + 1 2 p X m =1 c  π ∇ M X e m  c ( e m ) for al l X ∈ Γ ( T M ) is a wel l-define d C l ( Q ) -c onne ction and a metric c onne c t ion on E with r esp e ct to the c onne c tion ∇ Q . F urthermor e, f ∇ E is a C l ( L ) -c o nne c tion and a metric c onne c- tion on E with r esp e ct to the c onne ction ∇ L = (1 − π ) ∇ M . 3. Transvers e Dira c Ope ra tors f or Distributions W e sho w ed in Section 2 tha t, f o r a giv en distribution Q ⊂ T M , it is alw ay s p ossible to o bt a in a bundle of C l ( Q )-mo dules with Clifford connection from a bundle of C l ( T M )- Clifford mo dules. In this section, we will assume mor e generally that a C l ( Q )-mo dule structure on a complex Hermitian v ector bundle E is g iv en and will define transv erse D irac op erators on sections of E . As in Section 2, M is a closed Riemannian manifold with metric NA TURAL EQU IV ARIANT DIRAC OPERA TORS 5 h · , · i , c : Q → End ( E ) is the Clifford m ultiplication on E , and ∇ E is a C l ( Q ) connection that is compatible with t he metric on M ; that is, Clifford m ultiplication by eac h v ector is sk ew-Hermitian, and w e require ∇ E X ( c ( V ) s ) = c  ∇ Q X V  s + c ( V ) ∇ E X s for all X ∈ Γ ( T M ), V ∈ Γ Q , and s ∈ Γ E . Note that the connection f ∇ E from Section 2 is an example of suc h a connection, but not all suc h C l ( Q ) connections are o f that t yp e. Let L = Q ⊥ , let ( f 1 , ..., f q ) b e a lo cal orthonormal frame for Q , and let π : T M → Q b e the orthogonal pro jection. W e define the Dirac op erator A Q corresp onding to the distribution Q as A Q = q X j =1 c ( f j ) ∇ E f j . (3.1) This definition is indep enden t of the c hoices made; in fact it is the comp osition of the maps Γ ( E ) ∇ E → Γ ( T ∗ M ⊗ E ) ∼ = → Γ ( T M ⊗ E ) π → Γ ( Q ⊗ E ) c → Γ ( E ) . W e calculate the for ma l adjoint. Letting ( s 1 , s 2 ) denote the p oint wise inner pro duct of sections of E , w e hav e that ( A Q s 1 , s 2 ) − ( s 1 , A Q s 2 ) = q X j =1  c ( f j ) ∇ E f j s 1 , s 2  −  s 1 , c ( f j ) ∇ E f j s 2  = q X j =1  ∇ E f j ( c ( f j ) s 1 ) , s 2  −  c  π ∇ M f j f j  s 1 , s 2  +  c ( f j ) s 1 , ∇ E f j s 2  = q X j =1 ∇ M f j ( c ( f j ) s 1 , s 2 ) − c q X j =1 π ∇ M f j f j ! s 1 , s 2 ! = − q X j =1 ∇ M f j i f j ω + ω q X j =1 π ∇ M f j f j ! , where ω is the one-for m defined by ω ( X ) = − ( c ( X ) s 1 , s 2 ) for X ∈ Γ Q and is zero for X ∈ Γ L . Con tin uing, ( A Q s 1 , s 2 ) − ( s 1 , A Q s 2 ) = − q X j =1  i f j ∇ M f j + i ∇ M f j f j  ω + ω q X j =1 π ∇ M f j f j ! = − q X j =1  i f j ∇ M f j + i “ π ∇ M f j f j ”  ω + ω q X j =1 π ∇ M f j f j ! = − q X j =1 i f j ∇ M f j ω = − q X j =1 i f j π ∇ M f j ω , where the orthogo nal pro jection T ∗ M → Q ∗ is denoted by π as w ell. In what follow s, let ( e 1 , ..., e p ) b e an ortho no rmal frame of L , and let ∇ M = ∇ Q + ∇ L = π ∇ M + (1 − π ) ∇ M on 6 IGOR PROKHORENKO V AND KEN RICHARD SON forms. The div ergence of a general one-form β that is zero on L is δ β = − q X j =1 i f j ∇ M f j β − p X m =1 i e m ∇ M e m β = − q X j =1 i f j π ∇ M f j β − p X m =1 i e m ∇ L e m β , Letting β = P q k =1 β k f ∗ k , then δ β + q X j =1 i f j π ∇ M f j β = − p X m =1 i e m ∇ L e m q X k =1 β k f ∗ k ! = − q X k =1 p X m =1 β k i e m ∇ L e m ( f ∗ k ) = − q X k =1 p X m =1 β k i e m p X j =1  ∇ M e m ( f ∗ k ) , e ∗ j  e ∗ j ! = q X k =1 p X m =1 β k i e m p X j =1  ∇ M e m  e ∗ j  , f ∗ k  e ∗ j ! = q X k =1 β k p X m =1  ∇ M e m ( e ∗ m ) , f ∗ k  ! = i H L β , where H L is the mean curv at ure vec tor field of L . Th us, fo r ev ery one-form β that is zero on L , − q X j =1 i f j π ∇ M f j β = δ β − i H L β . Applying this result to the form ω defined ab o v e, we hav e ( A Q s 1 , s 2 ) − ( s 1 , A Q s 2 ) = δ ω − i H L ω = δ ω −  s 1 , c  H L  s 2  . Th us, t he formal adjoint A ∗ Q of A Q is A ∗ Q = A Q − c  H L  , and the op erator D Q = A Q − 1 2 c  H L  (3.2) is forma lly self-adjoint. A quic k lo ok at [9] yields the following. Theorem 3.1. F or e ach distribution Q ⊂ T M and every bund le E of C l ( Q ) -mo dules, the tr ansversal ly el liptic op er ator D Q define d by (3.1) and (3.2) is essential ly self-adjoint. It is not necessarily t he case that the sp