The Bernstein-Gelfand-Gelfand complex and Kasparov theory for SL(3,C)

The Bernstein-Gelfand-Gelfand complex and Kasparo v theory for SL(3 , C ) Rob ert Y unc k en No v em ber 3, 2018 Abstract F or G = S L ( 3 , C ), w e construct an elemen t of G -eq uiv a riant analytic K -h omolo gy from th e Bernstein-Gelfand-Gelfand complex for G . This furnishes an explicit splitting of the restriction map from th e Kasparov representa tion rin g R ( G ) to the represen tation ring R ( K ) of its maximal compact subgroup SU(3), and the splitting factors through t h e e quiv arian t K -h omolo gy of th e ﬂag v arie ty X of G . In particular, w e obtain a new mod el for the γ -element of G . The constru ction is made using SU(3)-harmonic analysis associated to the canonical ﬁbrations of X . On this matter, w e prov e results which demonstrate the compatibility of b oth the G -action and the order zero longitudinal p seudodiﬀerential operators with the SU(3)-harmonic analy- sis. 1 In tro duction A typical source for the construction of a Kaspa rov K -homology cycle is an elliptic diﬀerential complex. If the elliptic complex is equiv arian t with resp ect to the a ction of a gro up G , and if moreov er the group actio n satisﬁes an additional conformality prop ert y (see b elo w), then one can o btain an ele ment o f equiv arian t K -homolo gy . But if G is a semisimple Lie g roup o f rank greater than one, non- trivial examples o f such complexes cannot ex ist ([Pus08]). This pap er des c ribes a means of constructing a n equiv arian t K -ho mo logy cla ss from the Ber nstein- Gelfand-Gelfand complex for SL(3 , C )—a diﬀerential complex which is neither elliptic no r conforma l, but which satisﬁes some weaker (‘dir e ctional’) for m of these co nditions. The motiv ation for this construction comes from the Baum-Connes conjec- ture. Although an understanding of the conjecture is not e ssen tial to this pap er, it is use ful for p ersp ective. The co njecture a sserts that for a second countable lo cally compact gr oup G , the assembly map µ Γ : K Γ ( E Γ) → K ( C ∗ r Γ) . is an isomorphism, th us g iving a ‘to p ologica l computation’ o f the K -theory of the r educed gr oup C ∗ -algebra . F or a fuller desc ription of the conjectur e and its man y conseq uences, we refer the re a der to the exp ository article [Hig98] and the foundationa l paper [BCH94]. 1 The conjecture ha s been proven for a wide clas s o f groups, amongst which we men tion in particular the discr ete subgro ups of s imple Lie groups of real rank one. A no table unknown, howev er, is the gr o up SL(3 , Z ). Mo r e broadly , the conjecture is unknown for gener a l discrete subgr oups of semisimple Lie groups of r a nk greater than o ne. F or subgroups of rank one semisimple gro ups G , the pro ofs in each case cen tre on a canonica l idempotent γ in the representation ring R ( G ) := K K G ( C , C ). (See [Kas84] for G = SO 0 ( n, 1), [JK95] for G = SU( n, 1), [Jul02] 1 for Sp( n, 1)). F or our purp oses, the most conv enient wa y to descr ib e this idemp oten t γ is via the following fact. Theorem 1.1 (Kasparov) . L et G b e a semisimple Li e gr oup and K a m aximal c omp act sub gr oup. The r estriction map R ( G ) → R ( K ) is a split surje ction of rings. The unit in R ( K ) is the clas s of the trivial K -r epresen tation, and its ima ge under the splitting is an idemp otent in R ( G ). This is γ . If γ = 1 ∈ R ( G ) then the restriction ma p is an iso morphism. In this case, the ‘Dirac-dual Dirac method’ of Kaspa ro v implies that the Baum-Connes con- jecture ho lds for all dis c r ete subgro ups of G . This is the appro ac h taken in the pap ers c ited ab o ve, although in the case of Sp( n, 1) a weak er notion of ‘triviality’ for γ must be used. The idemp oten t γ was o r iginally deﬁned via equiv ar ian t K -homology for the prop er G -space G / K ([Ka s 88]). In the rank-one pro ofs men tioned ab ov e, ho w- ever, γ is mo r e co n veniently constructed using the compact space G / B , where B is a minimal parab olic subg roup. This can be explained by the fact that the in- duced repres e n tatio ns from B give a natural top ologica l parameteriz ation of (the relev ant s ubs et of ) repr esen tations of G , namely the gene r alized principa l ser ies, including the complementary series. It is also p ertinent that B is amenable, so itself sa tis ﬁes Baum-Connes. It is instructive to consider the construction of γ in the simple example G = SL(2 , C ). One b egins with the Dolb eault complex for the homogeneous space G / B ∼ = C P 1 : Ω 0 , 0 C P 1 ∂ − → Ω 0 , 1 C P 1 . This is a G -equiv ariant elliptic co mplex. Importantly , though, C P 1 do es not admit a G -inv ariant Riemannian metric. The action is confor mal (with r espect to the natural K -equiv ariant metric), a nd the translatio n repre s en ta tion of G on L 2 Ω 0 , • C P 1 can b e made unitary by the intro duction of a scalar Rado n- Nikodym factor . But the op erator D := ∂ + ∂ ∗ will no t be G -equiv ariant, not even in the weak sense of deﬁning an unbounded equiv ariant F redholm mo dule. Somewhat magically though, replacing D b y its op erator phase results in a bo unded equiv ariant F redholm mo dule. F or this to work it is crucial that the G -action is conformal 2 on the Hermitian bundles Ω 0 ,p C P 1 . In order to maintain this cr ucial conforma lity pro perty for the other r a nk one cases, one m us t use increasingly complicated sub ellitpic diﬀerential complexes — the Rumin complex for SU( n, 1); a quater nionic analo gue thereof for Sp( n, 1) 1 The ﬁrst proof of Baum-Connes for discrete subgroups of Sp( n, 1) was due to V. Laﬀorgue, but used a somewhat diﬀerent approac h. 2 In general, the conformality r equiremen t is even stronger: the ratio of the Radon-Nik odym factors in degrees p and p + 1 must b e indep ende nt of p . W e will not explain this further. 2 — and cor respo nding nonstandard pseudo diﬀeren tial calculi. W e remark that K -homolo gical constructions using even nonstandard pseudo diﬀeren tial calculi t ypically result in ﬁnitely-summable F r edholm mo dules. Pus chnigg [Pus08] has shown that simple Lie groups of higher r ank do not admit any nontrivial ﬁnitely summable F redholm mo dules. This motiv ates o ur construction using the Be r nstein-Gelfand-Gelfand (‘BGG’) complex. Theorem 1.2 (Bernstein-Gelfand-Gelfand) . L et G b e a c omplex semisimple gr oup and B a minimal p ar ab olic sub gr oup. F or any ﬁnite dimensional holomor- phic r epr esentation V of G ther e is a diﬀer ential c omplex, c onsisting of dir e ct sums of homo gene ous line bund les over G / B and G -e quivariant diﬀer ential op- er ators b etwe en them, whi ch r esolves V . The bundles in eac h degr ee here are not co nfo r mal, but their component line bundles are individually co nformal. (T rivially , a n y group action o n a Her mitian line bundle is co nformal.) The q uestion is whether this structure is eno ugh to pro duce an element of equiv ar ian t K -ho mo logy . In this paper , we answer this question a ﬃrmativ ely in the case of G = SL(3 , C ). W e thereb y obtain a n explicit construction of the splitting map R ( K ) → R ( G ), and in particular a construction o f γ , w hich factors through K K G ( C ( G / B ) , C ). The constructio n is based upo n ha rmonic analysis o f SU(3) ra ther than some nonstandard pse udo diﬀerential calculus. An indication of the diﬃculties of a purely pseudo diﬀeren tial approach is given in Chapter 5 of [Y un06]. In fact, our construction could be made without an y reference to pseudo diﬀeren tial op- erators at all, thoug h pseudo diﬀerential theory has b ecome so central to index theory that to do so might seem somewha t eccen tric. Much of the requir e d