# Representation theoretic embedding of twisted Dirac operators

aa r X i v : . [ m a t h . R T ] F e b REPRESENTATION THEORETIC EMBEDDINGOF TWISTED DIRAC OPERATORS

S. MEHDI AND P. PANDˇZI´C

Abstract.

Let G be a non-compact connected semisimple real Lie group withﬁnite center. Suppose L is a non-compact connected closed subgroup of G acting transitively on a symmetric space G/H such that L ∩ H is compact. Westudy the action on L/L ∩ H of a Dirac operator D G/H ( E ) acting on sectionsof an E -twist of the spin bundle over G/H . As a byproduct, in the case of ( G, H, L ) = ( SL (2 , R ) × SL (2 , R ) , ∆( SL (2 , R ) × SL (2 , R )) , SL (2 , R ) × SO (2)) ,we identify certain representations of L which lie in the kernel of D G/H ( E ) . Introduction

In a recent paper [MO1] and an ongoing project [MO2], Mehdi and Olbrich studyrepresentation theoretic features of the spectrum of locally symmetric spaces arisingas follows. Let G be a non-compact connected semisimple real Lie group with ﬁnitecenter and H a non-compact connected closed subgroup of G so that G/H is asymmetric space with respect to an involution σ of G . Suppose there exists anon-compact connected real closed subgroup L of G which acts transitively andcocompactly on G/H , i.e.,

G/H ≃ L/L ∩ H with L ∩ H compact. In particular, if D ( G/H ) (resp. D ( L/L ∩ H )) denotes the algebra of G -invariant (resp. L -invariant)diﬀerential operators on G/H (resp.

L/L ∩ H ), one gets an embedding of algebras(1.1) ı : D ( G/H ) ֒ → D ( L/L ∩ H ) . Now pick a discrete subgroup Γ in G which is contained in L and such thatΓ \ G/H is a smooth compact manifold, i.e., a compact Cliﬀord-Klein form of

G/H . Using (1.1), Mehdi and Olbrich describe the joint spectral decomposi-tion of the (commutative) algebra D ( G/H ) on the Hilbert space L (Γ \ G/H ) ofsquare integrable functions on Γ \ G/H in terms of the spectrum of D ( L/L ∩ H ) on L (Γ \ L/L ∩ H ) ≃ L (Γ \ L ) L ∩ H . In particular, they prove the L -admissibility ofcertain G -representations involved in the spectrum.In this paper, we extend the embedding (1.1) to the case of diﬀerential operatorsacting on sections of spin bundles over G/H and

L/L ∩ H twisted by a ﬁnite-dimensional representation E . Then we show that the image, under this embedding,of the Dirac operator D G/H ( E ) on G/H splits as combination of both geometric andalgebraic Dirac operators attached to the homogeneous spaces

L/L ∩ H , L/L ∩ K , L ∩ K/L ∩ H and H/H ∩ K (here K is a maximal compact subgroup of G ﬁxed bya Cartan involution θ , and H and L are assumed to be θ -stable). Note that while Key words and phrases.

Dirac operators; Lie groups; Spin; Representations; SL (2 , R ). . Primary: 22E46; Secondary: 43A85.P. Pandˇzi´c was supported by the QuantiXLie Center of Excellence, a project coﬁnanced by theCroatian Government and European Union through the European Regional Development Fund -the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). most of these spaces are symmetric, L/L ∩ H is not, and for this space we use thecubic Dirac operator deﬁned in [G] and [K].Finally, we use this splitting formula to relate the kernels of the various Diracoperators and we identify certain representations of L occuring in the kernel of D G/H ( E ) when G = SL (2 , R ) × SL (2 , R ), H = ∆( SL (2 , R ) × SL (2 , R )) is thediagonal in G and L = SL (2 , R ) × SO (2).The paper is organized as follows. In Section 2 we deﬁne the triples ( G, H, L )and we collect their main features. In Section 3 we review some facts about Cliﬀordalgebras and spin modules. In Section 4, we describe the geometric and the algebraicDirac operators, as well as the various bundles and sections on which they act. InSection 5, we compute the transfer of the cubic Dirac operator from

G/H to L/L ∩ H in terms of “smaller” Dirac operators. Finally, in Section 6, we use this transferformula when ( G, H, L ) = ( SL (2 , R ) × SL (2 , R ) , ∆( SL (2 , R ) × SL (2 , R )) , SL (2 , R ) × SO (2)) to relate the kernels of various Dirac operators. Moreover, we identify someof the representations of L involved in the kernel of the twisted cubic Dirac operatoron G/H . 2.

Transitive triples

Let G be a non-compact connected semisimple real Lie group with ﬁnite centerand Lie algebra g and let h , i be the Killing form on g . Fix a Cartan involution θ on G and let K be the corresponding maximal compact subgroup of G with Liealgebra k . The associated Cartan decomposition of g is g = k ⊕ s , with [ k , s ] ⊂ s and [ s , s ] ⊂ k . Let H be a non-compact connected semisimple closed subgroup of G with Liealgebra h such that the homogeneous space G/H is a symmetric space with respectto some involution σ . There is therefore a decomposition of g : g = h ⊕ q , with [ h , q ] ⊂ q and [ q , q ] ⊂ h . Recall the following invariance properties of the Killing form. For all X , Y and Z in g , one has: h θ ( X ) , θ ( Y ) i = h X, Y ih σ ( X ) , σ ( Y ) i = h X, Y ih ad( X )( Z ) , Y i = −h Z, ad( X ) Y i Moreover, the restrictions of h , i to k × k , s × s , h × h and q × q are non-degenerateso that k ⊥ s and h ⊥ q with respect to h , i . We assume that θ and σ commute, so that k and s are σ -stable: k = ( k ∩ h ) ⊕ ( k ∩ q ) and s = ( s ∩ h ) ⊕ ( s ∩ q )Next, let L be a non-compact connected semisimple closed subgroup of G withLie algebra l such that(i) L is reductively embedded in G ,(ii) L acts transitively on G/H ,(iii) L ∩ H is compact.We will refer to triples ( G, H, L ) satisfying (i),(ii) and (iii) as transitive triples . L need not be σ -stable in general, however we may assume that L is θ -stable: l = ( l ∩ k ) ⊕ ( l ∩ s ) . EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 3

The restriction to l of the Killing form h , i of g remains non-degenerate and g = h + l and q ∩ l ⊥ = { } where l ⊥ is the orthogonal of l in g with respect to the Killing form. Transitivetriples were classiﬁed in the late 1960s by Oniˇsˇcik in a more general setting [O]. Acomplete list can be found in [KY] or [MO2]. Examples 2.1. (1a) G = G ′ × G ′ , H = ∆( G ′ × G ′ ) , L = G ′ × { e } . (1b) G = G ′ × G ′ , H = ∆( G ′ × G ′ ) , L = G ′ × K ′ where G ′ is a non-compact connected semisimple real Lie group with ﬁnitecenter and K ′ is a maximal compact subgroup of G ′ . (2) G = SO e (2 , n ) , H = SO e (1 , n ) , L = U (1 , n ) , n ≥ . (3) G = SO e (2 , n ) , H = U (1 , n ) , L = SO e (1 , n ) , n ≥ . (4) G = SO e (4 , n ) , H = SO e (3 , n ) , L = Sp (1 , n ) , n ≥ . (5) G = SU (2 , n ) , H = SU (1 , n ) , L = Sp (1 , n ) , n ≥ . (6) G = SU (2 , n ) , H = Sp (1 , n ) , L = SU (1 , n ) , n ≥ . (7) G = SO e (8 , , H = SO e (7 , , L = Spin e (1 , . (8) G = SO e (4 , , H = SO e (4 , × SO (3) , L = Spin e (4 , . (9) G = SO e (4 , , H = SO e (4 , × SO (2) , L = G . (10) G = SO (8 , C ) , H = SO e (1 , , L = Spin (7 , C ) . (11) G = SO (8 , C ) , H = SO (7 , C ) , L = Spin e (1 , . We now recall some features of transitive triples described in [MO2]. Fix atransitive triple (