ectrum of D Q is discrete, as the followin g example sho ws. NA TURAL EQU IV ARIANT DIRAC OPERA TORS 7 Example 3.2. We c onsider the torus M = ( R 2 π Z ) 2 with the metric e 2 g ( y ) dx 2 + dy 2 for some 2 π -p erio dic smo oth function g . Consider the ortho gonal distributions L = span { ∂ y } and Q = span { ∂ x } . L et E b e the trivial c om plex line bund le ove r M , and let C l ( Q ) and C l ( L ) b oth act on E via c ( ∂ y ) = i = c  e − g ( y ) ∂ x  . The c onne c t ions ∇ L and ∇ Q satisfy ∇ L ∂ y ∂ y = ∇ L ∂ x ∂ y = 0; ∇ Q ∂ x ∂ x = 0; ∇ Q ∂ y ∂ x = g ′ ( y ) ∂ x . The trivial c onne ction ∇ E is a C l ( L ) c on ne ction with r esp e ct to ∇ L and is also a C l ( Q ) c onne ction with r esp e ct to ∇ Q . O bserve that the me an curvatur e s of these distributions ar e H Q = (1 − π ) ∇ M e − g ( y ) ∂ x e − g ( y ) ∂ x = − g ′ ( y ) ∂ y and H L = π ∇ M ∂ y ∂ y = ∇ M ∂ y ∂ y = 0 F r om formulas (3.1) and (3 . 2 ), A L = i∂ y , and D L = i  ∂ y + 1 2 g ′ ( y )  . The sp e ctrum σ ( D L ) = Z is a s e t c onsisting of eigenvalues of infinite multiplicity, and thus σ ( D L ) c onsists entir ely of pur e p oint sp e ctrum. The eigensp ac e E n c orr esp onding to the eigenvalue n is E n = n e − iny − g ( y ) 2 f ( x ) : f ∈ L 2  S 1  o , and S n ∈ Z E n is d e nse in L 2 ( M ) . On the other hand , the op er ator D Q = A Q − 1 2 c  H L  = A Q = ie − g ( y ) ∂ x has only one eig envalue, 0 , c o rr esp onding to the eigen sp ac e { h ( y ) : h ∈ L 2 ( S 1 ) } . Next, note that F n = { e − inx ψ ( y ) : ψ ∈ L 2 ( S 1 ) } i s an invariant subsp ac e for D Q . The sp e ctrum of the r estriction of D Q to F n is n [ a, b ] , wher e [ a, b ] ⊂ (0 , ∞ ) is the r ange of e − g ( y ) . Thus, the sp e ctrum σ ( D Q ) is σ ( D Q ) = [ n ∈ Z n [ a, b ] , and the pur e p oint sp e ctrum of D Q is { 0 } . Example 3.3. Supp ose a c lose d manifold M is endowe d with a Riemannian folia tion F such that the metric is bund lelike, me aning that the le aves ar e lo c al ly e quidistant. I f the orbits of a G -m a nifold h ave the same dimension, then they form a R iemannian foliation. I n such foliations, ther e is a n a tu r al c onstruction of tr a nsversal D ir ac op er ators (se e [8] , [11] , [15] ), which is a sp e cial c ase of the c onstruction in this se ction. Cho ose a lo c al adapte d fr a m e field { e 1 , ..., e n } for the tangent bund le of M , such that { e 1 , ..., e q } is a lo c al b asis of the normal bund le N F for the foliation and such that e ach e j is a b asi c ve ctor field for 1 ≤ j ≤ q . The wor d basic me ans that the flows of those ve ctor fields m a p le aves to le aves, and such a b asis c an b e chos e n ne ar every p oint if and o n ly if the folia tion is R iemannian . Next, assume that we have a c omplex Hermitian ve ctor bund le E → M that is a bund le of C l ( N F ) mo dules 8 IGOR PROKHORENKO V AND KEN RICHARD SON that is e quivariant with r esp e ct to the G action, and let ∇ b e the c orr esp ondin g e quivariant, metric, Cliffor d c onne ction. We define the tr ansversal D ir ac op er a t or by A N F = q X j =1 c ( e j ) ∇ e j , as i n the notation of this se ction. As b efor e, the op er ator D N F = A N F − 1 2 c ( H ) is an e ssential ly self-a d joint op er ator, wher e H is the me an curvatur e v e ctor field of the orbits. 4. Equiv ariant opera t ors on the frame bundle 4.1. Equiv arian t structure of the orthonormal frame bundle. Giv en a complete, connected G - manif old, the action of g ∈ G on M induces an a ction of dg on T M , whic h in turn induces an action of G on the principal O ( n )-bundle F O p → M of ort ho normal fra mes o v er M . Lemma 4.1. The action of G on F O is r e gular, i.e. the isotr opy sub gr oups c orr esp ondi n g to any two p oints of M ar e c onjugate. Pr o o f. Let H b e the isotrop y subgroup o f a frame f ∈ F O . Then H also fixes p ( f ) ∈ M , and since H fixes t he frame, its differen tials fix the entire tangen t space at p ( f ) . Since it fixes the ta ngen t space, ev ery elemen t of H also fixes ev ery frame in p − 1 ( p ( f )); thus ev ery frame in a giv en fib er m ust ha v e the same isotro py subgroup. Since the elemen ts of H map geo desics to geo desics and preserv e distance, a neigh b orho o d of p ( f ) is fixed by H . Th us, H is a subgroup of the isotropy subgroup at eac h p oin t of tha t neigh b orho o d. Conv ersely , if an elemen t of G fixes a neighborho o d of a p oin t x in M , then it fixes a ll frames in p − 1 ( x ), and th us all frames in the fib ers ab o v e that neigh b orho o d. Since M is connected, w e may conclude that ev ery p oin t of F O has the same isotr o p y subgroup H , and H is the subgroup of G that fixes eve ry p oin t of M .  