harmonic analysis has b een developed in [Y un] in the broader context of SU( n ) ( n ≥ 2). W e exp ect that the results o f this pa p er should b e extendable the groups SL( n, C ), and indeed to complex semisimple groups in g eneral. The main tec hnical diﬃculty in the ca se of SL( n, C ) is a n appropria te version of the the op erator partition of unity of Lemma 4.14 of this pap er. F or general semisimple groups, the require d dir ectional harmonic analysis is yet to b e developed. As for the Ba um-Connes Co njecture itse lf, it is kno wn that γ 6 = 1 for an y group G which ha s Ka zhdan’s prop erty T . Therefor e , a direct transla tion of Kaspar o v’s method cannot prov e the Baum-Connes co njecture for simple Lie groups of rank greater than one—some subtle v ariation of Kasparov’s argument would be required. Nevertheless, it is exp ected that the present construction will b e useful for further study of the Ba um-Connes conjecture. Let us now describ e the BGG complex in more detail. In fact, knowledge of the cohomolog ical version of the BGG co mplex is unnecessary for the presen t pap er, since our K - homological version will be pro duced from scra tc h. But it is such a stro ng motiv ation that it is w orth sp ending some time e xplaining it. Finite dimensional holomor phic r epresentations of G a re parameterized b y their hig hest w eights. Let V λ denote the repres e n tatio n with highest weigh t λ . An y w eight µ of G extends to a holomorphic c harac ter of B (see Section 2.2), and w e denote b y L hol µ the corresp onding induced holo mo rphic line bundle ov er X := G / B . The Borel-W eil Theorem states that V λ is equiv ar ian tly isomorphic to the s pace of glo bal holomorphic sections of L hol λ . 3 Recall that the W eyl gro up W is a g r oup of reﬂections on the weight space. It is gener a ted by the simple r eﬂe ct ions —r eﬂections in the w a lls orthog onal to a choice of simple ro ots for G . W or d le ng th in these generators deﬁnes a length function l : W → N . W e need the shifte d action of the W eyl group deﬁned b y the form ula w ⋆ µ := w ( µ + ρ ) − ρ , where ρ is the half-sum of the p ositiv e roo ts. Bernstein, Gelfa nd and Gelfand [BGG75] showed that ther e is a holomorphic G -equiv aria nt diﬀerent ial opera tor from L hol µ to L hol ν if a nd only if µ = w ⋆ λ and ν = w ′ ⋆ λ for some dominan t weigh t λ and so me w, w ′ ∈ W with l ( w ′ ) ≥ l ( w ). What is mo r e, these oper ators can be assembled into an exact complex as follows 3 . One deﬁnes the degree p cocy cle spa ce C p := L l ( w )= p C ∞ ( X , L hol w⋆λ ). The colle c tio n of eq uiv ariant diﬀerential op erators betw een a n y L hol w⋆λ and L hol w ′ ⋆λ with l ( w ) = p , l ( w ′ ) = p + 1 deﬁnes a matrix o f op erators C p → C p +1 . With an appr opriate choice of sig ns these operato rs resolve the Borel-W eil inclusion V λ ֒ → C ∞ ( X ; L hol λ ). In the ca se of SL(3 , C ), we get a complex C ∞ ( X ; L hol w α 1 ⋆λ ) / / " " F F F F F F F F F F F F F F F F F F F F F F L C ∞ ( X ; L hol w α 1 w α 2 ⋆λ ) " " F F F F F F F F L V λ ֒ → C ∞ ( X ; L hol λ ) < < x x x x x x x x x " " F F F F F F F F F C ∞ ( X ; L hol w ρ ⋆λ ) C ∞ ( X ; L hol w α 2 ⋆λ ) / / < < x x x x x x x x x x x x x x x x x x x x x x C ∞ ( X ; L hol w α 2 w α 1 ⋆λ ) < < x x x x x x x x (1.1) where α 1 , α 2 and ρ = α 1 + α 2 are the p ositive ro o ts, a nd w α denotes the reﬂection in the wall orthog onal to α . In this pa per, we deﬁne a ‘nor malized’, i.e. , L 2 -b ounded, version of this complex which is analo gous to the equiv a rian t F redholm mo dule constructed ab o ve fro m the Dolb eault complex of C P 1 . T o complete this ov erview, we give a very brief description of the ha rmonic analysis upo n whic h our K -ho mological BGG construction is based. The space X := G / B is the complete ﬂag v a riet y of C 3 . Corresp onding to the simple ro ots α 1 and α 2 , there are G -equiv ariant ﬁbrations X → X i ( i = 1 , 2 ) where X 1 and X 2 are the Grassmannia ns of lines and planes in C 3 . As descr ibed in [Y un], a s socia ted to each of thes e ﬁbratio ns is a C ∗ -algebra K α i of op erator s on the L 2 -section s pace of an y homog e neous line bundle over X . This algebra contains, in particular, the longitudinal pseudodiﬀer en tia l op erators o f negative order tangent to the giv en ﬁbration. A k ey prop ert y is that the intersection K α 1 ∩ K α 2 consists of co mpa ct opera tors. Ultimately , this a llows us to apply the Kaspa rov T echnical The o rem to construct a F redholm mo dule fro m the normalized B GG op erators. The str uc tur e of the pa per is a s follows. Section 2 gives the background o n the structur e theory o f the semisimple Lie gro up G = SL(3 , C ), the ﬂa g v ariety X and its homogeneous line bundles, mainly for the pur p ose o f s etting notation. 3 Strictly sp eaking, Bernstein, Gelfand and Gelfand made a homological complex by assem- bling intert wi ners b et ween V erma modules. What we are calling the BGG complex here is a dual cohomological co mplex. See the appendix of [ ˇ CSS01] for an explanation of this. 4 In Section 3 we review the C ∗ -algebra s K α i of [Y un ] and their rela tion to longitudinal pseudodiﬀerential oper ators on the ﬂag v a riet y X . W e also prov e t wo importa n t new results co ncerning these algebra s. F or the sake of stating these res ults elegantly , it is conv enient to place the C ∗ -algebra s K α i in the context of C ∗ -categor ies (see Section 3 .1 for details). Theorem 1.3. L et E , E ′ b e G -homo gene ous line bund les over X . L et A denote the simultane ous multiplier c ate gory of K α 1 and K α 2 (se e Deﬁnition 3.8). (i) The t r anslation op er ators g : L 2 ( X ; E ) → L 2 ( X ; E ) b elong to A , for al l g ∈ G . (ii) If T : L 2 ( X ; E ) → L 2 ( X ; E ′ ) is a longitudinal pseudo diﬀer ential op er ator of or der zer o tangent to one of t he ﬁbr ations X → X i ( i = 1 , 2 ), then T ∈ A . Theorem 1.3(i) is prov en in Section 3.2. Part (ii) is restated in Theo rem 3.1 8 . The pro of requires so me lengthy computations in SU(3) harmonic analysis which are pre sen ted in Appe ndix A. In Section 4, w e co m bine the abov e results to constr uct an elemen t of K K G ( C ( X ) , C ) from the BGG complex. W e also explain wh y this yields the splitting o f the restr iction morphism R ( G ) → R ( K ). Part of this w o rk appe ared in the author ’s do ctoral dis sertation [Y un06]. I would lik e to tha nk my thesis adviser , Nigel Higson. I would also like to thank Erik K oelink fo r several informative conversations. 2 Notation and Preli minaries 2.1 Lie groups Throughout this pape r G will denote the gro up SL(3 , C ). W e ﬁx notatio n for the following subgr o ups: K = SU(3), its maximal compact subgro up; H , the Cartan subgr oup of diagonal matrices; A , the subgroup of diago nal matric e s with p ositive r eal en tries; M = H ∩ K , the max imal torus of K ; N , the subgroup of upper triangula r unip o ten t ma tr ices; and B = MAN the subgroup of upp er triangular matr ices. Their Lie a lgebras are denoted g , k , h , a , m , n and b . W e use V † to denote the dual of a complex v ecto r space V . W e make the usual iden tiﬁcations o f the c o mplexiﬁcations m C and a C with h by extending the inclusions a , m ֒ → h to C -linear maps. W e thereby identif y characters of A and M with elements of h † . Chara cters o f h will be denoted b y χ = χ M ⊕ χ A , where χ M and χ A are the restrictions of χ to m and a , resp ectively . The corre s ponding group character of H will be denoted e χ . The weigh t lattice in m † C ∼ = h † will b e denoted by Λ W . The set of ro ots of K is denoted ∆. W e ﬁx the notation X α 1 =   0 1 0 0 0 0 0 0 0   , X α 2 =   0 0 0 0 0 1 0 0 0   , X ρ =   0 0 1 0 0 0 0 0 0   ∈ k C ∼ = g , which ar e ro ot vectors for the ro ots α 1 , α 2 and ρ := α 1 + α 2 . W e ﬁx these as our set of p ositiv e ro ots ∆ + , so Σ := { α 1 , α 2 } is the set of simple ro ots. F or 5 each α ∈ ∆ + , Y α will denote the transp ose of X α . W e abbreviate X α i and Y α i to X i and Y i , whenever conv enient. W e put H i := [ X i , Y i ] ∈ m C . The element s X i , Y i , H i span a Lie subalgebr a isomorphic to sl (2 , C ), which we denote by s i . W e also put H ′ 1 :=   1 0 0 0 1 0 0 0 − 2   , H ′ 2 :=   − 2 0 0 0 1 0 0 0 1   ∈ m C ∼ = h , so that for ﬁxed i = 1 , 2, H i and H ′ i span h and H ′ i commutes with s i . The W eyl group of G is W ∼ = S 3 . W e let w α denote the r eﬂection in the wall orthogo nal to the ro ot α . The s imple r eﬂections w α 1 and w α 2 are genera tors of W , and the minimal word leng th in these genera tors deﬁnes the leng th function l on W . 