G, H, L ). Let q l be the orthogonal of l ∩ h in l with respect to theKilling form: l = ( l ∩ h ) ⊕ q l with q l = ( l ∩ h ) ⊥ l and [ l ∩ h , q l ] ⊂ q l . Note that as l ∩ h -modules, one has the decomposition:(2.2) q l = q ′ l ⊕ ( l ∩ s )where q ′ l is the orthogonal of l ∩ h in l ∩ k with respect to the Killing form. Since L acts transitively on G/H , the L ∩ H -equivariant map p − : q l ≃ −→ q , X X − σ ( X )2is an isomorphism of L ∩ H -modules. However, the map p + : q l −→ h , X X + σ ( X )2need not be injective or surjective. In order to turn the isomorphism p − into anisometry, we diagonalize the symmetric bilinear form l × l ∋ ( X, Y )

7→ h σ ( X ) , Y i with respect to the Killing form h , i . More precisely, for a real number ν , deﬁne thevector subspace l ( ν ) of l : l ( ν ) := (cid:8) X ∈ l | h σ ( X ) , Y i = ν h X, Y i ∀ Y ∈ l (cid:9) . S. MEHDI AND P. PANDˇZI´C

Then l ( ν ) is invariant under the adjoint action of L ∩ H and l ( ν ) ⊥ l ( ν ′ ) whenever ν = ν ′ . One can check that ν ∈ [ − , l (1) = l ∩ h and l ( −

1) = l ∩ q . In fact,each l ( ν ) is σ -stable and θ -stable so that q ′ l = M ν =1 l ( ν ) ∩ k and l ∩ s = M ν l ( ν ) ∩ s . We will denote by X ( ν ) an element in l ( ν ). For ν = 1, set d ν = (cid:16) − ν (cid:17) − and consider the L ∩ H -equivariant maps ρ − : q l → q , X ( ν ) d ν p − ( X ( ν )) ρ + : q l → h , X ( ν ) d ν p + ( X ( ν )) . Then ρ − is an isometry, and ρ + is a partial isometry but it need not be injectiveor surjective. One easily checks that for all X ( ν ), whenever ν = 1 [MO2]:[ X ( ν ) , σ ( X ( ν ))] = 0 ρ + ( X ( ν )) + ρ − ( X ( ν )) = d ν X ( ν )[ ρ + ( X ( ν )) , ρ − ( X ( ν ))] = 0 . We will write λ for the compact ν , i.e the ν involved in the decomposition of q ′ l ,and µ for the non-compact ν , i.e the ν involved in the decomposition of l ∩ s .Transitive triples split into three families, depending on the map ρ + being in-jective but not surjective (type I), surjective but not injective (type S) or neither(type N). Examples 2.3. (1a) G = G ′ × G ′ , H = ∆( G ′ × G ′ ) , L = G ′ × { e } is of type I. (1b) G = G ′ × G ′ , H = ∆( G ′ × G ′ ) , L = G ′ × K ′ is of type S. (2) G = SO e (2 , n ) , H = SO e (1 , n ) , L = U (1 , n ) is of type S. (3) G = SO e (2 , n ) , H = U (1 , n ) , L = SO e (1 , n ) is of type S. (4) G = SO e (4 , n ) , H = SO e (3 , n ) , L = Sp (1 , n ) is of type I. (5) G = SU (2 , n ) , H = SU (1 , n ) , L = Sp (1 , n ) is of type S. (6) G = SU (2 , n ) , H = Sp (1 , n ) , L = SU (1 , n ) is of type S. (7) G = SO e (8 , , H = SO e (7 , , L = Spin e (1 , is of type I. (8) G = SO e (4 , , H = SO e (4 , × SO (3) , L = Spin e (4 , is of type N. (9) G = SO e (4 , , H = SO e (4 , × SO (2) , L = G is of type N. (10) G = SO (8 , C ) , H = SO e (1 , , L = Spin (7 , C ) is of type N. (11) G = SO (8 , C ) , H = SO (7 , C ) , L = Spin e (1 , is of type I. We ﬁx an orthonormal basis { Z j ( λ ) } (resp. { T k ( µ ) } ) of l ( λ ) (resp. l ( µ )) so that { Z j } = ∪ λ { Z j ( λ ) } (resp. { T k } = ∪ µ { T k ( µ ) } ) is an orthonormal basis of q l (resp. l ∩ s ) and [ Z j ( λ ) , σ ( Z j ( λ ))] = 0 , ∀ j, λ. [ T k ( µ ) , σ ( T k ( µ ))] = 0 , ∀ k, µ. (2.4)It is worth to mention that transitive triples of type S have the following additionalfeatures [MO2]: EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 5 (a) L ∩ K is σ -stable, i.e., L ∩ K/L ∩ H is a symmetric space with respect tothe involution σ .(b) q ′ l is irreducible as a L ∩ H -module and λ = − λ ).(c) l ∩ s is irreducible as a L ∩ H -module and µ = 0 (only one value for µ ).3. Clifford algebras and spin modules

Let (

G, H, L ) be a transitive triple, and let as before q denote the orthogonalcomplement of h is g with respect to the form h , i . Then h , i is nondegenerate onboth h and q .Let C ( q ) be the Cliﬀord algebra of q with respect to h , i . In other words, it isthe associative algebra with unit, generated by q , with relations XY + Y X = h X, Y i , X, Y ∈ q . It is well known that the category of complex C ( q )-modules is semisimple, withonly one irreducible module if dim q is even, and two irreducible modules if dim q isodd. These modules are called spin modules and they can be constructed as follows.Let q + C and q − C be maximal isotropic subspaces of q C , nondegenerately paired by h , i . Then q C = q + C ⊕ q − C if dim q is even; q C = q + C ⊕ q − C ⊕ C Z if dim q is odd , where in the odd case Z ∈ q C is a vector orthogonal to q + C ⊕ q − C such that h Z, Z i = 1.We deﬁne S q = V q + C , with elements of q + C acting by wedging and elements of q − C acing by contracting. Ifdim q is even, this determines a C ( q )-module structure on the spin module S q , andif dim q is odd we still need to deﬁne the action of Z . There are two choices: Z canact by 1 / √ V even q + C and by − / √ V odd q + C , or by − / √ V even q + C andby 1 / √ V odd q + C . We ﬁx one of these choices. We denote by(3.1) γ q : C ( q ) → End( S q )the action map for the C ( q )-module S q .Besides the obvious embedding of q into C ( q ), there is also an embedding of so ( q ) into C ( q ) induced by the skew symmetrization (Chevalley isomorphism): so ( q ) ≃ Λ q j ֒ → C ( q ) , j ( X ∧ Y ) = 12 ( XY − Y X ) . The image of this map is equal to the Lie subalgebra C ( q ) of C ( q ), consisting of“pure degree 2” elements. Its main property is that the natural action of so ( q ) on q corresponds to the action of C ( q ) on q ⊂ C ( q ) by Cliﬀord algebra commutators.Composing the above map with the adjoint action map h → so ( q ), we get a map(3.2) α g , h = α h : h → C ( q )such that(3.3) [ α h ( X ) , Y ] C ( q ) = [ X, Y ] g , X ∈ h , Y ∈ q . Moreover, α h is a Lie algebra morphism, i.e.,[ α h ( X ) , α h ( Z )] C ( q ) = [ X, Z ] g , X, Z ∈ h . S. MEHDI AND P. PANDˇZI´C