Remark 4.2. S inc e this s ub gr oup H is n o rmal, we often r e duc e the gr oup G to the gr oup G/H so that our action is effe ctive, in which c ase the isotr opy sub gr oups on F O ar e al l trivial. In an y case, the G orbits on F O are diffeomorphic and form a Riemannian fib er bundle, in the natural metric on F O defined as f o llo ws. The Levi-Civita connection on M determines the horizon tal subbundle H of T F O . W e construct the lo cal pro duct metric on F O using a biin v ariant fib er metric and the pullbac k of t he metric on M to H ; with this metric, F O is a compact Riemannian G × O ( n )-manifo ld. The lifted G -action commute s with the O ( n )- action. Let F denote the fo liation of G -orbits on F O , a nd observ e that F O π → F O  G = F O  F is a Riemannian submersion of compact O ( n )-manifolds. Let E → F O b e a Hermitian v ector bundle that is equiv ariant with resp ec t to the G × O ( n ) action. Let ρ : G → U ( V ρ ) and σ : O ( n ) → U ( W σ ) b e irreducible unitary represe n tations. W e define the bundle E σ → M by E σ x = Γ  p − 1 ( x ) , E  σ , where the sup erscript σ is defined for a O ( n )-mo dule Z by Z σ = ev al  Hom O ( n ) ( W σ , Z ) ⊗ W σ  , NA TURAL EQU IV ARIANT DIRAC OPERA TORS 9 where ev a l : Hom O ( n ) ( W σ , Z ) ⊗ W σ → Z is the ev aluation map φ ⊗ w 7→ φ ( w ). The space Z σ is the vec tor subspace o f Z on whic h O ( n ) acts as a direct sum of represen t ations of t yp e σ . The bundle E σ is a Hermitian G - v ector bundle of finite ra nk ov er M . The metric on E σ is chos en as follows . F or an y v x , w x ∈ E σ x , we define h v x , w x i := Z p − 1 ( x ) h v x ( y ) , w x ( y ) i y , E dµ x ( y ) , where dµ x is the measure on p − 1 ( x ) induced from the metric on F O . See [8] for a similar construction. Similarly , we define the bundle T ρ → F O  G by T ρ y = Γ  π − 1 ( y ) , E  ρ , and T ρ → F O  G is a Hermitian O ( n )-equiv a r ia n t bundle of finite rank. The metric on T ρ is h v z , w z i := Z π − 1 ( y ) h v z ( y ) , w z ( y ) i z ,E dm z ( y ) , where d m z is the measure on π − 1 ( z ) induced from the metric on F O . The v ector spaces of sections Γ ( M , E σ ) and Γ ( F O , E ) σ can b e identifie d via the isomor- phism i σ : Γ ( M , E σ ) → Γ ( F O , E ) σ , where fo r an y section s ∈ Γ ( M , E σ ), s ( x ) ∈ Γ ( p − 1 ( x ) , E ) σ for each x ∈ M , and w e let i σ ( s ) ( f x ) := s ( x ) | f x for ev ery f x ∈ p − 1 ( x ) ⊂ F O . Then i − 1 σ : Γ ( F O , E ) σ → Γ ( M , E σ ) is giv en b y i − 1 σ ( u ) ( x ) = u | p − 1 ( x ) . Observ e that i σ : Γ ( M , E σ ) → Γ ( F O , E ) σ extends to an L 2 isometry . Giv en u, v ∈ Γ ( M , E σ ), h u, v i M = Z M h u x , v x i dx = Z M Z p − 1 ( x ) h u x ( y ) , v x ( y ) i y , E dµ x ( y ) d x = Z M  Z p − 1 ( x ) h i σ ( u ) , i σ ( v ) i E dµ x ( y )  dx = Z F O h i σ ( u ) , i σ ( v ) i E = h i σ ( u ) , i σ ( v ) i F O , where dx is the Riemannian measure on M ; w e ha v e used the fact that p is a Riemannian submersion. Similarly , w e let j ρ : Γ ( F O  G, T ρ ) → Γ ( F O , E ) ρ b e the na tural iden t ific ation, whic h extends to an L 2 isometry . Let Γ ( M , E σ ) α = ev al (Hom G ( V α , Γ ( M , E σ )) ⊗ V α ) . Similarly , let Γ ( F O  G, T ρ ) β = ev al (Hom G ( W β , Γ ( F O  G, T ρ )) ⊗ W β ) . 