2.2 Homogeneous v ector bundles Throughout, X will denote the ho mogeneous space X = G / B = K / M . Let χ = χ M ⊕ χ A be a c har acter o f h . As usual, we extend it trivially on n to a character of b . W e use L χ to denote the G -homog eneous line bundle over X which is induced fro m χ . That is , contin uous sections of L χ are identiﬁed with B -equiv aria n t functions on G as follows: C ( X ; L χ ) = { s : G → C contin uous | s ( g man ) = e χ M ( m − 1 ) e χ A ( a − 1 ) s ( g ) ∀ g ∈ G , m ∈ M , a ∈ A , n ∈ N } . (2.1) The G -action on sectio ns is b y left translation: g ′ · s ( g ) := s ( g ′ − 1 g ). Restricting to K , we ha ve the ‘compac t picture’ of C ( X ; L χ ): C ( X ; L χ ) ∼ = { s : K → C contin uous | s ( k m ) = e χ M ( m − 1 ) s ( k ) ∀ k ∈ K , m ∈ M } . (2.2) Note that, as a K -homoge ne o us bundle, L χ depe nds only on χ M . The co mpact picture gives a Hermitian metr ic on L χ . Sp eciﬁcally , the p oint - wise inner pro duct of sec tions is given b y h s 1 ( k ) , s 2 ( k ) i = s 1 ( k ) s 2 ( k ) ∈ C ( X ) . The L 2 -section s pace L 2 ( X ; L χ ) is the co mpletion of C ( X ; L χ ) with resp ect to the inner pro duct h s 1 , s 2 i = Z K s 1 ( k ) s 2 ( k ) dk . (2.3) Some cas es w arrant specia l notation. If µ is a weigh t for K , we let L hol µ denote the holomorphic line bundle L µ ⊕ µ . W e als o le t E µ denote the ‘unitarily induced’ bundle L µ ⊕ ρ . O n E µ the tra ns lation action U µ : G → L ( L 2 ( X ; E µ )) is a unitary representation. Thes e will b e the main fo cus o f our attention. Restricting U µ to K , L 2 ( X ; E µ ) b ecomes a subrepr esen tation of the left r eg- ular representation K . If R denotes the rig ht regular representation, then the 6 equiv aria nc e c ondition of Equation (2.2) b ecomes R ( m ) s = e − µ ( m ) s for all m ∈ M . Inﬁnitesimally , L 2 ( X ; E µ ) = { s ∈ L 2 ( K ) | R ( M ) s = − µ ( M ) s for all M ∈ m } = p − µ L 2 ( K ) , (2.4) where p − µ denotes the orthogonal pro jection on to the ( − µ )-weigh t spa ce of the right regula r representation of K on L 2 ( K ). Let χ , χ ′ be character s o f B . If f ∈ C ( X ; L χ ′ − χ ) then p oint wise multiplica- tion by f , denoted M f , maps C ( X ; L χ ) to C ( X ; L χ ′ ). This gives a G -eq uiv ariant bundle iso morphism End( L χ , L χ ′ ) ∼ = L χ ′ − χ . In pa rticular, E nd ( E µ , E µ ′ ) ∼ = L ( µ ′ − µ ) ⊕ 0 for any weigh ts µ , µ ′ . Moreover, for any f ∈ C ( X ; L ( µ ′ − µ ) ⊕ 0 ), U µ ′ ( g ) M f U µ ( g − 1 ) = M g · f . (2 .5) In this pictur e, a loca lly trivializing partition of unit y on E µ takes the following form. Lemma 2.1 . F or any weight µ , ther e exists a ﬁ n ite c ol le ction of c ontinu ous se ctions ϕ 1 , . . . , ϕ n ∈ C ( X ; L µ ⊕ 0 ) such that P n j =1 M ϕ j M ϕ j = 1 . Pr o of. Le t f 1 , . . . , f n ∈ C ( X ) be a partition o f unit y sub ordinate to a lo cally trivializing c o ver of E µ . Comp osing f 1 2 j with the corresp onding lo cal trivializa - tion L 0 ∼ = − → L µ ⊕ 0 gives the sections ϕ j . 2.3 P arab olic subgroups and equiv arian t ﬁbrations Let P b e a par abolic s ubg roup, B ≤ P ≤ G , with Lie algebra p . Let S ⊆ Σ be the se t of simple r oo ts α such that the ro ot space g − α is con ta ined in p . This set cla ssiﬁes P , and we therefor e in tro duce the notation P Σ := G , P { α 1 } :=      ∗ ∗ ∗ ∗ ∗ ∗ 0 0 ∗      , P { α 2 } :=      ∗ ∗ ∗ 0 ∗ ∗ 0 ∗ ∗      , P ∅ := B . (Here ∗ denotes po ssibly nonzero entries.) W e will simplify this by writing P i := P { α i } whenever con venien t. F or i = 1 , 2, let X i := G / P i . The natural maps ϕ i : X → X i are equiv ariant ﬁbrations with ﬁbres P i / B ∼ = C P 1 . W e will denote the corre s ponding foliations of X by F i := ker D ϕ i . Denote the co mpact part o f P S by K S := P S ∩ K . Explicitly , K Σ := K , K 1 := P 1 ∩ K =      0 A 0 0 0 z       A ∈ U(2) , z = (det A ) − 1    , K 2 := P 2 ∩ K =      z 0 0 0 0 A       A ∈ U(2) , z = (det A ) − 1    , K ∅ := M . 7 Then X i = K / K i ( i = 1 , 2). The co mplexiﬁed Lie a lgebra ( k i ) C of K i decomp oses a s s i ⊕ z i , where s i := span { X i , H i , Y i } ∼ = sl (2 , C ) and z i := span { H ′ i } ⊂ m C . (Notation as in Section 2.1.) F or the sake of ﬁxing no tation, we recall the re pr esen tation theor y of s i ∼ = sl (2 , C ). The weight s of sl (2 , C ) are parameter ized by the int egers. The restriction of a weigh t µ of K to a weigh t o f s i is µ i := µ ( H i ) ∈ Z . The dominant weigh ts are the nonneg ativ e in tegers N . Let X , H , Y ∈ sl (2 , C ) b e the basis elements c o rresp onding to X i , H i .Y i ∈ s i . The ir r educible repres en ta tion of sl (2 , C ) with highest weight δ ∈ N will b e de- noted V δ . It has a n orthonormal basis of w eight vectors { e δ , e δ − 2 , . . . , e − δ +2 , e − δ } , such that X · e j = 1 2 p ( δ − j )( δ + j + 2) e j +2 (2.6) H · e j = j e j (2.7) Y · e j = 1 2 p ( δ − j + 2)( δ + j ) e j − 2 (2.8) 2.4 Harmonic analysis F or any compac t group C , we will use ˆ C to denote the set o f ir r educible represen- tations of C , often r eferred as C -typ es . F or a n y unitary repr esen tation π of C , we use V π to denote its representation space , a nd π † to denote its contragredient representation. F or a representation π of K = SU(3) a nd elements ξ ∈ V π , η † ∈ V π † , w e use c η † ,ξ to denote the matr ix unit c η † ,ξ ( k ) := ( η † , π ( k ) ξ ). Reca ll the Peter-W eyl isomorphism M π ∈ ˆ K V π † ⊗ V π ∼ = L 2 ( K ) η † ⊗ ξ 7→ (dim V π ) 1 2 c η † ,ξ . which in tertwines L π and L π † with the left and right regular r epresent a- tions, resp ectively . If p µ denotes the pro jection onto the µ -weight space of a representation then from E quation 2.4, L 2 ( X ; E µ ) ∼ = M π ∈ ˆ K V π † ⊗ p − µ V π . 3 Harmonic analysis on the ﬂag v ariet y 3.1 Harmonic C ∗ -categories W e will make muc h use of the results of [Y un] reg arding ha r monic a nalysis on ﬂag manifolds for SL( n, C ). In this section, w e r eview the ma jor deﬁnitions and results o f that pap er. Because we are only interested in n = 3 her e, w e will simplify the notation so mewhat. Let K ′ be a closed subgroup o f K = SU(3). Let H b e a Hilb ert s pace equipp ed with a unitary representation of K . F or σ ∈ ˆ K ′ , w e let p σ denote the o rthogonal pro jection on to the σ -isot ypical subspace of H (with representation restricted to K ′ ). If F ⊆ ˆ K ′ is a set of K ′ -types, we let p F := P σ ∈ F p σ . 