In the last two formulas, the subscript of a bracket denotes the Lie algebra in whichthe bracket is taken.The following lemma describes α h more explicitly. The proof can be found forexample in [HP], Section 2.3.3 (but note that the conventions there are diﬀerent,so a factor shows up in the formula). Lemma 3.4.

Let { e i } be an orthonormal basis of q , and let ε i = h e i , e i i ∈ {± } .Then for any X ∈ h , α h ( X ) = − X i

With notation as above, ρ − ( α l ∩ h ( X )) = α h ( X ) , X ∈ l ∩ h . In particular, the isomorphism of S q l and S q induced by ρ − is l ∩ h -equivariant.Proof. Recall that ρ − : q l → q is an l ∩ h -equivariant isometry. This implies thatfor any X ∈ l ∩ h and Y, Z ∈ q l ,(3.6) h [ X, ρ − ( Y )] , ρ − ( Z ) i = h ρ − ([ X, Y ]) , ρ − ( Z ) i = h [ X, Y ] , Z i . Let now { e i } be an orthonormal basis of q l with h e i , e i i = ε i = ±

1. By Lemma 3.4,if X ∈ l ∩ h , then α l ∩ h ( X ) = − X i

We also consider the spin modules S q ′ l and S l ∩ s for the pairs ( l ∩ k , l ∩ h ) and( l , l ∩ k ) respectively. Recall that by (2.2) q l = l ∩ s ⊕ q ′ l . This implies that C ( q l ) = C ( l ∩ s ) ¯ ⊗ C ( q ′ l ) , where ¯ ⊗ denotes the graded tensor product of superalgebras, i.e.,( a ¯ ⊗ b )( c ¯ ⊗ d ) = ( − deg b deg c ac ¯ ⊗ bd, with deg being 0 for even elements and 1 for odd elements.To get an analogous decomposition of the spin module, we ﬁrst note that if( l ∩ s ) + C and ( q ′ l ) + C are maximal isotropic subspaces of ( l ∩ s ) C respectively ( q ′ l ) C , EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 7 then ( l ∩ s ) + C ⊕ ( q ′ l ) + C is a maximal isotropic subspace of ( q l ) C , unless dim( l ∩ s )and dim( q ′ l ) are both odd. (In case they are both odd, there is an isotropic vectoroutside of ( l ∩ s ) + C ⊕ ( q ′ l ) + C .) To simplify matters, we make the following assumption. Assumption 3.7.

The dimension of l ∩ s is even. Since the triples of type S other than ( G, H, L ) = ( G ′ × G ′ , ∆( G ′ × G ′ ) , G ′ × K ′ )all have L of equal rank, Assumption 3.7 is satisﬁed for all of them. For thetriples ( G ′ × G ′ , ∆( G ′ × G ′ ) , G ′ × K ′ ), we have l ∩ s = s ′ ×

0, so Assumption 3.7is equivalent to dim s ′ being even. This is certainly true if G ′ is of equal rank, andif G ′ is of unequal rank it is still often true. For example, if G ′ = SL ( n, R ), thendim s ′ = n ( n +1)2 −

1, which is even if n is congruent to 1 or 2 modulo 4 (and oddotherwise).The above discussion about isotropic subspaces implies that under Assumption3.7 we have S q l = S l ∩ s ⊗ S q ′ l as vector spaces. The isomorphism S l ∩ s ⊗ S q ′ l → S q l can be realized as the exterior algebra multiplication map.This vector space isomorphism is also an isomorphism of C ( q l ) = C ( l ∩ s ) ¯ ⊗ C ( q ′ l )-modules. To see this, we ﬁrst note that since dim l ∩ s is even, the C ( l ∩ s )-module S l ∩ s is Z -graded, with S l ∩ s = V even ( l ∩ s ) + C ; S l ∩ s = V odd ( l ∩ s ) + C . We can now deﬁne an action of C ( q l ) = C ( l ∩ s ) ¯ ⊗ C ( q ′ l ) on S l ∩ s ⊗ S q ′ l by(3.8) ( c ⊗ d ) · ( s ⊗ t ) = ( − deg d deg s cs ⊗ dt. To see that this is a well deﬁned algebra action, we check(3.9) ( a ⊗ b ) · [( c ⊗ d ) · ( s ⊗ t )] = [( a ⊗ b )( c ⊗ d )] · ( s ⊗ t ) , for any a, c ∈ C ( l ∩ s ), b, d ∈ C ( q ′ l ), s ∈ S l ∩ s and t ∈ S q ′ l such that b, c, d and s arehomogeneous.Up to sign, both sides of (3.9) are equal to acs ⊗ bdt . The sign on the left sideis − d deg s + deg b deg( cs ), while the sign on the right side is − b deg c + deg( bd ) deg s . Both exponents are equal todeg b deg c + deg b deg s + deg d deg s, so (3.9) is true and the action (3.8) is well deﬁned. Moreover, we have Lemma 3.10.

The exterior algebra multiplication map m : S l ∩ s ⊗ S q ′ l → S q l is an isomorphism of C ( q l ) = C ( l ∩ s ) ¯ ⊗ C ( q ′ l ) -modules, where the action of C ( q l ) on S q l is the usual one, and the action on S l ∩ s ⊗ S q ′ l is the one deﬁned by (3.8) .Proof. We already know that m is a vector space isomorphism, so we only need tocheck it respects the actions. This follows immediately from the deﬁnitions. (cid:3) It is now not diﬃcult to see that we also have(3.11) S q l = S l ∩ s ⊗ S q ′ l on the level of l ∩ h -modules, where S q l is viewed as an l ∩ h -module through themap α l , l ∩ h : l ∩ h → C ( q l ), S l ∩ s is viewed as an l ∩ h -module through the restrictionof the map α l , l ∩ k : l ∩ k → C ( l ∩ s ), and S q ′ l is viewed as an l ∩ h -module throughthe map α l ∩ k , l ∩ h : l ∩ h → C ( q ′ l ). Indeed, we have S. MEHDI AND P. PANDˇZI´C

Lemma 3.12.