10 IGOR PROKHORENKO V AND KEN RICHARD SON Theorem 4.3. F or any irr e ducible r epr esentations ρ : G → U ( V ρ ) and σ : O ( n ) → U ( W σ ) , the map j − 1 ρ ◦ i σ : Γ ( M , E σ ) ρ → Γ ( F O  G, T ρ ) σ is an isom orphism (w ith inverse i − 1 σ ◦ j ρ ) that extends to a n L 2 -isometry. Pr o o f. Observ e that i σ implemen ts the isomorphism Γ ( M , E σ ) = Γ  M , Γ  p − 1 ( · ) , E | p − 1 ( · )  σ  ∼ = Γ  M , Γ  p − 1 ( · ) , E | p − 1 ( · )  σ = Γ ( F O , E ) σ to the space of sections of E of O ( n ) represen tation t yp e σ . Its restriction to Γ ( M , E σ ) ρ is Γ ( M , E σ ) ρ = Γ  M , Γ  p − 1 ( · ) , E | p − 1 ( · )  σ  ρ ∼ =  Γ  M , Γ  p − 1 ( · ) , E | p − 1 ( · )  σ  ρ = (Γ ( F O , E ) σ ) ρ = Γ ( F O , E ) σ ,ρ , where the sup erscript σ, ρ denotes restriction first to sections of O ( n )-represen ta tion t yp e [ σ ] and then to the subspace of sections of G - repres en tation t yp e [ ρ ]. Since the O ( n ) and G actions commute, w e may do this in the o t he r o rder, so that Γ ( F O , E ) σ ,ρ = (Γ ( F O , E ) ρ ) σ ∼ = Γ  F O  G, Γ  π − 1 ( y ) , E | π − 1 ( · )  ρ  σ = Γ ( F O  G, T ρ ) σ , where the isomorphism is the in v erse of the restriction of j ρ to Γ ( F O  G, T ρ ) σ . Since i σ and j ρ are L 2 isometries, the result follows.  4.2. Dirac-t yp e op erators on the frame bundle. Let E → F O b e a Hermitian v ector bundle of C l ( N F ) mo dules that is equiv a r ian t with resp ect to t he G × O ( n ) action. With notation as in Example 3.3, we ha v e the transv ersal Dirac op erator A N F defined b y the comp osition Γ ( F O , E ) ∇ → Γ ( F O , T ∗ F O ⊗ E ) pro j → Γ ( F O , N ∗ F ⊗ E ) c → Γ ( F O , E ) . As explained previously , the op erator D N F = A N F − 1 2 c ( H ) is a essen tially self-adjoin t G × O ( n )-equiv arian t op erator, where H is the mean curv ature v ector field of the G -orbits in F O . F rom D N F w e no w construct equiv ariant differen tial op erators on M and F O  G , as fo l- lo ws. W e define the op erators D σ M := i − 1 σ ◦ D N F ◦ i σ : Γ ( M , E σ ) → Γ ( M , E σ ) , and D ρ F O  G := j − 1 ρ ◦ D N F ◦ j ρ : Γ ( F O  G, T ρ ) → Γ ( F O  G, T ρ ) . F or an irreducible represen tation α : G → U ( V α ), let ( D σ M ) α : Γ ( M , E σ ) α → Γ ( M , E σ ) α NA TURAL EQU IV ARIANT DIRAC OPERA TORS 11 b e the restriction of D σ M to sections of G -represen tation type [ α ]. Similarly , fo r an irreducible represen tat io n β : G → U ( W β ), let  D ρ F O  G  β : Γ ( F O  G, T ρ ) β → Γ ( F O  G, T ρ ) β b e the restriction of D ρ F O  G to sections of O ( n )-represen tation t yp e [ β ]. The prop osition b elo w follows from Theorem 4 .3. Prop osition 4.4. The op er ator D σ M is tr ansversal ly el liptic and G -e quivariant, an d D ρ F O  G is el liptic an d O ( n ) - e quivariant, and the closur e s of these op er ators ar e self-a djoint. The op er ators ( D σ M ) ρ and  D ρ F O  G  σ have iden tic al discr ete sp e ctrum, and t he c orr esp onding eigensp ac es ar e c onj ugate via Hilb ert sp a c e isomo rphisms. Th us, questions ab out the transve rsally elliptic op erator D σ M can b e r educed to questions ab out the elliptic o perators D ρ F O  G for each irreducible ρ : G → U ( V ρ ). 5. Topological proper ties of the lifted Dira c opera t ors In this section, we will pro v e that if F O is G -tr a ns v ersally spin c , then the sym b ols of the lifted tra nsve rsal Dirac op erators generate all the p ossible equiv ariant indices. T o sho w this, w e generalize the standar d m ultiplicativ e pro perty of K - t he ory to the equiv arian t setting of our pap er. 5.1. Equiv arian t Multiplicativ e P ropert ies of K -theory. Let H b e a compact Lie group. Supp ose that P is a principal H -bundle ov er a compact manifold M . Supp ose that the compact L ie group G acts on M and lifts to P , suc h that the G -action on P comm utes with the H - action. Let Z π → M b e a fib er bundle asso ciated to P with H -fib er Y ; that is, Z = P × H Y = P × Y  ( p, y ) ∼  ph, h − 1 y  . Then G acts on Z via g [( p, y )] = [( g p, y ) ]. F or an y v in the L ie algebra g of G , let v denote the fundamen tal vector field on M asso ciated to v . As in [1], let T ∗ G M = { ξ ∈ T ∗ M : ξ ( v ) = 0 for all v ∈ g } , and let T ∗ G Z b e defined similarly . Let K cpt,G ( T ∗ G M ) denote the G -equiv ariant, compactly sup- p orted K-g r o up of T ∗ G M , whic h is isomorphic to the gro up of stable G -equiv ariant homoto p y classes of transv ersally elliptic first-order sym b ols under direct sum. Lik ewise, K cpt,H ( T ∗ Y ) is isomorphic to the gr oup of the