8 W e are particularly in teres ted in the four subgr oups K ≥ K 1 , K 2 , ≥ M abov e. Note that the iso t y pic a l subspaces of M a re the weigh t spaces. If K ′′ is a subgroup of K ′ , then the isotypical pro jections of K ′ and K ′′ commute. In particular, the isotypical pro jectio ns of K , K 1 and K 2 commute with the weight-space pro jections. Thes e iso t ypica l pro jections can therefore b e restricted to any w e igh t-space of a unitary K -representation. Deﬁnition 3.1. A harmonic K -sp ac e H is a direct sum of weight spaces o f unitary K -r e pr esen tations: H = L k p µ k H k for some weigh ts µ k and unitary K -repres en ta tions on H k . A harmonic K -space H is called ﬁnite multiplicity if for ev ery π ∈ ˆ K , p π H is ﬁnite dimensiona l. Example 3.2. The (right) regular representation is a ﬁnite m ultiplicit y ha r- monic K -space by the Peter-W eyl Theo rem, as is L 2 ( X ; E µ ) for an y weigh t µ . More gener ally , an y homogeneous vector bundle E ov er X decomp oses equiv ari- antly in to line bundles, so L 2 ( X ; E ) is a harmonic K -spa ce. Deﬁnition 3.3. Let S ⊆ Σ. Let A : H → H ′ be a bounded linear op erator betw een harmonic K -spaces. F or σ ′ , σ ∈ ˆ K S , let A σ ′ σ := p ′ σ Ap σ , so that ( A σ ′ σ ) is the matrix deco mposition of A with resp ect to the decomp ositions o f H, H ′ int o K S -types. (i) W e say A is K S -harmonic al ly pr op er if the matrix ( A σ ′ σ ) is row- and column-ﬁnite, i.e. , if for every σ ∈ ˆ K S , there are o nly ﬁnitely many σ ′ ∈ ˆ K S for which either A σ ′ σ or A σσ ′ is nonzero . (ii) W e say A is K S -harmonic al ly ﬁnite if the matrix ( A σ ′ σ ) has only ﬁnitely many nonzero ent ries. Deﬁne A S ( H, H ′ ), resp. K S ( H, H ′ ), to b e the op erator-nor m closure of the K S -harmonica lly pro p er, resp. K S -harmonica lly ﬁnite, op erator s fro m H to H ′ . If H = H ′ , we write A S ( H ) and K S ( H ) for A S ( H, H ) and K S ( H, H ), re s pec- tively . These are C ∗ -subalgebra s o f the algebra s L ( H ) of bounded oper a tors on H . Letting H a nd H ′ v ary , we consider A S and K S as deﬁning C ∗ -categor ies of op erators b et w een har mo nic K -spaces. W e also use K and L to denote the C ∗ - categorie s of compact ope r ators and bo unded o p erator s, res pectively , betw een Hilber t spaces. Lemma 3.4 ([Y un , Lemma 3 .2]) . If S ⊆ S ′ ⊆ Σ t hen K S ′ ⊆ K S . The following t wo results are r e statemen ts of Le mmas 3.4 and 3 .5 of [Y un]. Prop osition 3.5. L et K : H → H ′ b e a b ounde d line ar op er ator b etwe en har- monic K -sp ac es. The fol lowing ar e e quivalent: (i) K ∈ K S , (ii) F or any ǫ > 0 , ther e is a ﬁnite set F ⊂ ˆ K S of K S -typ es such that k p ⊥ F K k < ǫ and k K p ⊥ F k < ǫ . (iii) F or any ǫ > 0 , ther e is a ﬁnite set F ⊂ ˆ K S of K S -typ es such that k K − p F K p F k < ǫ . 9 If A and K are b o unded linear o perato rs, we s ay K is rig ht-c omp osable for A if the codo main of K is the domain of A . L eft-c omp osability is deﬁned similarly . Prop osition 3.6. L et A : H → H ′ b e a b ounde d line ar op er ator b etwe en har- monic K -sp ac es. The fol lowing ar e e quivalent: (i) A ∈ A S , (ii) F or any σ ∈ ˆ K S , and any ǫ > 0 , ther e is a ﬁnite set F ⊂ ˆ K S of K S -typ es such that k p ⊥ F Ap σ k < ǫ and k p σ Ap ⊥ F k < ǫ . (iii) F or any σ ∈ ˆ K S , Ap σ and p σ A ar e in K S . (iv) A is a t wo-side d multiplier of K S , me aning that AK ∈ K S for al l right- c omp osable K ∈ K S , and K A ∈ K S for al l left-c omp osable K ∈ K S . W e now descr ib e some considerable simpliﬁca tio ns from [Y un] in the case of homogeneous vector bundles for SU(3). Lemma 3.7. L et E , E ′ b e K -homo gene ous ve ct or bund les over X , and put H = L 2 ( X ; E ) , H ′ = L 2 ( X ; E ′ ) . Then K Σ ( H, H ′ ) = K ( H , H ′ ) and A Σ ( H, H ′ ) = K ∅ ( H, H ′ ) = A ∅ ( H, H ′ ) = L ( H , H ′ ) . Pr o of. Since H and H ′ are direct sums of ﬁnitely man y w eig h t spaces for the right regular r epresentation of K , any bo unded op erator fro m H to H ′ is M - harmonically ﬁnite. Hence, K ∅ ( H, H ′ ) = A ∅ ( H, H ′ ) = L ( H , H ′ ). Lemma 3.3 of [Y un] shows that K Σ ( H, H ′ ) = K ( H , H ′ ). By P r opos ition 3.6 ab o ve, a n y bo unded op erator A : H → H ′ is in A Σ . The only non trivial cases, then, a re K { α i } and A { α i } , which we abbrev iate as K α i and A α i . Deﬁnition 3.8. As in [Y un ], w e put A := ∩ S ⊆ Σ A S , the simultaneous multiplier category of all K S ( S ⊆ Σ). Note, thoug h, that by Lemma 3.7 this reduces to A ( H , H ′ ) = A α 1 ( H, H ′ ) ∩ A α 2 ( H, H ′ ) when H , H ′ are L 2 -section s paces of homogeneous vector bundles. In the generality of [Y un], it is nece s sary to adjust the opera to r spaces K S by deﬁning J S := K S ∩ A . The next lemma shows tha t this is not necessa ry for the curr en t application. Lemma 3.9. With H , H ′ as in L emma 3.7, K α i ( H, H ′ ) ⊆ A ( H , H ′ ) , for i = 1 , 2 . Thus, J α i ( H, H ′ ) = K α i ( H, H ′ ) . Pr o of. Le t i = 1. It is immediate that K α 1 ( H, H ′ ) ⊆ A α 1 ( H, H ′ ). Lemma 5.4 o f [Y un ] implies that o n H and H ′ , p σ 1 p σ 2 is compact for any σ 1 ∈ ˆ K 1 and σ 2 ∈ ˆ K 2 . Thus, if K : H → H ′ is K 1 -harmonica lly ﬁnite, then K p σ 2 ∈ K ( H , H ′ ) ⊆ K α 2 ( H, H ′ ). By Pro p osition 3.6, K ∈ A α 2 ( H, H ′ ). T aking the norm-closur e, K α 1 ( H, H ′ ) ⊆ A α 2 ( H, H ′ ), which pr oves the result. The case i = 2 is analog ous. W e therefore av oid the notatio n J α i altogether. Theorem 3.10 ([Y un , Theo r em 1.1 1]) . L et E b e a K -homo gene ous ve ctor bun- d le over X , and H := L 2 ( X ; E ) . Then 10 (i) K α i ( H ) is an ide al in A ( H ) , fo r i = 1 , 2 . (ii) K α 1 ( H ) ∩ K α 2 ( H ) = K ( H ) . Lemma 3.11 ([Y un, Lemma 8.1]) . L et µ , ν b e weights. F or any f ∈ C ( X ; E µ − ν ) , the multiplic ation op er ator M f : L 2 ( X ; E ν ) → L 2 ( X ; E µ ) is in A . R emark 3.12 . Lemma 3.1 1 dep ends on K -equiv ariant structure only , so that f may be (the restr iction to K of ) a s ection of L ( µ − ν ) ⊕ χ A for any χ A ∈ m † C . 3.2 Principal series represen t ations The purp ose of this section is to prov e the follo wing impor tan t fact, the ﬁr st of t wo rather technical harmonic analy sis results. Prop osition 3 .13. L et µ ∈ Λ W . F or any g ∈ G , U µ ( g ) ∈ A ( L 2 ( X ; E µ )) . W e will use the notation for the elemen ts of k C from Sectio n 2.1, noting that the elements X α , Y α ( α ∈ ∆ + ) and H i , H ′ i (for either i = 1 or 2) form a basis for g . W e let X † α , Y † α , H † i , H ′ i † denote the dual basis elements o f g † . W e also recall the notation c η † ,ξ for matrix units. Lemma 3. 14. L et A ∈ a . L et π ∈ ˆ K and η † ∈ V π † , ξ ∈ ( V π ) − µ . Then U µ ( A ) c η † ,ξ = c η † ⊗ A, Ξ( ξ ) , wher e Ξ( ξ ) := ρ ( H i ) ξ ⊗ H † i + ρ ( H ′ i ) ξ ⊗ H ′ i † + X α ∈ ∆ sign( α ) π ( X α ) ξ ⊗ X † α ∈ V π ⊗ g † . Note that c η † ⊗ A, Ξ( ξ ) is a matrix unit for the non-irr e ducible repres en ta tio n π ⊗ Ad † , hence a s um o f matrix units for the ir reducible comp onen ts of π ⊗ Ad † . Pr o of. Deﬁne functions κ , a , n on G using the Iwasaw a decompo sition: g =: κ ( g ) a ( g ) n ( g ) ∈ KAN , for g ∈ G . The deriv atives D κ e , D a e and D n e at the identit y are the ( R -linea r) pr o jections of g on to the compo ne nts of the dec o mposition g = k ⊕ a ⊕ n . If P ∈ g , let us write P = P + + P 0 + P − where P + , P 0 , P − are strictly upp er-triangular , diagonal, and strictly lower-triangular, resp ectiv ely . If P is self-a djo in t, the k ⊕ a ⊕ n decomp osition of P is P = ( − P + + P − ) ⊕ P 0 ⊕ 2 P + . Thus, D κ e ( P ) = − X α ∈ ∆ sign( α ) X α ⊗ X † α ! P, (3.1) D a e ( P ) =  H i ⊗ H † i + H ′ i ⊗ H ′ i †  P. (3.2) F or a ∈ A , k ∈ K , a − 1 k = k k − 1 a − 1 k = k κ ( k − 1 a − 1 k ) a ( k − 1 a − 1 k ) n ( k − 1 a − 1 k ) . In order to de s cribe the G -actio n on a K -matrix unit, o ne must extend c η † ,ξ to a B -equiv ar ian t function o n G . Equation (2.1)) gives U µ ( a ) c η † ,ξ ( k ) := c η † ,ξ ( a − 1 k ) = e ρ ( a ( k − 1 ak )) c η † ,ξ ( k κ ( k − 1 a − 1 k )) = e ρ ( a ( k − 1 ak ))  η † , π ( k ) π ( κ ( k − 1 a − 1 k )) ξ  . (3.3) 11 Let a = exp( tA ), and ta ke the deriv ative with resp ect to