Let ( G, H, L ) be a triple of type S satisfying Assumption 3.7. Thenfor any X ∈ l ∩ h , α l , l ∩ h ( X ) decomposes under C ( q l ) = C ( l ∩ s ) ¯ ⊗ C ( q ′ l ) as α l , l ∩ h ( X ) = α l , l ∩ k ( X ) ⊗ ⊗ α l ∩ k , l ∩ h ( X ) . In particular, the isomorphism of Lemma 3.10 is an isomorphism of l ∩ h -modules.Proof. Let { Z i } respectively T r be orthonormal bases of q ′ l respectively l ∩ s . Since h Z i , Z i i = − h T k , T k i = 1 for any i and k , Lemma 3.4 implies that for any X ∈ l ∩ h ,(3.13) α l , l ∩ h ( X ) = − X i

X, Z i ] ∈ l ∩ k ⊥ l ∩ s , the second sum in (3.13) is 0, i.e.,(3.14) α l , l ∩ h ( X ) = − X i

1, the lemma follows. (cid:3) Twisted Dirac operators

As before, let G be a non-compact connected semisimple real Lie group with ﬁnitecenter and with Lie algebra g , K a maximal compact subgroup of G with respect toa Cartan involution θ and H a non-compact connected semisimple closed subgroupof G with Lie algebra h such that G/H is a symmetric space with respect to aninvolution σ commuting with θ .Let U ( g C ) (resp. U ( h C )) be the enveloping algebra of the complexiﬁcation of g (resp. h ). For i = 1 ,

2, let τ i : H → GL ( F i )be ﬁnite dimensional smooth complex representations of H . Write Hom( F , F )for the vector space of complex homomorphisms from F to F . The envelopingalgebra U ( g C ) is both a right U ( h C )-module and an H -module: X · A = AXh · A = Ad( h ) A for all h ∈ H , X ∈ U ( h C ) and A ∈ U ( g C ). Write X X for the anti-automorphism of U ( h C ) deﬁned by( X · · · X n ) = ( − n X n · · · X for X j ∈ h . In particular, the vector space Hom( F , F ) is equipped with a structure of U ( h C )-module:(4.1) X · T = T ◦ dτ ( X ) , X ∈ U ( h C ) , T ∈ Hom( F , F )where dτ denotes the diﬀerential of τ extended naturally to U ( h C ). Let F i → G/H be the homogeneous vector bundle over

G/H induced by the H -module F i andwrite C ∞ ( G/H, F i ) for the space of smooth sections on which the group G actsby left translations. As a G -module, C ∞ ( G/H, F i ) is isomorphic to the space of H -invariant vectors (cid:0) C ∞ ( G ) ⊗ F i (cid:1) H . Here the G -action on C ∞ ( G ) ⊗ F i is givenby left translations on C ∞ ( G ) and trivial on F i , while the H -action is given bythe right translations on C ∞ ( G ) and by τ i on F i . Let D G/H ( F , F ) be the vector EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 9 space of left-invariant diﬀerential operators C ∞ ( G/H, F ) → C ∞ ( G/H, F ). Onehas the following isomorphism: D G/H ( F , F ) ≃ n U ( g C ) ⊗ U ( h C ) Hom( F , F ) o H where the H -action on U ( g C ) ⊗ U ( h C ) Hom( F , F ) is given by(4.2) h · ( A ⊗ T ) = (Ad( h ) A ) ⊗ τ ( h ) ◦ T ◦ τ ( h ) − When F = F , D G/H ( F , F ) is an algebra and in the case when F = F = C ,one has an isomorphism of algebras D G/H ( C , C ) ≃ U ( g C ) H /U ( g C ) H ∩ U ( g C ) h C where D G/H ( C , C ) coincides with the commutative algebra D ( G/H ) of left-invariant diﬀerential operators acting on smooth functions on

G/H .Let ( β, E ) be a ﬁnite dimensional representation of h such that the tensor product S q ⊗ E lifts to a representation ( τ, F ) of the group H . There is an associated smoothhomogeneous vector bundle over G/H , which we denote by S q ⊗ E , whose space ofsmooth sections is C ∞ ( G/H, S q ⊗ E ) ≃ n C ∞ ( G ) ⊗ ( S q ⊗ E ) o H ≃ { f : G → S q ⊗ E | f is smooth and f ( gh ) = τ ( h ) − ( f ( g )) , ∀ h ∈ H } . For X ∈ q and φ ∈ C ∞ ( G ), deﬁne the right diﬀerential of φ along X as follows:( R ( X )) ϕ )( g ) = ddt ϕ ( g exp( tX )) | t =0 . Pick an orthonormal basis { X j } of q and consider the operator(4.3) b D G/H ( E ) : C ∞ ( G/H, S q ⊗ E ) −→ C ∞ ( G/H, S q ⊗ E )deﬁned by b D G/H ( E ) = X j h X j , X j i R ( X j ) ⊗ ( γ q ( X j ) ⊗ . Note that b D G/H ( E ) is independent of the basis { X j } . One checks that b D G/H ( E )belongs to the space n U ( g C ) ⊗ U ( h C ) Hom( S q ⊗ E, S q ⊗ E ) o H and therefore deﬁnesa G -invariant diﬀerential operator acting on C ∞ ( G/H, S q ⊗ E ). b D G/H ( E ) is knownas the twisted geometric Dirac operator associated with E .There is an algebraic analog of the geometric Dirac operator. Namely, attachedto a g -module ( π, V ) there is the Dirac operator D g , h ( V ) : V ⊗ S q → V ⊗ S q with(4.4) D g , h ( V ) := X j h X j , X j i π ( X j ) ⊗ γ q ( X j ) . The operator D g , h ( V ) is independent of the choice of basis { X j } and is h -invariant. Embedding of Dirac operators

Fix a transitive triple (

G, H, L ). Recall that the spin modules S q l and S q for l ∩ h and h respectively are isomorphic as l ∩ h -modules. Let ( β, E ) be a ﬁnitedimensional representation of h such that the tensor product S q ⊗ E lifts to arepresentation ( τ, F ) of the group H . There is an associated smooth homogeneousvector bundle over both G/H and

L/L ∩ H , which we denote by S q ⊗ E → G/H and S q l ⊗ E → L/L ∩ H respectively. The spaces of smooth sections are, as above,denoted respectively by C ∞ ( G/H, S q ⊗E ) and C ∞ ( L/L ∩ H, S q l ⊗E ). The transitiveaction of L on G/H implies that restriction to L is an isomorphism of L -modules:(5.1) : C ∞ ( G/H, S q ⊗ E ) ≃ −→ C ∞ ( L/L ∩ H, S q l ⊗ E ) , f f | L . On the other hand, if U ( l C ) denotes the enveloping algebra of the complexiﬁca-tion l , the transitive action of L on G/H and the Poincar´e-Birkhoﬀ-Witt theoreminduce an isomorphism of algebras U ( g C ) ≃ U ( h C ) ⊗ U ( l C ∩ h C ) U ( l C ) . One deduces the following L ∩ H -equivariant embedding of invariant diﬀerentialoperators: ı : D G/H ( S q ⊗ E , S q ⊗ E ) ֒ → D L/L ∩ H ( S q l ⊗ E , S q l ⊗ E ) ı ( D )( ( f )) = ( D ( f )) . (5.2)Moreover, recall that under Assumption 3.7, the spin modules S q ′ l for the pair( l ∩ k , l ∩ h ) and S l ∩ s for the pair ( l , l ∩ k ) satisfy the isomorphism (3.11) of l ∩ h -modules: S q l ≃ S l ∩ s ⊗ S q ′ l . We get the following isomorphism of L -modules:Φ : C ∞ ( L/L ∩ H, S q l ⊗ E ) ≃ −→ C ∞ ( L/L ∩ K, S l ∩ s ⊗ C ∞ ( L ∩ K/L ∩ H, S q ′ l ⊗ E ))Φ( f )( l )( k ) = ( k ⊗ f ( lk )) ∀ l ∈ L, k ∈ L ∩ K (5.3)where k acts on the ﬁrst factor of S l ∩ s ⊗ ( S q ′ l ⊗ E ). Deﬁne the L ∩ K -module e E and e E the corresponding homogeneous vector bundle over L ∩ K/L ∩ H :(5.4) e E := C ∞ ( L ∩ K/L ∩ H, S q ′ l ⊗ E ) and e E → L ∩ K/L ∩ H. Next, as in (4.3), one can deﬁne the operator(5.5) b D L/L ∩ H ( E ) : C ∞ ( L/L ∩ H, S q l ⊗ E ) → C ∞ ( L/L ∩ H, S q l ⊗ E ) . Let c l , l ∩ h be the degree three element in C ( q l ) deﬁned as the image under theChevalley isomorphism of the 3-form on q l given by q l × q l × q l ∋ ( X, Y, Z )