stable H -equiv a r ian t homotopy classes of first order elliptic sym b ols ov er Y . W e define a m ultiplication K cpt,G ( T ∗ G M ) ⊗ K cpt,H ( T ∗ Y ) → K cpt,G ( T ∗ G Z ) as fo llo ws. Let u b e a transve rsally elliptic, G -equiv arian t sym b ol o v er M taking v a lues in Hom ( E + , E − ), a nd let v be a H -equiv aria n t elliptic sym b ol ov er Y taking v alues in Hom ( F + , F − ). First, w e lift the sym b ol u to the H × G - eq uiv ariant sym b ol b u o n P . Let b u ∗ v b e the standard K-theory m ultiplication (similar to [1, L emma 3.4]) K cpt,H × G ( T ∗ G P ) ⊗ K cpt,H ( T ∗ Y ) → K cpt,H × G  T ∗ H × G ( P × Y )  . An elemen t ( h, g ) ∈ H × G acts on ( p, y ) ∈ P × Y by ( h, g ) ( p, y ) =  phg , h − 1 y  =  pg h, h − 1 y  . (5.1) 12 IGOR PROKHORENKO V AND KEN RICHARD SON Since t he action of H × { e } is f r ee , w e ha v e K cpt,H × G  T ∗ H × G ( P × Y )  ∼ = K cpt,G ( T ∗ G ( P × H Y )) = K cpt,G ( T ∗ G Z ) . Finally we define u · v = ] b u ∗ v to b e the image of b u ∗ v in K cpt,G ( T ∗ G Z ) under the isomorphism ab o v e. Giv en an y finite-dimensional unitary virtual H -represen tation τ o n V , w e ma y form t he asso ciated G -virtual bundle f V τ = P × τ V ov er M , defining a class in K G ( M ). The tensor pro duct mak es K cpt,G ( T ∗ G M ) naturally in to a K G ( M )-mo dule; fo r eac h [ u ] ∈ K cpt,G ( T ∗ G M ), the sym b ol u ⊗ τ := u ⊗ 1 f V τ defines an elemen t of K cpt,G ( T ∗ G M ). W e let ind H ( · ) denote the virtual represen tatio n-v alued index as explained in [1]; no t e that the result is a finite-dimensional virtual represen tation if the input is a sym b ol of an elliptic op erator. Theorem 5.1. L et Z = P × H Y as ab ove, with P a H -bund le over M . L et u b e a tr ansve r- sal ly el liptic, G - e quivariant symb o l over M takin g values in Ho m ( E + , E − ) , and let v b e a H -e quivariant el liptic symb ol over Y taking values in Hom ( F + , F − ) , s o that u and v define classes [ u ] and [ v ] in K cpt,G ( T ∗ G M ) and K cpt,H ( T ∗ Y ) , r esp e ctively. Then u · v defines an element of K cpt,G ( T ∗ G Z ) , and ind G ( u · v ) = ind G  u ⊗ ind H ( v )  . Pr o o f. W e adopt the arg ument in [16, 1 3 .6] to our situation. Let L b e a transv ersally elliptic, G -equiv ariant first or der op erator represen ting u , and let Q b e an elliptic, H -equiv aria n t first-order op erator represen ting v . Let b u b e the lift o f u to a H × G -transv ersely elliptic sym b ol o v er P , and let b L b e a tr a ns v ersally elliptic, H × G - eq uiv aria nt first order o perator represen ting b u . Next, consider op erator pro duct D = b L ∗ Q ov er P × Y , whic h represen ts b u ∗ v . This op erator is H × G equiv ariant with resp ect to the action (5.1 ). Then k er ( D ∗ D ) = h k er  b L ⊗ 1  ∩ k er ( 1 ⊗ Q ) i ⊕ h k er  b L ∗ ⊗ 1  ∩ k er ( 1 ⊗ Q ∗ ) i and k er ( D D ∗ ) = h k er  b L ∗ ⊗ 1  ∩ k er ( 1 ⊗ Q ) i ⊕ h k er  b L ⊗ 1  ∩ k er ( 1 ⊗ Q ∗ ) i Let e D and e D ∗ b e the restrictions of the o perators D and D ∗ to sections that are pullbac ks of sections o v er the base Z = P × H Y , i.e. those that are H - in v a rian t. Let τ + denote the H -represen tation k er Q , and let τ − b e the represen tation ke r Q ∗ . By the definition of the H -action in (5.1) , the decomp osition yields the asso ciated ke rnels k er  e D ∗ e D  = k er  b L ⊗ τ +  ⊕ k er  b L ∗ ⊗ τ −  , k er  e D e D ∗  = k er  b L ∗ ⊗ τ +  ⊕ k er  b L ⊗ τ −  . W e next decomp ose the ab o v e as G -repres en tations, and we obtain ind G  e D  = ind G  b L ⊗ ind H ( Q )  . The r esult follows , since u · v = ] b u ∗ v is stably homotopic to the principal symbol o f e D .  