t at t = 0: U µ ( A ) c η † ,ξ ( k ) = ρ ( D a e (Ad k − 1 ( A )))  η † , π ( k ) ξ  −  η † , π ( k ) π ( D κ e (Ad k − 1 ( A )) ξ  . Since Ad k − 1 ( A ) is self-adjoint, Equations (3.1) and (3.2) give U µ ( A ) c η † ,ξ ( k ) = ρ ( H i )  H † i , Ad k − 1 ( A )  η † , π ( k ) ξ  + ρ ( H ′ i )  H ′ i † , Ad k − 1 ( A )  η † , π ( k ) ξ  + X α ∈ ∆ sign( α )  η † , π ( k ) π ( X α )  X † α , Ad k − 1 ( A )  ξ  =  A, Ad † k ( H † i )  η † , π ( k ) ρ ( H i ) ξ  +  A, Ad † k ( H ′ i † )  η † , π ( k ) ρ ( H ′ i ) ξ  + X α ∈ ∆ sign( α )  A, Ad † k ( X † α )  η † , π ( k ) π ( X α ) ξ  = c η † ⊗ A, Ξ( ξ ) ( k ) . Recall the decomp osition ( k i ) C = s i ⊕ z i of Section 2.3. Let µ ∈ Λ W . Since z i ⊆ h , the action of z i on the ( − µ )-weight space o f a ny K -repre sen tation is completely determined b y µ . Thus, the K i -isotypical subs paces of L 2 ( X ; E µ ) are the s i -isotypical subspac es. Mor eo ver, since L 2 ( X ; E µ ) ha s s i -weigh t − µ i := − µ ( H i ), the s i -types which o ccur must ha ve highest weigh ts | µ i | , | µ i | + 2 , . . . In what follows, w e ﬁx i = 1 or 2 and let σ l denote the s i -type with hig hest weigh t l ∈ N . W e abbrev iate p l := p σ l . Note that p l = 0 on L 2 ( X ; E µ ) if l 6≡ µ i (mo d 2) or l < | µ i | . The next lemma shows that U µ ( A ) is tridiagonal with r espect to K i -types, a nd that the oﬀ-diagonal entries hav e at most linear growth. Lemma 3.15. Fix µ ∈ Λ W and let A ∈ a . Ther e exists a c onstant C > 0 such that for any m, l ∈ N , k p m U µ ( A ) p l k = 0 if | m − l | > 2 , k p m U µ ( A ) p l k ≤ C ( l + 1) if | m − l | = 2 . Pr o of. Le t us take i = 1, with the case of i = 2 b eing entirely analo gous. Suppo se c η † ,ξ ∈ p l L 2 ( X ; E µ ), which is to say that η † ∈ V π † , ξ ∈ ( V π ) σ l for some π ∈ ˆ K . By L e mma 3.14, w e need to understand the decomp o sition of Ξ( ξ ) int o s 1 -types. The adjoint representation of g decomp oses into the s 1 -representations span { X 1 , H 1 , Y 1 } , s pan { H ′ 1 } , s pan { X 2 , X 3 } , s pan { Y 2 , Y 3 } , and g † decomp oses dually . W e break up the expre ssion for Ξ( ξ ) into co rresp ond- ing parts. Firstly , H ′ 1 has trivial s 1 -type, so ρ ( H ′ 1 ) ξ ⊗ H ′ 1 † has s i -type l . Next, note that the vector X 2 ⊗ X † 2 + X 3 ⊗ X † 3 ∈ g ⊗ g † also has tr ivial s 1 -type, since it corres p onds to the identit y map on the subrepre s en ta tion spa n { X 2 , X 3 } . The map V π ⊗ g ⊗ g † → V π ⊗ g † ζ ⊗ Z ⊗ Z † 7→ π ( Z ) ζ ⊗ Z † 12 is a morphis m o f K -representations, in par ticular of s i -representations, so π ( X 2 ) ξ ⊗ X † 2 + π ( X 3 ) ξ ⊗ X † 3 also has s 1 -type l . Similarly , − π ( Y 2 ) ξ ⊗ Y † 2 − π ( Y 3 ) ξ ⊗ Y † 3 has s 1 -type l . Thu s, all the oﬀ-diag onal compo nents of U µ ( A ) are due to the comp onent s Ξ 1 ( ξ ) := ρ ( H 1 ) ξ ⊗ H † 1 + π ( X 1 ) ξ ⊗ X † 1 − π ( Y 1 ) ξ ⊗ Y † 1 (3.4) of Ξ( ξ ). The coadjoint representation of s 1 on span { X † 1 , H † 1 , Y † 1 } has highest weigh t 2, so the fusio n rules for SU(2)-repr esen tations imply that (3.4) contains s i -types l − 2 , l , l + 2 only . It rema ins to prove the norm estimate on the o ﬀ-dia gonal terms. By Equa - tions (2.6)–(2 .8), k ρ ( H 1 ) ξ k = 2 k ξ k ≤ ( l + 1) k ξ k , k π ( X 1 ) ξ k = 1 2 p ( l − µ i )( l + µ i + 2) k ξ k ≤ ( l + 1 ) k ξ k , k π ( Y 1 ) ξ k = 1 2 p ( l − µ i + 2)( l + µ i ) k ξ k ≤ ( l + 1) k ξ k , so the no rm of Ξ 1 ( ξ ) is bounded b y C 0 ( l + 1) k ξ k for s ome co nstan t C 0 . W e need to conv ert this into a bound on the norm of the ma tr ix units. Decomp ose π ⊗ Ad † int o irreducible K -subr epresent ations. Suppose π ′ is an irreducible subrepresentation of π ⊗ Ad † . By orthogona lit y of c haracters , π is a subrepresentation of π ′ ⊗ Ad. Therefore dim π ≤ dim( π ′ ⊗ Ad) = 8 dim π ′ , so that dim π ′ ≥ 1 8 dim π . This also shows that the num b er of irreducible comp onents of π ⊗ Ad † is at most 64. F or ea c h irreducible s ubrepresentation π ′ of π ⊗ Ad † , let y † π ′ denote the π ′ † -comp onen t of η † ⊗ A , a nd x π ′ the π ′ -comp onen t of Ξ 1 ( ξ ). W e get k p l ± 2 U µ ( A ) p l c η † ,ξ k 2 ≤ k c η ⊗ A, Ξ 1 ( ξ ) k 2 = X π ′ 1 dim π ′ k y † π ′ k 2 k x π ′ k 2 ≤ X π ′ 1 dim π ′ k η † ⊗ A k 2 k Ξ 1 ( ξ ) k 2 ≤ X π ′ 1 dim π ′ k η † k 2 k A k 2 C 2 0 ( l + 1) 2 k ξ k 2 ≤ k A k 2 C 2 0 ( l + 1) 2 X π ′ 8 dim π k η † k 2 k ξ k 2 ≤ 8 . 64 . k A k 2 C 2 0 ( l + 1) 2 k c η † ,ξ k 2 . Putting C = √ 512 k A k C 0 gives the result. Pr o of of Pr op osition 3.13 . W e need to show U µ ( g ) ∈ A α i for i = 1 , 2. F or k ∈ K , the left tr a nslation action U µ ( k ) commutes with the deco mposition into right K i -types, so that U µ ( k ) ∈ A α i trivially . By the KAK -dec o mposition, it suﬃces to prov e the prop osition for g = a ∈ A . W e contin ue with the no tation of the pre vious lemma. Put P m := P m j =0 p j . W e will show that for a n y l ∈ N and a n y ǫ > 0, ther e e x ists m ∈ N such 13 that k P ⊥ m U µ ( a ) p l k < ǫ and k p l U µ ( a ) P ⊥ m k < ǫ , fr o m which Lemma 3.6 g iv es U µ ( a ) ∈ A α i . Let A ∈ a such that e A = a . Deﬁne φ : N → [0 , 1 ] b y φ ( n ) := ( 1 , n ≤ l , max { 0 , 1 − ǫ 2 4 C log( n + 3) } , n > l , where C is the constant of the previous lemma. Deﬁne Φ := P n ∈ N φ ( n ) p n , an op erator on L 2 ( X ; E µ ) which is sca lar on each K i -type. W e now decomp ose U µ ( A ) in to its diagonal a nd oﬀ-diag onal comp onent s. F or con venience of notatio n, we put U := U µ ( A ), then write U = U − + U 0 + U + , where U − = ∞ X n =2 p n − 2 U p n , U 0 = ∞ X n =0 p n U p n , U + = ∞ X n =0 p n +2 U p n . The diag onal compo nen t U 0 commutes with Φ. On the other ha nd, k [ p n − 2 U p n , Φ] k = k ( φ ( n ) − φ ( n − 2)) p n − 2 U p n k ≤ ǫ 2 4 C (log( n + 3) − lo g( n + 1)) ≤ ǫ 2 2 C 1 ( n + 1 ) ≤ ǫ 2 2 , by Lemma 3.15. Thus, k [ U − , Φ] k = sup n ∈ N k [ U n − 2 ,n , Φ] k ≤ 1 2 ǫ 2 . Similarly , k [ U + , Φ] k ≤ 1 2 ǫ 2 . Therefor e, k [ U µ ( A ) , Φ] k ≤ ǫ 2 . Let s ∈ p l L 2 ( X ; E µ ) hav e norm o ne. P ut s t := U µ ( e tA ) s for 0 ≤ t ≤ 1. Then | d dt h Φ s t , s t i| = |h Φ U µ ( A ) s t , s t i + h Φ s t , U µ ( A ) s t i| = |h [Φ , U µ ( A )] s t , s t i| ≤ ǫ 2 , for all t . Therefore, |h Φ s 1 , s 1 i| =     h Φ s 0 , s 0 i + Z 1 t ′ =0 d dt h Φ s t , s t i dt ′     ≥ 1 − ǫ 2 . Let m be the smalles t integer for whic h φ ( m ) = 0. Put v := P m s 1 and w := P ⊥ m s 1 . Then k v k 2 + k w k 2 = 1 , but als o k v k 2 > h Φ v , v i = h Φ v , v i + h Φ w , w i = h Φ s 1 , s 1 i ≥ 1 − ǫ 2 . It follows that k w k < ǫ , ie , k P ⊥ m U µ ( a ) s k < ǫ . Since s ∈ p l L 2 ( X ; E µ ) was arbitrar y , k P ⊥ m U µ ( a ) p l k < ǫ . Replacing a with a − 1 , there exists m ′ ∈ N suc h that k P ⊥ m ′ U µ ( a − 1 ) p l k < ǫ . Thu s, after enlarg ing m to b e at lea st m ′ , we ha ve k p l U µ ( a ) P ⊥ m k = k P ⊥ m U µ ( a − 1 ) p l k < ǫ. 14 In fact, P ropo sition 3.13 holds for a n y generalize d principa l series repr esen- tation. Although w e do n’t actually need this here, it is now trivial to prove. Corollary 3.1 6. F or any G -homo gene ous line bund le L 2 ( X ; L χ ) over X , the tr anslation op er ators s 7→ g · s b elong to A . Pr o of. Le t χ = χ M ⊕ χ A . A computation of the form o f Eq. (3.3) gives g · s ( k ) = e χ A ( a ( k − 1 g k )) s ( k κ ( k − 1 g − 1 k )) , for any k ∈ K , while U χ M ( g ) s ( k ) = e ρ ( a ( k − 1 g k )) s ( k κ ( k − 1 g − 1 k )) . Note that a ( m − 1 g m ) = a ( g ) for a ny m ∈ M , g ∈ G . Therefore, g · s = M f U χ M ( g ) s , where f ( k ) := e χ A − ρ ( a ( k − 1 g k )) is in C ( K / M ) = C ( X ). Since M f and U χ M ( g ) are in A , we are done. 3.3 Longitudinal pseudodiﬀeren tial op erators Let X ∈ k C be a ro ot vector, of weigh t α . Via the rig h t regular r epresentation, X deﬁnes a