7→ h

X , [ Y, Z ] i . If { e j } is an orthonormal basis of q l with h e j , e j i = ε j = ±

1, then(5.6) c l , l ∩ h = X i

The L -invariant diﬀerential operator b D L/L ∩ H ( E ) is the non-cubic geometric Diracoperator and(5.7) D L/L ∩ H ( E ) := b D L/L ∩ H ( E ) − ⊗ γ q l ( c l , l ∩ h ) ⊗ D l , l ∩ h ( E ) and b D l , l ∩ h ( E ) on S q l ⊗ E : D l , l ∩ h ( E ) = b D l , l ∩ h ( E ) − ⊗ γ q l ( c l , l ∩ h ) . The algebraic Dirac operator for ( h , h ∩ k ) is deﬁned analogously. Note that since G/H (resp.

K/K ∩ H ) is a symmetric space, the cubic term c g , h (resp. c h , h ∩ k ) forthe pair ( g , h ) (resp. ( h , h ∩ k )) vanishes, so that cubic and non-cubic Dirac operatorscoincide. In particular, one may write equivalently D G/H ( E ) or b D G/H ( E ). Usingthe isomorphism Φ, the operator D L/L ∩ H ( E ) may be pushed over to the right sideof (5.3) to an L -invariant diﬀerential operator:(5.8) e D L/L ∩ H ( E )(Φ( f )) = Φ( D L/L ∩ H ( E )( f )) . We will omit the tilde and still write D L/L ∩ H ( E ) for the pushed operator e D L/L ∩ H ( E ) when there is no confusion. Theorem 5.9.

For transitive triples of type S , one has: ı ( D G/H ( E )) = √ D L/L ∩ H ( E ) + (1 − √ D L ∩ K/L ∩ H ( E )+( √ − ⊗ γ q l ( c l , l ∩ h ) ⊗ ⊗ D h , h ∩ k ( E ) . Another way to express ı ( D G/H ( E )) is ı ( D G/H ( E )) = √ D L/L ∩ K ( e E )+ D L ∩ K/L ∩ H ( E ) − ⊗ γ q l ( c l , l ∩ h ) ⊗ ⊗ D h , h ∩ k ( E ) . The actions of D L/L ∩ H ( E ) and c l , l ∩ h were deﬁned above, while D L ∩ K/L ∩ H ( E ) actson e E = C ∞ ( L ∩ K/L ∩ H, S q l ⊗ E ) , D L/L ∩ K ( e E ) acts on C ∞ ( L/L ∩ K, S l ∩ s ⊗ e E ) and D h , h ∩ k ( E ) acts on S h ∩ s ⊗ E ≃ S l ∩ s ⊗ E .Proof. Recall that for triples of type S the only λ is − d λ = 1 and the only µ is 0 with d µ = √

2. We denote by { Z j } and { T k } the orthonormal bases of q ′ l and l ∩ s deﬁned in (2.4). The Dirac operator D G/H ( E ) is then given by D G/H ( E ) = − X j ρ − ( Z j ) ⊗ γ q ( ρ − ( Z j )) ⊗ X k ρ − ( T k ) ⊗ γ q ( ρ − ( T k )) ⊗ . It is an H -invariant element of U ( g C ) ⊗ U ( h C ) End( S q ⊗ E ) = U ( g C ) ⊗ U ( h C ) End( S q ) ⊗ End( E ) . We know from Section 2 that ρ − ( Z j ) = Z j , ρ + ( Z j ) = 0; ρ − ( T k ) = √ T k − ρ + ( T k ) . Using (4.1) and remembering that the h -action on S q is through the map α h of(3.2), we get D G/H ( E ) = − X j Z j ⊗ γ q ( ρ − ( Z j )) ⊗ √ X k T k ⊗ γ q ( ρ − ( T k )) ⊗ | {z } D + 1 ⊗ γ q (cid:16) X k ρ − ( T k ) α h ( ρ + ( T k )) | {z } D (cid:17) ⊗ ⊗ (cid:16) X k γ q ( ρ − ( T k )) ⊗ β ( ρ + ( T k )) (cid:17)| {z } D , where β ( ρ + ( T k )) ∈ End( E ) is the action of ρ + ( T k ) ∈ h on the h -module E .It follows that ı ( D G/H ( E )) = ı ( D ) + ı (1 ⊗ γ q ( D ) ⊗

1) + ı (1 ⊗ D ) . Recall that the embedding ı includes using the isometry ρ − : q l → q to identify theCliﬀord algebra and spin module for q l with the Cliﬀord algebra and spin module for q . Eﬀectively this means that in the Cliﬀord algebra factor of the above expressionswe replace γ q ( ρ − ( Z j )) respectively γ q ( ρ − ( T k )) with γ q l ( Z j ) respectively γ q l ( T k ).In particular, ı ( D ) = − X j Z j ⊗ γ q l ( Z j ) ⊗ √ X k T k ⊗ γ q l ( T k ) ⊗ . We now recall that b D L/L ∩ H ( E ) = − X j Z j ⊗ γ q l ( Z j ) ⊗ X k T k ⊗ γ q l ( T k ) ⊗ D L/L ∩ H ( E ) = b D L/L ∩ H ( E ) − ⊗ γ q l ( c l , l ∩ h ) ⊗ D L ∩ K/L ∩ H ( E ) = − X j Z j ⊗ γ q l ( Z j ) ⊗ D L/L ∩ K ( e E ) = X k T k ⊗ γ q l ( T k ) ⊗ . (Note that D L ∩ K/L ∩ H ( E ) = b D L ∩ K/L ∩ H ( E ) since L ∩ K/L ∩ H is symmetric fortriples of type S . Also, the deﬁnition of this operator has γ l ∩ s in place of γ q l , butthe two can be identiﬁed via the embedding of C ( l ∩ s ) into C ( q l ).)It follows that(5.10) ı ( D ) = √ (cid:0) D L/L ∩ H ( E ) + 1 ⊗ γ q l ( c l , l ∩ h ) ⊗ (cid:1) + (1 − √ D L ∩ K/L ∩ H ( E ) , and also that(5.11) ı ( D ) = √ D L/L ∩ K ( e E ) + D L ∩ K/L ∩ H ( E ) , We next compute the summand ı (1 ⊗ γ q ( D ) ⊗

1) of ı ( D G/H ). Since ρ − ( Z r ) and ρ − ( T i ) form an orthonormal basis of q , with k ρ − ( Z r ) k = − k ρ − ( T i ) k = 1, EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 13

Lemma 3.4 gives α h ( ρ + ( T k )) = − X r~~
~~

~~1) = 1 ⊗ γ q l (cid:16) X k,r,i h [ ρ + ( T k ) , ρ − ( Z r )] , ρ − ( T i ) i T k Z r T i (cid:17) ⊗ . The following lemma will help us to simplify the coeﬃcients in the above expression.~~

Lemma 5.14.