NA TURAL EQU IV ARIANT DIRAC OPERA TORS 13 5.2. Index of Lifted Dirac op erators. S upp ose that D : Γ ( M , E + ) → Γ ( M , E − ) is an y transv ersally elliptic, G -equiv ariant op erator w ith tr a ns v ersally elliptic symbol u ∈ Γ ( M , Hom ( E + , E − )), so that [ u ] ∈ K cpt,G ( T ∗ G M ). If 1 denotes the t rivial O ( n ) repre- sen tation ov er t he identit y , then let v b e a n y elemen t of the class i ! ( 1 ) ∈ K cpt,O ( n ) ( T ∗ O ( n )) induced from the inclusion o f the identit y in O ( n ) via an extension of the Thom isomor- phism (see [3]). Observ e that the equiv a rian t index ind O ( n ) ( v ) of the elliptic sym b ol v is equal to one cop y of the trivial represen tation (see axioms of the equiv ariant index in [3, 13.6]). By Theorem 5.1, G -equiv ariant tra ns v ersally elliptic sym b ol u · v defines an elemen t of K cpt,G ( T ∗ G F O ) such that ind G ( u · v ) = ind G  u ⊗ ind O ( n ) ( v )  = ind G ( u ⊗ 1 ) = ind G ( u ) . Supp ose further that F O is G -tra nsve rsally spin c . Then the class [ u · v ] ∈ K cpt,G ( T ∗ G F O ) ma y b e represen ted by the sym b ol of a transv ersally-elliptic, G -equiv ar ia n t op erator D N F of Dirac type. Th us, t he op erator D 1 M = i − 1 1 ◦ D N F ◦ i 1 satisfies ind G  D 1 M  = ind G ( D N F ) = ind G ( u · v ) = ind G ( u ) = ind G ( D ) . The result b elo w follows. Theorem 5.2. Supp ose that F O is G - t r a nsversal ly spin c . Then for every tr ansve rs a l ly el liptic symb ol c l a ss [ u ] ∈ K cpt,G ( T ∗ G M ) , ther e exists an op er ator of typ e D 1 M such that ind G ( u ) = ind G ( D 1 M ) . 6. Example 6.1. A transv ersal Dirac op erator on the sphere. Let G = S 1 act on S 2 ⊂ R 3 b y rotations ab out t he z -axis. Let p : F O → S 2 b e the oriente d orthonormal frame bundle. W e will iden tif y F O with S O (3) b y letting the first row denote the p oin t on S 2 and the last tw o ro ws denote the framing of the tangent space. W e choose the metric on F O to b e h A, B i = tr  A t B  . The a ction o f S 1 lifted to F O is given b y m ultiplication on t he right: R t ( A ) = A   cos t − sin t 0 sin t c os t 0 0 0 1   . T angen t vec tors to F O are elemen ts of the Lie algebra o (3) =      0 a b − a 0 c − b − c 0         a, b, c ∈ R    , 14 IGOR PROKHORENKO V AND KEN RICHARD SON and the tang ent space to t he S 1 action is the span of the left-in v ariant v ector field T induced b y   0 1 0 − 1 0 0 0 0 0   at the iden tit y . Thus , the no r mal bundle o f the corresp onding foliation on F O is trivial. It is the subbundle N S 1 of T F O that is given at A ∈ F O b y N S 1 | A =    A   0 0 b 0 0 c − b − c 0         b, c ∈ R    . The v ectors V 1 = A   0 0 1 0 0 0 − 1 0 0   , V 2 = A   0 0 0 0 0 1 0 − 1 0   and the or bit direction T = A   0 1 0 − 1 0 0 0 0 0   are m utually or thogonal. Let E = F O × C 2 → F O b e the trivial bundle. The action of C l ( N S 1 ) ∼ = C l ( R 2 ) on fib ers of E is defined b y c ( V 1 ) =  0 − 1 1 0  , c ( V 2 ) =  0 i i 0  . W e identify the v ectors  1 0  and  0 1  ∈ C 2 with the left-in v ar ia n t fields V 1 and V 2 in Γ ( N S 1 ). W e assume that the S 1 -action on E is trivial. As in Example 3.3, the transv ersal Dirac op erator is D N F = A N F = 2 X j =1 c ( V j ) ∇ V j , where ∇ V j is the directional deriv at ive in the direction V j . Since the length of eac h orbit of the S 1 action is constan t, the mean curv ature ve ctor is zero. The bundle F O → S 2 is an S O (2) principal bundle and comes equipped with an action of S O (2) on the frames o v er a p oint. The left action of  cos α − sin α sin α cos α  ∈ S O (2 ) on a frame A ∈ S O (3 ) is giv en by L α ( A ) =   1 0 0 0 cos α sin α 0 − sin α cos α   A. Again, w e extend this action trivially to the C 2 bundle. Note that D N F is equiv ariant with resp ec t to b oth the ab o v e left S O (2) action and the rig h t S 1 action. W e c ho ose the standard spherical co ordinates x ( θ , φ ) ∈ S 2 . Let P θ , φ denote parallel transp ort in the tangen t bundle from the north po le a lo ng the minimal g eodesic connected to x ( θ , φ ). Then P θ , φ v =   cos θ − sin θ 0 sin θ cos θ 0 0 0 1     cos φ 0 sin φ 0 1 0 − sin φ 0 cos φ     cos θ sin θ 0 − sin θ cos θ 0 0 0 1   v . NA TURAL EQU IV ARIANT DIRAC OPERA TORS 15 W e parallel transp ort the standar d frame ( e 1 , e 2 ) at the north p ole to get X θ , φ = P θ , φ e 1 , Y θ , φ = P θ , φ e 2 , and the w e rotate b y α to get all possible frames. The result is a co ordinate c hart U 1 : [0 , 2 π ] ×  0 , π 2  × [0 , 2 π ] → S O (3) defined b y U 1 ( θ , φ, α ) = L α   x ( θ , φ ) X θ , φ Y θ , φ   . A section u is defined to b e of irreducible represen tation type σ n : S O ( 2) → C if it satisfies  ( L β ) ∗ u  = e inβ u . Since