left K - in v a riant diﬀerential op erator on C ∞ ( K ). F or e a c h weigh t µ , X maps p − µ L 2 ( K ) to p − µ + α L 2 ( K ), so it deﬁnes a K -inv ariant diﬀeren tial op erator X : L 2 ( X ; E µ ) → L 2 ( X ; E µ − α ) . The principa l sym b ol of this diﬀer en tia l op erator is a K -equiv a rian t linear map from the cotangent bundle T ∗ X ∼ = K × M ( k / m ) ∗ to End( E µ , E µ − α ) ∼ = E − α . (Here ( k / m ) ∗ denotes the r e al dual of k / m .) By equiv a r iance, this ma p is de- termined by its v alue o n the cotang en t ﬁbre a t the identit y co set e ∈ X , whic h is Symb( X ) : T ∗ e X = ( k / m ) ∗ → C (3.5) ξ 7→ ξ ( X ) . If X ∈ ( k i ) C ( i = 1 o r 2), then the diﬀerential o p erator X : C ∞ ( X ; E µ ) → C ∞ ( X ; E µ − α ) is ta ngen tial to the foliation F i of Section 2.3 . W e will refer to such an o p erator as an F i -longitudinal diﬀerential o perato r. Its longitudinal principal symbol is the K -equiv a riant map Symb F i : F i ∗ → E − α which, at the ident ity coset, is g iv en by Symb F i : ( F i ∗ ) e = ( k i / m ) ∗ → C ξ 7→ ξ ( X ) . An F i -longitudinal diﬀerential o perato r is longitudinal ly el liptic if its longi- tudinal principal symbol is inv ertible o ﬀ the zer o section of T ∗ F i . No te that X i = − 1 2 ( X ′ i + √ − 1 X ′′ i ) ∈ ( k i ) C where X ′ i =  0 − 1 1 0  , X ′′ i =  0 √ − 1 √ − 1 0  15 span k i / m , s o that X i is F i -longitudinally elliptic. Similarly , Y i is F i -longitudinally elliptic. Moreover, X i and Y i are fo rmal adjo ints. W e s hall use X i , Y i also to denote their closures as un b ounded op erators on the L 2 -section spa ces. Fix µ ∈ Λ W . L et E := E µ ⊕ E µ − α i , and deﬁne D i :=  0 Y i X i 0  on L 2 ( X ; E ). The s i -isotypical subspaces of L 2 ( X ; E ) ar e eigenspaces for D i , and by the rep- resentation theory of s i —sp eciﬁcally Equa tions (2.6) and (2.8)—its sp ectrum is discrete. F or the deﬁnition a nd basic prop erties of long itudinal pseudo diﬀerential op- erators , we r efer the rea de r to [MS06] 4 . If E , E ′ are vector bundles ov er X , we denote the set of F i -longitudinal pseudodiﬀerential op erators of order at most p by Ψ p F i ( E , E ′ ). If E = E ′ , we abbreviate this to Ψ p F i ( E ). Let C ( S ∗ F i ; End( E )) denote the algebr a of contin uous sections of the pull- back of E nd( E ) to the cosphere bundle of the folia tion F i . The longitudinal principal s y m b ol map Sym b F i : Ψ 0 F i ( E ) → C ( S ∗ F i ; End( E )) extends to the op erator-nor m closur e Ψ 0 F i ( E ), and we hav e Connes’ shor t exact sequence, 0 / / Ψ − 1 F i ( E ) / / Ψ 0 F i ( E ) Symb F i / / C ( S ∗ F i ; End( E )) / / 0 . (3.6) F or a n y closed, densely deﬁned, un bo unded o perato r T betw een Hilber t spaces, we let P h T denote the phase in the po lar de c o mposition: T = (Ph T ) | T | . W e also use Ph z to denote the phase of a complex nu mber z ∈ C × . Lemma 3.17. F or any weight µ , Ph X i : L 2 ( X ; E µ ) → L 2 ( X ; E µ − α i ) and Ph Y i : L 2 ( X ; E µ − α i ) → L 2 ( X ; E µ ) ar e F i -longitudinal pseudo diﬀer ential op- er ators. Their longitudinal princip al symb ols at t he identity c oset ar e Symb F i (Ph X i )( ξ ) = Ph ( ξ ( X i )) , Symb F i (Ph Y i )( ξ ) = Ph ( ξ ( Y i )) = Ph ( ξ ( X i )) . for ξ in t he unit spher e of ( k i / m ) ∗ ∼ = ( F i ∗ ) e . Pr o of. Le t E := E µ ⊕ E µ − α i . Fix ǫ > 0 s uc h tha t Sp ec( D i ) ∩ ( − ǫ, ǫ ) = { 0 } . Let f : R → [ − 1 , 1] be smo oth with f (0) = 0 and f ( x ) = sig n( x ) for all | x | ≥ ǫ . A ﬁbrewise applicatio n of [T a y81, Theo rem 1.3] sho ws that f ( D i ) = Ph D i ∈ Ψ 0 F i ( X ; E ). Mor eo ver the pro of of the theor em shows that its full sym b ol has an asymptotic expansion with leading ter m f (Symb F i D i ). Note that (Sym b F i D i )( ξ ) =  0 ξ ( X i ) ξ ( X i ) 0  has sp ectrum { ±| ξ ( X i ) |} , so if ξ is larg e enough that | ξ ( X i ) | > ǫ , then f (Symb F i D i )( ξ ) = Ph (Symb F i D i ( ξ )) =  0 Ph ( ξ ( X i )) Ph ( ξ ( X i )) 0  . This is radially constan t on ( k i / m ) ∗ for | ξ ( X i ) | > ǫ . The principal sy m b ol is the limit at the spher e at inﬁnit y . 4 In this r eference, they ar e call ed tang ent ial pseudodiﬀerent ial operators. 16 Theorem 3.18. L et E , E ′ b e K -homo gene ou s ve ctor bund les over X . Then (i) Ψ − 1 F i ( E , E ′ ) ⊆ K α i , (ii) Ψ 0 F i ( E , E ′ ) ⊆ A , Part (i) is prov e n in Pro position 1.12 of [Y un]. It is als o shown there that Ψ 0 F i ( E , E ′ ) ⊆ A i . The more diﬃcult question o f s ho wing Ψ 0 F i ( E , E ′ ) ⊆ A j for j 6 = i requires some leng th y computations in no ncomm utative harmo nic analysis. In order not to disrupt the ﬂow of ideas to o severely , w e hav e pre sen ted the pro of in App endix A. As an indication o f the subtleties inv olved, w e rema r k that the longitudinally elliptic diﬀerential op erator X 1 is not an un b ounded multiplier of K α 2 . T o see this, note that (1 + X ∗ 1 X 1 ) − 1 2 ∈ Ψ − 1 F 1 ( E µ ) ⊆ K α 1 . Since K α 1 . K α 2 ⊆ K , the range of (1 + X ∗ 1 X 1 ) − 1 2 as a multiplier of K α 2 is not dense. T hus, X 1 is not regular with resp ect to K α 2 (see [Lan9 5, Chapter 10]). Hence, proving that Ph X 1 m ultiplies K 2 can no t be achiev ed by direct functional ca lculus. Lemma 3.19. L et i = 1 , 2 and let µ, ν b e weights. F or any f ∈ C ( X ; E ν − µ ) , the diagr am L 2 ( X ; E µ ) M f / / Ph X i   L 2 ( X ; E ν ) Ph X i   L 2 ( X ; E µ − α i ) M f / / L 2 ( X ; E ν − α i ) c ommutes mo dulo K α i . R emark 3.20 . W e abbr eviate this result by writing [Ph X i , M s ] ∈ K α i . By taking adjoints, we also hav e [Ph Y i , M s ] ∈ K α i . Pr o of. As an element of C ( S ∗ F i ; E α i ), the principal symbo l of Ph X i : L 2 ( X ; E µ ) → L 2 ( X ; E µ − α i ) is indep enden t of the weigh t µ . Thus, the ab ov e diagr am com- m utes at the level of principal symbols. 4 The normalized BGG c omplex 4.1 G -con tinuit y Before embarking on the main co nstruction, we need to mak e some rema rks regar ding the issue of G -contin uity . Recall that a bounded op erator A betw een unitary G -represent ations is G -con tinuous if the map g 7→ g .A.g − 1 is co ntin uous in the op erator-nor m top ology . Rather tha n burden the no ta tion with extra decor ations, we choose to make the conv ention that througho u t this section, w e us e K α i ( i = 1 , 2 ) to denote its C ∗ -sub category of G -contin uous eleme n ts. This is reasonable, since almost every op erator we deal with is G -contin uous. F rom [AS71], we know that for a n y homog e neous vector bundles E , E ′ ov er X , the set of longitudinal pseudo diﬀeren tial op erators Ψ 0 F i ( E , E , ′ ) consists of G -contin uous op erators . This includes cont inuous multiplication op erators, in the sense of Section 2.2 (whic h are G -contin uous for muc h simpler rea s ons). 