In the above setting, we have h [ ρ + ( T k ) , ρ − ( Z r )] , ρ − ( T i ) i = h [ T k , Z r ] , T i i . We postpone the proof until after we prove Theorem 5.9. Using Lemma 5.14,we rewrite (5.13) as(5.15) ı (1 ⊗ γ q ( D ) ⊗

1) = 1 ⊗ γ q l (cid:16) X k,r,i h [ T k , Z r ] , T i i T k Z r T i (cid:17) ⊗ . On the other hand, by (5.6) we know c l , l ∩ h = − (cid:16) X r~~
~~

1) = − ⊗ γ q l ( c l , l ∩ h ) ⊗ . Finally we consider the third summand of ı ( D G/H ), ı (1 ⊗ D ) = 1 ⊗ X k γ q l ( T k ) ⊗ β ( ρ + ( T k )) . We know from Section 2 that ρ + maps l ∩ s isometrically onto h ∩ s . In partic-ular, ρ + ( T k ) form an orthonormal basis of h ∩ s . Also, we can identify C ( q l ) = C ( q ′ l ) ¯ ⊗ C ( l ∩ s ) with C ( q ′ l ) ¯ ⊗ C ( h ∩ s ), and S q l = S q ′ l ⊗ S l ∩ s with S q ′ l ⊗ S h ∩ s . In thisway we see that we can identify(5.17) ı (1 ⊗ D ) = D h , h ∩ k ( E ) , with D h , h ∩ k ( E ) acting on S h ∩ s ⊗ E .We now add up (5.10), (5.16) and (5.17) to get the ﬁrst claim of Theorem 5.9,or (5.11), (5.16) and (5.17) to get the second claim of Theorem 5.9. This ﬁnishesthe proof of Theorem 5.9 modulo Lemma 5.14. (cid:3) Proof of Lemma 5.14.

Let ω denote the invariant trilinear alternating form on g given by ω ( X, Y, Z ) = h [ X, Y ] , Z i . Suppose X ∈ l ( ν ), Y ∈ l ( ν ′ ) and Z ∈ l ( ν ′′ ). We claim that(5.18) ω ( ρ + ( X ) , ρ − ( Y ) , ρ − ( Z )) = 14 d ν d ν ′ d ν ′′ (1 + ν − ν ′ − ν ′′ ) ω ( X, Y, Z ) . If we prove this claim, the lemma follows immediately by applying the claim to X = T k , Y = Z r , Z = T i , since in that case ν = ν ′′ = 0; ν ′ = − d ν = d ν ′′ = √ d ν ′ = 1 . To prove (5.18), we ﬁrst recall that ρ + ( X ) = d ν X + σ ( X )) , ρ − ( Y ) = d ν ′ Y − σ ( Y )) , ρ − ( Z ) = d ν ′′ Z − σ ( Z )) , so (5.18) will follow if we prove(5.19) ω ( X + σ ( X ) , Y − σ ( Y ) , Z − σ ( Z )) = 2 ω ( X, Y, Z ) . By deﬁnition of l ( ν ),(5.20) ω ( σ ( X ) , Y, Z ) = h [ σ ( X ) , Y ] , Z i = h σ ( X ) , [ Y, Z ] | {z } ∈ l i = ν h X, [ Y, Z ] i = ν ω ( X, Y, Z ) . Since ω is skew symmetric, this implies that also(5.21) ω ( X, σ ( Y ) , Z ) = ν ′ ω ( X, Y, Z ) , ω ( X, Y, σ ( Z )) = ν ′′ ω ( X, Y, Z ) . The σ -invariance of ω now implies(5.22) ω ( σ ( X ) , σ ( Y ) , Z ) = ω ( X, Y, σ ( Z )) = ν ′′ ω ( X, Y, Z ) , so by the skew symmetry of ω also(5.23) ω ( σ ( X ) , Y, σ ( Z )) = ν ′ ω ( X, Y, Z ) , ω ( X, σ ( Y ) , σ ( Z )) = ν ω ( X, Y, Z ) . The equation (5.19) follows immediately from (5.20) – (5.23). (cid:3)

Remark . Similar formulas for ı ( D G/H ( E )) can also be obtained for triples( G, H, L ) of type I or N . They are however more complicated, and therefore lesslikely to have nice applications. Especially, we do not have a nice interpretation asabove for the operator D and its action seems hard to understand.If however the subgroup H is simply connected, the spin module S q admits anaction of H and one can choose the h -module E to be the trivial representation.In this case, the operator D is zero and the above problem with interpretationdisappears. EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 15 Example: ( G, H, L ) = ( G ′ × G ′ , ∆( G ′ × G ′ ) , G ′ × K ′ ) for G ′ = SL (2 , R ) and K ′ = SO (2)Let ( G, H, L ) be a triple ( G ′ × G ′ , ∆( G ′ × G ′ ) , G ′ × K ′ ), where G ′ is a non-compactconnected semisimple real Lie group with ﬁnite center, with maximal compactsubgroup K ′ , G = G ′ × G ′ , H = ∆( G ′ × G ′ ) and L = G ′ × K ′ . Note that themaximal compact subgroup of G is K = K ′ × K ′ . Moreover, L ∩ H = ∆( K ′ × K ′ ),and L ∩ K = K .If g ′ = k ′ ⊕ s ′ is the Cartan decomposition of the Lie algebra of G ′ , then we have g = g ′ × g ′ , h = ∆( g ′ × g ′ ) , l = g ′ × k ′ , and furthermore q = ∆ − ( g ′ × g ′ ) , q l = q ′ l ⊕ ( l ∩ s ) = ∆ − ( k ′ × k ′ ) ⊕ ( s ′ × h ∩ k = ∆( k ′ × k ′ ) = l ∩ h , h ∩ s = ∆( s ′ × s ′ ); q ∩ k = ∆ − ( k ′ × k ′ ) , q ∩ s = ∆ − ( s ′ × s ′ ); l ∩ k = k , l ∩ s = s ′ × − denotes the anti-diagonal. From now on we let G ′ = SL (2 , R ) and K ′ = SO (2). Let h = (cid:18) − ii (cid:19) , e = 12 (cid:18) ii − (cid:19) , f = 12 (cid:18) − i − i − (cid:19) be the usual sl (2)-basis of g ′ C = sl (2 , C ), with commutators[ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h. We consider the trace form h , i on g ′ and the direct sum form on g . It is easy tocheck that ˜ h = i √ h, ˜ e = 1 √ e + f ) , ˜ f = i √ e − f )form an orthonormal basis of g ′ , i.e., they are orthogonal to each other and satisfy h ˜ h, ˜ h i = − , h ˜ e, ˜ e i = h ˜ f , ˜ f i = 1 . So if we set Z = 1 √ h, − ˜ h ) , T = (˜ e, , T = ( ˜ f , , we see that Z forms an orthonormal basis of q ′ l (i.e., h Z, Z i = − T and T form an orthonormal basis of l ∩ s (i.e., h T , T i = h T , T i = 1, h T , T i = 0).Since L ∩ K is σ -stable, then λ = −