the a ctio n of S O (2 ) on the fib ers of E is trivial, w e ha v e  ( L β ) ∗ u  ( θ , φ, α ) =  u ◦ ( L β ) − 1  ( θ , φ, α ) = u ( θ , φ , α − β ) = e inβ u ( θ, φ, α ) . (6.1) Th us, it suffices to calculate u ( θ, φ, 0) = u at U 1 ( θ , φ, 0). The lo w er hemisphere co ordinates of the p oin t and ve ctors w ould hav e the opp osite third co ordinate, and the sign of X θ , φ is rev ersed in addition to ensure that the frame is orien ted. Note that the φ in the lo w er hemisphere is ( π − φ ) in the upp er hemisphere. Th us the second c hart is U 2 : [0 , 2 π ] ×  0 , π 2  × [0 , 2 π ] U 2 ( θ , φ, α ) = U 2 α   x ( θ , φ ) − X θ , φ Y θ , φ   diag (1 , 1 , − 1) One can c hec k that U 1  θ , π 2 , α  = U 2  θ , π 2 , α − 2 θ  = L − 2 θ U 2  θ , π 2 , α  . Th us, t he clutc hing function for the frame bundle is multiplic ation on the left by e 2 θi . Next, supp ose that u is a section suc h that  ( L β ) ∗ u  ( M ) = u ( L − β M ) = e inβ u ( M ) . This means in fact that u ( θ, φ, α − β ) = e inβ u ( θ, φ, α ) in b oth c harts. Th us w e ma y trivialize the bundle b y restricting to ( θ , φ, 0) in each c hart. W e observ e u 1  θ , π 2 , 0  = u 2  L − 2 θ  θ , π 2 , 0  = e i 2 nθ u 2  θ , π 2 , 0  , and thus the clutching function for E σ n is e 2 nθ i . One may express the v ector fields V 1 , V 2 in terms of the co ordinate v ector fields ∂ α , ∂ θ , ∂ φ . In the upp er hemisphere, V 1 1 = sin θ (cos φ − 1) sin φ ∂ α + sin θ cos φ sin φ ∂ θ − cos θ ∂ φ V 1 2 = cos θ (1 − cos φ ) sin φ ∂ α − cos θ cos φ sin φ ∂ θ − sin θ ∂ φ No w we wish to consider the op erator D σ n S 2 = i − 1 σ n ◦ D N F ◦ i σ n : Γ  S 2 , E σ n  → Γ  S 2 , E σ n  , 16 IGOR PROKHORENKO V AND KEN RICHARD SON where i σ n : Γ ( S 2 , E σ n ) → Γ ( F O , E ) σ n . W e hav e √ 2 D 1 N F  u 1 u 2  =  0 − 1 1 0  ∇ V 1 1  u 1 u 2  +  0 i i 0  ∇ V 1 2  u 1 u 2  =  − ( V 1 1 + iV 1 2 ) u 2 ( V 1 1 + iV 1 2 ) u 1  . There is a similar fo r mula for √ 2 D 2 N F in the low er hemisphere c hart. Observ e that  V 1 1 + iV 1 2  = − ie iθ (cot φ − csc φ ) ∂ α + − ie iθ cot φ ∂ θ + − e iθ ∂ φ W e easily c hec k that right m ult iplicatio n by β ∈ S 1 = R mo d 2 π on U 1 ( θ , φ, α ) satisfies R β U 1 ( θ , φ, α ) = U 1 ( θ + β , φ, α + β ) . If ψ 1 ( θ , φ, 0) is a section of F O × C 2 → F O of ty p e σ n (with resp ect to t he fib erwise action of S O (2)) ov er the upp er hemisphere, then  ( R β ) ∗ ψ 1  ( θ , φ, 0) = ψ 1 ◦ R − β ( θ , φ, 0) = ψ 1 ( θ − β , φ, − β ) = e inβ ψ 1 ( θ − β , φ, 0) , using the upp er hemisphere trivialization U 1 ( θ , φ, α ) and equation ( 6.1 ). If w e assume that ψ : F O → E σ n is a section of t yp e ρ m with resp ect to the lifted S 1 action, then  ( R β ) ∗ ψ 1  = e imβ ψ 1 . Th us,  ( R β ) ∗ ψ 1  ( θ , φ, 0) = e imβ ψ 1 ( θ , φ, 0) = e inβ ψ 1 ( θ − β , φ, 0) , whic h implies ψ 1 ( θ , φ, 0) = e i ( n − m ) θ ψ 1 (0 , φ, 0) . The a nalogous calculation in the low er hemisphere c hart yields ψ 2 ( θ , φ, 0) = e i ( − n − m ) θ ψ 2 (0 , φ, 0) . 6.2. Calculation of ker D σ n S 2 . Since D σ n S 2 = i − 1 σ n ◦ D N F ◦ i σ n , we seek solutions to the equation √ 2 D 1 N F  ψ 1 ψ 2  =  − ( V 1 1 + iV 1 2 ) ψ 2 ( V 1 1 + iV 1 2 ) ψ 1  =  0 0  in the upp er hemisphe re chart. F rom the equations ( V 1 1 + iV 1 2 ) ψ 1 = 0, ∂ α ψ 1 = − niψ 1 (since Ψ ∈ Γ ( S 2 , E σ n ) ), and ∂ θ ψ 1 = i ( n − m ) ψ 1 (since Ψ ∈ Γ ( S 2 , E σ n ) ρ m ), we ha v e 0 =  V 1 1 + iV 1 2  ψ 1 =  − ie iθ (cot φ − csc φ ) ∂ α + − ie iθ cot φ ∂ θ + − e iθ ∂ φ  ψ 1 =  − me iθ (cot φ ) + ne iθ (csc φ ) − e iθ ∂ φ  ψ 1 . Solving this equation, we obtain ψ 1 (0 , φ ) = C 2 (sin φ ) n − m (cos φ + 1 ) n . NA TURAL EQU IV ARIANT DIRAC OPERA TORS 17 This implies that ψ 1 is ψ 1 ( θ , φ ) = C 2 (sin φ ) n − m (cos φ + 1 ) n e i ( n − m ) θ , o r ψ 1 ( z ) = C 2 z n − m  q 1 − | z | 2 + 1  n in the complex co ordinates of the pro jection of the upp er hemisphere to the xy plane. Th us, ψ 1 is smo oth in the upp er hemisphere o nly if n ≥ m . Similarly , − ( V 1 1 + iV 1 2 ) ψ 2 = 0, ∂ α ψ 2 = − niψ 2 , a nd ∂ θ ψ 2 = i ( n − m ) ψ 2 implies ψ 2 ( θ , φ ) = C (cos φ + 1) n (sin φ ) m − n e i ( n − m ) θ , or ψ 2 ( z ) = C 2  q 1 − | z | 2 + 1  n z m − n . Hence, ψ 2 is