17 The notable exceptions, of course, are the repre sen tations U µ ( g ) of the group elements themselv es. In the ma jority of instances, where G -con tin uity is a trivial co ns equence o f the ab ov e r e marks, w e will not make sp eciﬁc ment ion of it in the pr oofs. 4.2 In tert wining op erators Let µ , µ ′ be weigh ts for K = SU(3). It is well known that the principal series representations U µ and U µ ′ are unitarily e quiv alen t if and only if µ ′ = w · µ for so me W e y l gro up element w ∈ W . When w = w α i is a simple reﬂection corres p onding to the ro ot α i , there is a very concise formula for the intert wining op erator. Prop osition 4.1. L et µ , µ ′ b e weights with µ ′ = w α i µ , so that µ − µ ′ = nα i for some n ∈ Z . If n > 0 , the op er ator (Ph X i ) n : L 2 ( X ; E µ ) → L 2 ( X ; E µ ′ ) intertwines U µ and U ′ µ . If n < 0 , then (Ph Y i ) n : L 2 ( X ; E µ ) → L 2 ( X ; E µ ′ ) is an intertwiner. This is essentially the for m ula given b y Duﬂo in [Duf75, Ch. I II]. Ho wever, Duﬂo’s formulation is suﬃciently diﬀerent that w e feel a brief compar ison is worth while. Pr o of. W e follow the notation for sl (2 , C )-repr esen tations fro m the end o f Sec- tion 2.3. Note that Equatio ns (2.6) and (2.8) imply that (Ph X ) e j = e j +2 and (Ph Y ) e j = e j − 2 . Secondly , with w =  0 − 1 1 0  , (Ph X ) j · e − j = e j = ( − 1) 1 2 ( δ + j ) w · e − j , (Ph Y ) j · e j = e − j = ( − 1) 1 2 ( δ − j ) w · e j , (4.1) for any j ≥ 0. (See [Duf75, § II I.3.5].) Recall that the restriction of µ to a w eight of s i is µ i := µ ( H i ) ∈ Z . The hypotheses of the pr opos itio n a re equiv alent to saying µ i = − µ ′ i = n . First co nsider the c a se n > 0 . Let A = A ( w i , µ, 0) : L 2 ( X ; E µ ) → L 2 ( X ; E µ ′ ) be the in tertwiner of [Duf75, § II I.3.1]. The action of A up on matrix units is given in [Duf75, § II I.3.3 and § II I.3.9] as follows. Let π ∈ ˆ K , η † ∈ V π † , ξ ∈ p − µ ( V π ) and s upp ose that ξ lies in an irreducible s i -subreprese n tatio n of V π with highest weigh t δ . Then, in the notation of Section 2.4, A : c η † ,ξ 7→ c η † ,ξ ′ where ξ ′ = ( − 1) 1 2 ( δ + | µ i | ) | µ i | − 1 π ( w i ) ξ = | µ i | − 1 (Ph X i ) n ξ . Hence, A = | µ i | − 1 (Ph X i ) n : L 2 ( X ; E µ ) → L 2 ( X ; E µ ′ ), where X i here denotes the right regula r action. Thus, (Ph X i ) n diﬀers from A b y the po sitiv e scalar | µ i | = n . The case n < 0 follows since Ph Y i = Ph X i ∗ . W e now recap the directed graph structure which underlies the BGG com- plex. F or our K -homolog ic a l purp oses, it will b e co n venient to mak e an undi- rected gr a ph, or more accur ately , to include also the reversal of each edge. 18 As b efore, if α is a po sitiv e ro ot, we use w α ∈ W to denote the reﬂection in the wall orthogo nal to α . F or w , w ′ ∈ W , w e write w α ← → w ′ if w ′ = w α w a nd l ( w ′ ) = l ( w ) ± 1 . W e will w r ite w ← → w ′ if w α ← → w ′ for some α ∈ ∆ + . An e dg e w α ← → w ′ will b e ca lle d simple if α is a simple ro ot. F or G = SL(3 , C ), this yields the g r aph w α 1 • o o ρ / / _ _ α 2   > > > > > > > > > > > > > > > > > > > w α 1 w α 2 • a a α 2 ! ! D D D D D D D D D 1 •   α 1 @ @         ^ ^ α 2   = = = = = = = = w ρ • w α 2 • o o ρ / /   α 1 ? ?                    w α 2 w α 1 • } } α 1 = = z z z z z z z z z (4.2) Deﬁnition 4.2. Fix a dominant w eight λ . If w α i ← → w ′ is a simple edge, w e denote by I λ,w → w ′ the intert wining op erator of Lemma 4.1: I λ,w → w ′ := ( (Ph X i ) n if n ≥ 0 , (Ph Y i ) − n if n ≤ 0 , where w λ − w ′ λ = nα i . Thes e will be referre d to as simple intertwiners . Note that I λ,w ′ → w = I ∗ λ,w → w ′ . F or the no n-simple edges , we deﬁne in tertwiners as compositio ns of simple int ertwiners: I λ,w α 1 → w α 1 w α 2 := I λ,w α 2 → w α 1 w α 2 .I λ, 1 → w α 2 .I λ,w α 1 → 1 I λ,w α 2 → w α 2 w α 1 := I λ,w α 1 → w α 2 w α 1 .I λ, 1 → w α 1 .I λ,w α 2 → 1 , (4.3) and I λ,w α 1 w α 2 → w α 1 := I ∗ λ,w α 1 → w α 1 w α 2 , I λ,w α 2 w α 1 → w α 2 := I ∗ λ,w α 2 → w α 2 w α 1 . R emark 4.3 . Duﬂo’s intertwiners form a commuting diag r am of the form L 2 ( X ; E w α 1 λ ) B B B B B B B B B B B B B B B B B B B B L 2 ( X ; E w α 1 w α 2 λ ) B B B B B B B B L 2 ( X ; E λ ) > > | | | | | | | | B B B B B B B B L 2 ( X ; E w ρ λ ) L 2 ( X ; E w α 2 λ ) > > | | | | | | | | | | | | | | | | | | | | L 2 ( X ; E w α 2 w α 1 λ ) > > | | | | | | | | (4.4) Since the s imple in tertwiners I λ,w → w ′ deﬁned here ar e positive scalar multip les of Duﬂo’s, the cor r esponding diagram of intert w ine r s I λ,w → w ′ commutes up to some p ositive scalar. But I λ,w → w ′ = (Ph X i ) n is unitary , so that scalar is 1. The non-simple in tertwiners deﬁned b y Equatio n (4.3) are pr ecisely those that complete (4 .4) to a commuting diagram of the form (4.2). Deﬁnition 4.4. Deﬁne K ρ := K α 1 + K α 2 . That is, K ρ ( H, H ′ ) := K α 1 ( H, H ′ ) + K α 2 ( H, H ′ ) for any ha r monic K -spaces H , H ′ . F ollowing the conv ention o f Section 4 .1, w e are including the c o ndition of G -contin uit y in this deﬁnition. 19 Lemma 4.5. L et λ b e a dominant wei ght. (i) F or e ach w ← → w ′ , I λ,w → w ′ ∈ A . (ii) If w α ← → w ′ , then [ I λ,w → w ′ , M f ] ∈ K α for any f ∈ C ( X ) . Pr o of. Part (i) is immediate from Theorem 3.18. If α is a s imple r oot, then (ii) follows from Lemma 3 .19. F or α = ρ , ther e are four in ter t winer s to be chec ked. The following calculation is representative of all of them: [ I λ,w α 1 → w α 1 w α 2 , M f ] = [ I λ,w α 2 → w α 1 w α 2 , M f ] .I λ, 1 → w α 2 .I λ,w α 1 → 1 + I λ,w α 2 → w α 1 w α 2 . [ I λ, 1 → w α 2 , M f ] .I λ,w α 1 → 1 + I λ,w α 2 → w α 1 w α 2 .I λ, 1 → w α 2 . [ I λ,w α 1 → 1 , M f ] . ∈ K α 1 + K α 2 + K α 1 = K ρ . 4.3 Normalized BGG op erators Deﬁnition 4. 6. Deﬁne the shifte d action o f the W eyl gr oup on weights by w ⋆ µ := w ( µ + ρ ) − ρ . F rom now on, λ will deno te a dominant weigh t. Deﬁnition 4.7 . If w α i ← → w ′ is a simple e dg e, then w ⋆ λ − w ′ ⋆ λ = nα i for some n ∈ Z . W e deﬁne the normal ize d BGG op er ator T λ,w → w ′ : L 2 ( X ; E w⋆λ ) → L 2 ( X ; E w ′ ⋆λ ) by T λ,w → w ′ := ( (Ph X i ) n if n ≥ 0 , (Ph Y i ) − n if n ≤ 0 . where w ⋆ λ − w ′ ⋆ λ = nα i . F or the non-s imple arrows, deﬁne T λ,w α 1 → w α 1 w α 2 := T λ,w α 2 → w α 1 w α 2 .T λ, 1 → w α 2 .T λ,w α 1 → 1 T λ,w α 2 → w α 2 w α 1 := T λ,w α 1 → w α 2 w α 1 .T λ, 1 → w α 1 .T λ,w α 2 → 1 T λ,w α 1 w α 2 → w α 1 := T ∗ λ,w α 1 → w α 1 w α 2 T λ,w α 2 w α 1 → w α 2 := T ∗ λ,w α 2 → w α 2 w α 1 . Obviously , the deﬁnitions o f the nor malized BGG op erator s T λ,w → w ′ are ident ical to the deﬁnitions of the intert w ining op erators I λ + ρ,w → w ′ , except tha t the weigh ts of the principal s e r ies representations on which they act diﬀer b y the s hift o f ρ . The next few lemmas describ e the consequences of this. T o b egin with, we hav e an exac t analo g ue of Lemma 4.5, w ith essentially identical pro of. Lemma 4.8. L et λ b e a dominant wei ght. (i) F or e ach arr ow w ← → w ′ , T λ,w → w ′ ∈ A . (ii) If w α ← → w ′ , then [ T λ,w → w ′ , M f ] ∈ K α for any f ∈ C ( X ) . 20 Lemma 4.9. L et ϕ 1 , . . . , ϕ k ∈ C ( X ; E ρ ) b e such t hat P k j =1 | ϕ j | 2 = 1 , as in L emma 2.1. If w α ← → w ′ , then T λ,w → w ′ ≡ k X j =1 M ϕ j I λ + ρ,w → w ′ M ϕ j (mo d K α ) . Pr o of. This is a n immediate conseque nce of Lemma 3.19. Lemma 4.10. If w α ← → w ′ , then T λ,w ′ → w T λ,w → w ′ − 1 ∈ K α . Pr o of. Le t ϕ 1 , . . . , ϕ k ∈ C ( X ; E ρ ) b e as in the previous lemma. By Lemmas 4.9 and 4.5, T λ,w ′ → w T λ,w → w ′ ≡ X j,j ′ M ϕ j I λ + ρ,w ′ → w ′ M ϕ j ϕ j ′ I λ + ρ,w → w ′ M ϕ j ′ (mo d K α ) ≡ X j,j ′ M ϕ j I λ + ρ,w ′ → w I λ + ρ,w → w ′ M ϕ j ϕ j ′ M ϕ j ′ (mo d K α ) = X j,j ′ M ϕ j ϕ j ϕ j ′ ϕ j ′ = 1 . Lemma 4.11. The diagr am of normalize d BG G op er ators L 2 ( X ; E w α 1 ⋆λ ) ` ` B B B B B B B B B B B B B B B B B B B B o o / / L 2 ( X ; E w α 1 w α 2 ⋆λ ) ` ` B B B B B B B B L 2 ( X ; E λ ) ~ ~ > > | | | | | | | | ` ` B B B B B B B B L 2 ( X ; E w ρ ⋆λ ) L 2 ( X ; E w α 2 ⋆λ ) ~ ~ > > | | | | | | | | | | | | | | | | | | | | o o / / L 2 ( X ; E w α 2 w α 1 ⋆λ ) ~ ~ > > | | | | | | | | (4.5) c ommutes mo dulo K ρ . Pr o of. F or adjacen t edges w α ← → w ′ α ′ ← → w ′′ , a calculation a nalogous to that of the prev io us proo f gives T λ,w ′ → w ′′ T λ,w → w ′ ≡ X j,j ′ M ϕ j ( I λ + ρ,w ′ → w ′′ I λ + ρ,w → w ′ ) M ϕ j ϕ j ′ M ϕ j ′ (mo d K α ′ ) . Note that K α ′ ⊆ K ρ . The commutativit y of (4.5) mo dulo K ρ is therefor e a consequence of the co mm utativity o f the corres ponding diagram o f intert winers I λ + ρ,w → w ′ (Remark 4.3). Lemma 4.12. L et w α ← → w ′ . F or any g ∈ G , U w ′ ⋆λ ( g ) T λ,w → w ′ U w⋆λ ( g − 1 ) − T λ,w → w ′ ∈ K α (4.6) 21 Pr o of. W e ﬁrst note that if A is a G -contin uous op erator, then so is g .A.g − 1 . Let ϕ 1 , . . . , ϕ k ∈ C ( X ; E ρ ) b e as in Lemma 4.9. Then, U w ′ ⋆λ ( g ) T λ,w → w ′ U w⋆λ ( g − 1 ) ≡ X j U w ′ ⋆λ ( g ) M ϕ j I λ + ρ,w → w ′ M ϕ j U w⋆λ ( g − 1 ) (mod K α ) = X j U w ′ ⋆λ ( g ) M ϕ j U w ′ ( λ + ρ ) ( g − 1 ) I λ + ρ,w → w ′ U w ( λ + ρ ) ( g ) M ϕ j U w⋆λ ( g − 1 ) = X j M g · ϕ j I λ + ρ,w → w ′ M g · ϕ j . (4.7) Since P k j =1 | g · ϕ j | 2 = 1, Lemma 4.9 shows that (4.7) equals T λ,w → w ′ mo dulo K α . 