1, i.e., Z ∈ q and d λ = 1. Moreover, l ∩ s isirreducible under the action of L ∩ H , with µ = 0 and d µ = √

2. We see immediatelythat ρ + ( Z ) = 0 , ρ − ( Z ) = Z ;(6.1) ρ + ( T ) = 1 √ e, ˜ e ) , ρ − ( T ) = 1 √ e, − ˜ e );(6.2) ρ + ( T ) = 1 √ f , ˜ f ) , ρ − ( T ) = 1 √ f , − ˜ f ) . (6.3)Since SL (2 , R ) acts on any ﬁnite-dimensional representation of sl (2 , C ), we maychoose E to be any ﬁnite-dimensional representation of ∆( sl (2 , C ) × sl (2 , C )). By the second statement of Theorem 5.9, the embedding of D G/H ( E ) can be writtenas(6.4) ı ( D G/H ( E )) = √ D L/L ∩ K ( e E ) + D L ∩ K/L ∩ H ( E ) − ⊗ γ q l ( c l , l ∩ h ) ⊗ ⊗ D h , h ∩ k ( E ) . This is an equality of operators acting on the space(6.5) C ∞ ( L, S l ∩ s ⊗ C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H ) L ∩ K . The L ∩ H -invariants are taken with respect to the right translation on C ∞ ( L ∩ K )and lift of l ∩ h -action on S q ′ l ⊗ E and the L ∩ K -invariants are taken with respectto the right translation on C ∞ ( L ) and lift of l ∩ k -action on S l ∩ s ⊗ e E . Recall thatwe are using the isometry ρ + : l ∩ s → h ∩ s to identify S l ∩ s with S h ∩ s .We ﬁrst consider the action of the summand − ⊗ γ q l ( c l , l ∩ h ) ⊗ c l , l ∩ h = − ZT T , so we have to determine the action of 2 ZT T on the spin module S q l = S l ∩ s ⊗ S q ′ l .To construct the spin module we use the dual isotropic vectors u = 1 √ T + iT ) , v = 1 √ T − iT ) . Then S q l is spanned by 1 and u , and the action of u and v is given by u · u, u · u = 0; v · , v · u = 1 . Since T = √ ( u + v ) and T = − i √ ( u − v ), they act by(6.6) T · √ u, T · u = 1 √ T · − i √ u, T · u = i √ . Finally, Z acts by ± i/ √ S ± , so Z · i √ , Z · u = − i √ u. A straightforward computation now proves

Lemma 6.7.

The summand − ⊗ γ q l ( c l , l ∩ h ) ⊗ ⊗ γ q l (2 ZT T ) ⊗ of the expression for ı ( D G/H ( E )) given by (6.4) acts by the scalar √ on the wholespace (6.5) . To understand the action of the other summands of (6.4), we describe e E = C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H more explicitly. The group L ∩ K = K ′ × K ′ = SO (2) × SO (2)is abelian, compact and connected, and its Lie algebra is l ∩ k = l ∩ h ⊕ q ′ l . EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 17

The Lie algebra l ∩ h is one-dimensional, spanned by W = 1 √ h, ˜ h ) . Furthermore, W and Z form an orthonormal basis for the abelian Lie algebra l ∩ k .We denote by C a,b the one-dimensional character of L ∩ K on which ( h,

0) acts bythe scalar a and (0 , h ) acts by the scalar b . Since h ∈ k ′ C acts by integers on thecharacters of K ′ = SO (2), a and b are integers. Furthermore, W = i ( h, h ) acts on C a,b by the scalar i ( a + b )2 , while Z = i ( h, − h ) acts on C a,b by the scalar i ( a − b )2 .Since the space of smooth vectors in L ( L ∩ K ) coincides with C ∞ ( L ∩ K ) andthe dual of C a,b is C − a, − b , Peter-Weyl theorem implies that C ∞ ( L ∩ K ) = M a,b C − a, − b ⊗ C a,b with the left regular action of L ∩ K realized on the ﬁrst factor of each summand,and the right regular action of L ∩ K realized on the second factor of each summand.Here L denotes the closure of the algebraic direct sum in the smooth topology (see[MO2] for more detail in a more general setting.) Since the L ∩ H -invariants aretaken with respect to the right regular action, we have (up to passing to a densesubspace)(6.8) C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H = M a,b C − a, − b ⊗ ( C a,b ⊗ S q ′ l ⊗ E ) L ∩ H . Since L ∩ H is connected, and l ∩ h is spanned by ( h, h ), the L ∩ H -invariants aresimply the 0-eigenspace of ( h, h ). The map α l ∩ k , l ∩ h : l ∩ h → C ( q ′ l )is zero since l ∩ k is abelian, so ( h, h ) acts by 0 on S q ′ l . Furthermore, ( h, h ) actsby a + b on C a,b . It follows that taking the L ∩ H -invariants in (6.8) amounts topairing each C a,b with the ( − a − b )-weight space of E . So (6.8) can be rewritten as(6.9) C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H = M a,b ; a + b ∈ ∆( E ) C − a, − b ⊗ C a,b ⊗ S q ′ l ⊗ E − a − b , with ∆( E ) denoting the set of weights of E . So the space (6.5) can be identiﬁedwith(6.10) C ∞ (cid:2) L, S l ∩ s ⊗ (cid:0) M a,b ; a + b ∈ ∆( E ) C − a, − b ⊗ C a,b ⊗ S q ′ l ⊗ E − a − b (cid:1)(cid:3) L ∩ K . We now consider the summand(6.11) D L ∩ K/L ∩ H ( E ) = − Z ⊗ Z of (6.4). Recall that the ﬁrst factor Z is acting on C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H by theright regular action, while the second factor Z is acting on S q ′ l . So the action of D L ∩ K/L ∩ H ( E ) on the space (6.10) is on C a,b ⊗ S q ′ l .Since the spin module S q ′ l is one-dimensional, with Z ∈ C ( q ′ l ) acting by the scalar i √ , and since Z operates on C a,b by the scalar i ( a − b )2 , the operator D L ∩ K/L ∩ H ( E ) given by (6.11) acts on C a,b ⊗ S q ′ l by the scalar a − b √ . In this way we have diagonalizedthe action of D L ∩ K/L ∩ H ( E ) on the space (6.10): the eigenvalues are k √ , k ∈ Z , and the eigenspace corresponding to k is the space(6.12) C ∞ (cid:2) L, S l ∩ s ⊗ (cid:0) M a,b ; a − b = k, a + b ∈ ∆( E ) C − a, − b ⊗ C a,b ⊗ S q ′ l ⊗ E − a − b (cid:1)(cid:3) L ∩ K . In particular, if we set k = − Corollary 6.13.