smo oth in the upp er hemisphere only if m ≥ n . W e need to see if the solutions ψ 1 and ψ 2 extend to solutions ov er the en tire sphere. In the low er hemisphere, we ha v e the equation √ 2 D 2 N F Ψ = √ 2 D 2 N F  ψ 1 ψ 2  =  ( − V 2 1 + iV 2 2 ) ψ 2 ( V 2 1 + iV 2 2 ) ψ 1  =  0 0  . Similar computat io ns sho w t ha t ψ 1 ( z ) = C 2  q 1 − | z | 2 + 1  n z − n − m , ψ 2 ( z ) = C 2  q 1 − | z | 2 + 1  − n z m + n in the complex co ordinates of the pro jection of the lo w er hemisphere t o the xy plane. Th us, ψ 1 is smo o th in the low er hemisphere only if n + m ≤ 0 , and ψ 2 is smo o th in the lo w er hemisphere o nly if m + n ≥ 0. In summary , we seek solutions to √ 2 D 2 N F Ψ = √ 2 D 1 N F Ψ = 0 restricted to sections of E σ n of t yp e ρ m . The clutc hing f unc tion o f E σ n is m ultiplication b y e 2 nθ i (i.e. z 2 n or z − 2 n ), so that Ψ 1  θ , π 2 , 0  = Ψ 2  A 2 θ  θ , π 2 , 0  = e i 2 nθ Ψ 2  θ , π 2 , 0  . The function ψ 1 is contin uous if and only if ψ 1 1  θ , π 2 , 0  = e i 2 nθ ψ 2 1  θ , π 2 , 0  and m ≤ − | n | . Similarly , ψ 2 is contin uous if and only if ψ 1 2  θ , π 2 , 0  = e i 2 nθ ψ 2 2  θ , π 2 , 0  and m ≥ | n | . F rom the equations in the previous section, the index of D σ n S 2 restricted to sections of t yp e ρ m is ind ρ m  D σ n S 2  =    − 1 if m > | n | o r m = | n | and n 6 = 0 0 if − | n | < m < | n | or m = n = 0 1 if m < − | n | o r m = − | n | a nd n 6 = 0 . (6.2) Note tha t the k ernel of D σ n S 2 is infinite-dimensional. The op erator D σ n S 2 fails t o b e elliptic precisely a t the p oin ts where cot φ = 0; that is, at the equator. 6.3. The op erator on F O  G . W e no w construct the op erator D ρ m F O  G : Γ ( F O  G, T ρ m ) → Γ ( F O  G, T ρ m ). First, observ e that F O  G = S O (3)  S 1 is ag a in the sphere S 2 . The orbits of the action of G on F O are of the form { R t M : t ∈ [0 , 2 π ] } , with M ∈ F O . The map R t 18 IGOR PROKHORENKO V AND KEN RICHARD SON rotates t he first a nd second columns of the matrix, so that the map F O → S 2 is the map to the third column. Th us, the pro jection of the v ector V 1 to T S 2 is V 1 = M   0 0 1 0 0 0 − 1 0 0   7→ first column of M Similarly , V 2 = M   0 0 0 0 0 1 0 − 1 0   7→ second column of M A section of t yp e ρ m of F O × C 2 → F O is one f or whic h the pa r t ial deriv a t iv e in direction T = M   0 1 0 − 1 0 0 0 0 0   is m ultiplication b y im . In the U 1 co ordinate c hart (corresp onding to 0 ≤ φ ≤ π 2 , 0 ≤ θ ≤ 2 π , 0 ≤ α ≤ 2 π ), the quotien t to F O  G go es t o   cos φ − cos ( θ − α ) sin φ − sin ( θ − α ) sin φ   , mapping to the en tire upp er hemisphere x ≥ 0, with fib ers of the form ( φ , θ , α ) = ( φ 0 , θ 0 + t, α 0 + t ). Th us, w e ma y fix α = 0 for the sake of ar g ume n t and a llo w θ = θ − α and φ to v ary . The gr o up S 1 acts as b efo re by U 1 ( θ , φ, α ) R β 7→ U 1 ( θ + β , φ, α + β ). In particular, if ψ is a sec tion of type ρ m , ( R β ψ 1 ) ( θ , φ , α ) = ψ 1 ◦ m − β ( θ, φ, α ) = ψ 1 ( θ − β , φ, α − β ) = e imβ ψ 1 ( θ , φ, α ). Th us, setting α = 0 , we hav e ψ 1 ( θ − β , φ, − β ) = e imβ ψ ( θ, φ, 0) . Th us, ∂ ∂ β  ψ 1 ( θ − β , φ, − β )  = imψ 1 ( θ − β , φ, − β ) =  − ∂ ∂ θ − ∂ ∂ α  ψ 1  ( θ − β , φ, − β ) So all sections of type ρ m satisfy  − ∂ ∂ θ − ∂ ∂ α  ψ 1 = imψ 1 , or ∂ α = − ∂ θ − im . Restricted t o this space of sections, w e hav e  V 1 1 + iV 1 2  = − ie iθ csc φ ∂ θ − e iθ ∂ φ − me iθ (cot φ − csc φ ) The interes ted reader may chec k that this op erator is elliptic a t all p oints of the hemi- sphere x ≥ 0. A similar statemen t is true in the other hemisphere. Thus , D ρ m F O  G : Γ ( F O  G, T ρ m ) → Γ ( F O  G, T ρ m ) is elliptic, as exp ected. The k ernel of D ρ m F O  G restricted to Γ ( F O  G, T ρ m ) σ n has dimension dim  k er σ n  D ρ m F O  G  = dim  k er ρ m  D σ n S 2  =    2 if m = n = 0 1 if | n | ≤ | m | , m 6 = 0 0 if | n | > | m | , NA TURAL EQU IV ARIANT DIRAC OPERA TORS 19 using the results preceding formula (6.2). 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Dep ar tment of Ma thema tics, Texas Christian U niversity, Box 298900, For t Wor th, Texas 76129 E-mail addr ess : i.pr okhorenkov @tcu.edu, k.rich ardson@tcu . edu