The part D L ∩ K/L ∩ H ( E ) − ⊗ γ q l ( c l , l ∩ h ) ⊗ of the expression for ı ( D G/H ( E )) given by (6.4) acts by 0 on the subspace C ∞ (cid:2) L, S l ∩ s ⊗ (cid:0) M a,b ; a − b = − , a + b ∈ ∆( E ) C − a, − b ⊗ C a,b ⊗ S q ′ l ⊗ E − a − b (cid:1)(cid:3) L ∩ K . of the space (6.10) . This subspace is nonzero if and only if E has even weights. We now turn our attention to the summand D h , h ∩ k ( E ) of the expression for ı ( D G/H ( E )) given by (6.4). Identifying h with g ′ = sl (2 , R ), we can use the basis e, f, h of g ′ C to write D h , h ∩ k ( E ) = e ⊗ f + f ⊗ e with the second factor acting on S h ∩ s = C ⊕ C e in the usual way. If E has highest weight 2 m , m ∈ Z + , then it is well known andeasy to see that ker D h , h ∩ k ( E ) = ( E m ⊗ e ) ⊕ ( E − m ⊗ . Identifying S h ∩ s with S l ∩ s , the basis of the spin module becomes { , u } , and weconclude Corollary 6.14.

If the highest weight of E is m , m ∈ Z + , then the operators ı ( D G/H ( E )) and √ D L/L ∩ K ( e E ) are equal on the subspace C ∞ (cid:0) L, u ⊗ C m +1 ,m − ⊗ C − m − , − m +1 ⊗ S q ′ l ⊗ E m L ⊗ C − m +1 , − m − ⊗ C m − ,m +1 ⊗ S q ′ l ⊗ E − m (cid:1) L ∩ K of the space (6.10) . In particular, ı ( D G/H ( E )) and D L/L ∩ K ( e E ) have the samekernel on this subspace. We now discuss the kernel of D L/L ∩ K ( e E ) on the subspace of the space (6.10)described in Corollary 6.14. We ﬁrst recall that in the deﬁnition of the space C ∞ ( L, S l ∩ s ⊗ e E ) L ∩ K ∼ = ( C ∞ ( L ) ⊗ S l ∩ s ⊗ e E ) L ∩ K , the L ∩ K -invariants are taken with respect to the right regular action on C ∞ ( L ),the spin action on S l ∩ s , and the left regular action on e E = C ∞ ( L ∩ K, S q ′ l ⊗ E ) L ∩ H . EPRESENTATION THEORETIC EMBEDDING OF TWISTED DIRAC OPERATORS 19

This means that we can identify the subspace described in Corollary 6.14 with(6.15) C ∞ (cid:0) L, u ⊗ C m +1 ,m − L ⊗ C − m +1 , − m − (cid:1) L ∩ K ;the omitted factors are one-dimensional and carry no relevant action.We now use L = G ′ × K ′ and L ∩ K = K ′ × K ′ to rewrite the space (6.15) as(6.16) C ∞ (cid:0) G ′ , u ⊗ C m +1 ) K ′ ⊗ C ∞ (cid:0) K ′ , C m − ) K ′ L C ∞ ( G ′ , ⊗ C − m +1 ) K ′ ⊗ C ∞ (cid:0) K ′ , C − m − ) K ′ . Here and below for any k ∈ Z , C k denotes the character of K ′ on which h acts by k . Peter-Weyl Theorem for K ′ implies that for any such character C k , C ∞ (cid:0) K ′ , C k ) K ′ = M n ∈ Z C − n ⊗ ( C n ⊗ C k ) K ′ = C k . So (6.16) becomes(6.17) C ∞ (cid:0) G ′ , u ⊗ C m +1 ) K ′ ⊗ C m − L C ∞ ( G ′ , ⊗ C − m +1 ) K ′ ⊗ C − m − . Note that D L/L ∩ K ( e E ) = D G ′ /K ′ is acting on the ﬁrst factors in (6.17). To deter-mine the kernel of this operator, we use a reciprocity result of [MZ]. More precisely,let V be a smooth admissible representation of G ′ , V K ′ the space of K ′ -ﬁnite vec-tors and V ∗ K ′ the K ′ -ﬁnite dual of V K ′ . Let S s ′ be the spin module for k ′ and F aﬁnite dimensional representation of k ′ such that S s ′ ⊗ F lifts to a K ′ -representation.Proposition 1.8 of [MZ] says that the mapΨ : Hom G ′ ( V, C ∞ ( G ′ /K ′ , S s ′ ⊗ F )) → Hom K ′ ( F ∗ , S s ′ ⊗ V ∗ K ′ )deﬁned by Ψ( T )( f ∗ )( v ) = (id S s ′ ⊗ f ∗ )[ T ( v )(1)] , where we identify S s ′ ⊗ V ∗ K ′ with Hom( V K ′ , S s ′ ), induces an isomorphism(6.18) Hom G ′ ( V, ker( D G ′ /K ′ ( F ))) ≃ Hom K ′ ( F ∗ , ker( D V ∗ K ′ ))where F ∗ is the dual of F . This isomorphism does not involve the spin module, sowe can replace the spin module by any of its irreducible components and still havethe same statement. This means that the kernel of D G ′ /K ′ on C ∞ (cid:0) G ′ , u ⊗ C m +1 ) K ′ contains irreducible representations ( π, V π ) of G ′ with the property that the kernelof D g ′ , k ′ on u ⊗ V ∗ π contains C ∗ m +1 = C − m − . There is exactly one such π ∗ : theirreducible representation with highest weight − m −

2. So π is the unique irreduciblerepresentation with lowest weight m + 2, i.e., the discrete series representation withlowest weight m +2 (recall that m ≥

0, so m +2 ≥ DS + m +2 .Analogously, the kernel of D G ′ /K ′ on C ∞ (cid:0) G ′ , ⊗ C − m +1 ) K ′ contains irreduciblerepresentations ( π, V π ) of G ′ with the property that the kernel of D g ′ , k ′ on 1 ⊗ V ∗ π contains C ∗− m +1 = C m − . There is exactly one such π ∗ : the irreducible represen-tation with lowest weight m . The corresponding π is the irreducible representationwith highest weight − m , so it is the discrete series representation DS −− m with high-est weight − m if m ≥

2, the limit of discrete series representation

LDS − withhighest weight − m = 1, or the trivial representation C if m = 0.The above discussion and (6.17) imply Theorem 6.19.

Suppose that the highest weight of E is m , m ∈ Z + . Then thekernel of D L/L ∩ K ( e E ) on C ∞ ( L, S l ∩ s ⊗ e E ) L ∩ K contains the following representationsof L = G ′ × K ′ : ( DS + m +2 ⊗ C m − ) ⊕ ( DS −− m ⊗ C − m − ) if m ≥ DS +3 ⊗ C ) ⊕ ( LDS − ⊗ C − ) if m = 1;( DS +2 ⊗ C − ) ⊕ ( C ⊗ C − ) if m = 0 . It follows that the same L -representations appear in the kernel of ı ( D G/H ), andthus also in the kernel of D G/H . On the other hand, the kernel of D G/H is arepresentation of G and not only of L , so our L -representations inside ker( D G/H )generate G -representations inside ker( D G/H ). t is possible (but quite complicated)to identify the representations of G in ker( D G/H ) whose restrictions to L containthe representations described in Theorem 6.19. References [G] S. Goette,

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Math. USSR-Sb. (1969), 515–554 Institut Elie Cartan de Lorraine, CNRS - UMR 7502, Universit´e de Lorraine, France

Email address : [email protected] Department of Mathematics, Faculty of Science, University of Zagreb, Croatia

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