4.4 Construction of the gamma elemen t Fix a dominant weigh t λ . L e t H λ := L w ∈ W L 2 ( X ; E w⋆λ ). F or each w ∈ W , let Q w denote the orthogo na l pro jection o n to the summand L 2 ( X ; E w⋆λ ) of H λ . W e put a gr ading o n H λ by declar ing L 2 ( X ; E w⋆λ ) to be even or o dd accor ding to the pa rit y of l ( w ). F or f ∈ C ( X ), M f will denote the m ultiplica tion o pera to r on H λ , a cting diagonally on the s umma nds . W e let U denote the diago nal repr esen tation ⊕ w ∈ W U w⋆λ of G . F o r each w ← → w ′ , we extend the normalized BGG o perato r T λ,w → w ′ : L 2 ( X ; E w⋆λ ) → L 2 ( X ; E w ′ ⋆λ ) to an op erator ˜ T λ,w → w ′ : H λ → H λ by deﬁning it to b e zero on the comp onen ts L 2 ( X ; E w ′′ ⋆λ ) with w ′′ 6 = w . F or the remainder of this se c tio n, w e use K α , A , K , L to denote K α ( H λ ), A ( H λ ), K ( H λ ), L ( H λ ). Lemma 4.13 (Ka sparov T ec hnical Theorem) . Ther e exist p ositive G -c ontinuous op er ators N 1 , N 2 ∈ L with the fol lowing pr op erties: (i) N 2 1 + N 2 2 = 1 , (ii) N i · K α i ⊆ K for e ach i = 1 , 2 , (iii) N i c ommutes mo dulo c omp act op er ators with  M f for al l f ∈ C ( X ) ,  U ( g ) for al l g ∈ G ,  the normalize d BGG op er ators ˜ T λ,w → w ′ , for al l w ← → w ′ , (iv) N i c ommutes on the nose with U ( k ) for al l k ∈ K , (v) N i c ommutes on t he n ose with the pr oje ctions Q w for al l w ∈ W , i.e . , N i is diagonal with r esp e ct to t he di r e ct sum de c omp osition of H λ . Note also that N 1 and N 2 commute, b y (i) . Pr o of. See [Bla98, Theore m 20.1.5]. The K -inv ariance of (iv) is o btained b y av er- aging ov er the K -tr anslates U ( k ) N i U ( k − 1 ) of N i . Also , the op erators P w ± Q w (taking a ll po ssible choices of sig ns) form a ﬁnite group of unitaries, so that a similar averaging trick gives prop erty (v). 22 Lemma 4.1 4. Ther e exist mutu al ly c ommuting op er ators N w → w ′ ∈ L , indexe d by the e dges of t he gr aph (4.2), with the fol lowing pr op ert ies: (i) N w → w ′ = N w ′ → w (ii) If w α ← → w ′ for α ∈ { α 1 , α 2 , ρ } , then N w → w ′ K α ⊆ K . (iii) If w ↔ w ′ ↔ w ′′ with w 6 = w ′′ then N w ′ → w ′′ N w → w ′ K ρ ⊆ K . (iv) F or any w, w ′′ ∈ W , P w ′ N w ′ → w ′′ N w → w ′ = δ w, w ′′ , wher e the sum is over w ′ such that w ↔ w ′ ↔ w ′′ . (v) N w → w ′ satisﬁes (iii), (iv) and(v) of L emma 4.13. R emark 4.15 . T o cla rify a po ssibly mislea ding nota tio nal p oin t, N w → w ′ do es not designate an op erator b et ween L 2 ( X ; E w⋆λ ) and L 2 ( X ; E w ′ ⋆λ ). Ra ther it is an op erator on H λ which w e will use to mo dify the op erator T λ,w → w ′ . Pr o of. With N 1 , N 2 as in the pr evious lemma, assign op erators N w → w ′ to each arrow as follows: w α 1 • o o − N 1 N 2 / / d d − N 2 2 $ $ H H H H H H H H H H H H H H H H H H H H H H H H w α 1 w α 2 • d d − N 2 $ $ H H H H H H H H H H 1 • z z N 1 : : v v v v v v v v v v v d d N 2 $ $ H H H H H H H H H H H w ρ • w α 2 • o o N 1 N 2 / / z z N 2 1 : : v v v v v v v v v v v v v v v v v v v v v v v v w α 2 w α 1 • z z N 1 : : v v v v v v v v v v The asser ted prop erties ca n b e easily chec ked using the pr o perties of N 1 and N 2 from Lemma 4.13 and the dia g ram (4 .2). It is w orth noting particularly that N 1 N 2 m ultiplies K ρ int o the compact op erators. Deﬁnition 4.16. Deﬁne F λ,w → w ′ := N w → w ′ ˜ T λ,w → w ′ . Lemma 4.17. F or any w ← → w ′ , (i) F λ,w → w ′ − F ∗ λ,w ′ → w ∈ K . (ii) [ F λ,w → w ′ , M f ] ∈ K , for a ny f ∈ C ( X ) , (iii) U ( g ) F λ,w → w ′ U ( g − 1 ) − F λ,w → w ′ ∈ K , for any g ∈ G , (iv) F λ,w → w ′ is K -invariant, ie , [ F λ,w → w ′ , U ( k )] = 0 , for any k ∈ K , (v) F λ,w → w ′ is G -c ontinu ous. Also , (vi) F or any w , w ′′ ∈ W , ( P w ′ F λ,w ′ → w ′′ F λ,w → w ′ ) ≡ δ w, w ′′ Q w (mo d K ) , wher e the sum is over w ′ ∈ W such t hat w ↔ w ′ ↔ w ′′ . 23 Pr o of. Le t w α ← → w ′ . By deﬁnition, T λ,w → w ′ = T ∗ λ,w ′ → w , so F λ,w → w ′ − F ∗ λ,w ′ → w = [ N w → w ′ , ˜ T λ,w → w ′ ], which proves (i). Since N w → w ′ commutes modulo c o mpacts with multiplication op erators , [ F λ,w → w ′ , M f ] ≡ N w → w ′ [ ˜ T λ,w → w ′ , M f ] (mod K ) By Lemma 4.8, the la tter is in N w → w ′ K α ⊆ K , which pro ves (ii). Similarly , for (iii), U ( g ) F λ,w → w ′ U ( g − 1 ) − F λ,w → w ′ ≡ N w → w ′  U ( g ) ˜ T λ,w → w ′ U ( g − 1 ) − ˜ T λ,w → w ′  (mo d K ) and the la tter is in N w → w ′ K α ⊆ K b y Lemma 4.12. F or a n y weight µ , the diﬀerent ial o perato r X i : L 2 ( X ; E µ ) → L 2 ( X ; E µ − α i ) is K - in v a riant . Likewise for its esse ntial adjoint Y i : L 2 ( X ; E µ − α i ) → L 2 ( X ; E µ ). Hence, Ph X i : L 2 ( X ; E µ ) → L 2 ( X ; E µ − α i ) is K -eq uiv a riant . The normalized BGG op erators T λ,w → w ′ are comp ositions of such o p erator s, and N λ → w w ′ is K -inv a riant by deﬁnition. This prov e s (iv). Once ag a in, G -con tinuit y is trivial. W e prov e (vi) in tw o s eparate cas e s . Firstly , supp ose w = w ′′ . F or a n y w ′ with w ← → w ′ , Lemma 4.10 implies that ˜ T λ,w ′ → w ˜ T λ,w → w ′ ≡ Q w (mo d K α ). By Lemma 4.14(iv), X w ′ F λ,w ′ → w F λ,w → w ′ ≡ X w ′ N w ′ → w N w → w ′ ˜ T λ,w ′ → w ˜ T λ,w → w ′ (mo d K ) ≡ X w ′ N w ′ → w N w → w ′ Q w (mo d K ) = Q w . If w 6 = w ′ , the result is trivia l unless there ex ists at least o ne w ′ such that w ↔ w ′ ↔ w ′′ . If such a w ′ exists, Lemma 4.11 implies that the pro ducts T λ,w ′ → w ′′ T λ,w → w ′ are independent o f this intermediate v er tex w ′ , mo dulo K ρ . Let us ﬁx o ne suc h pro duct and denote it temp orarily by T λ,w →·→ w ′′ . Then by Lemma 4.14(iv), X w ′ F λ,w ′ → w ′′ F λ,w → w ′ ≡ X w ′ N w ′ → w ′′ N w → w ′ ˜ T λ,w ′ → w ′′ ˜ T λ,w → w ′ (mo d K ) ≡ X w ′ N w ′ → w N w → w ′ ! ˜ T λ,w →·→ w ′′ (mo d K ) = 0 . Deﬁnition 4.18. Deﬁne F λ := P F λ,w → w ′ , where the sum is ov er a ll directed edges in the gr aph (4.2). Theorem 4.19. The op er ator F λ ∈ L deﬁnes an element θ λ ∈ K G ( C ( X ) , C ) . That is, (i) F λ is o dd with r esp e ct t o t he gr ading of H λ , 24 (ii) F λ − F ∗ λ ∈ K , (iii) F 2 λ − 1 ∈ K , (iv) [ F λ , M f ] ∈ K , for any f ∈ C ( X ) , (v) [ F λ , U ( g )] ∈ K , for any g ∈ G , (vi) F λ is G -c ontinu ous, Mor e over, F λ is K -invariant: [ F λ , U ( k )] = 0 for al l k ∈ K . Pr o of. This is mostly immediate from the previous lemma. T o b e explicit a bout the pro of of (iii), Lemma 4.17(vi) gives F 2 λ = X w,w ′ ,w ′′ ∈ W w ↔ w ′ ↔ w ′′ F λ,w ′ → w ′′ F λ,w ′ → w ′′ ≡ X w Q w (mo d K ) = 1 . Deﬁnition 4. 20. Let π λ denote the irreducible repr esen tation o f K with highest weigh t λ . Deﬁne a homomorphism of ab elian groups θ : R ( K ) → K K G ( C ( X ) , C ) [ π λ ] 7→ θ λ . Let ι : C → C ( X ) denote the G -equiv ariant C ∗ -morphism induced b y the map o f X to a p oint. Theorem 4 .21. The map ι ∗ ◦ θ : R ( K ) → R ( G ) is a ring homomorph ism which splits the r estriction homomorphism Res G K : R ( G ) → R ( K ) . Pr o of. Le t λ b e a dominant w eight. W e hav e that Res G K ι ∗ ◦ θ ([ π λ ]) is the K -index of F λ . Since F λ is K -equiv ar ia n t, it decomp oses as a direct sum of o pera tors o n the K -isotypical subspaces of H λ , each of which is ﬁnite dimensional (Example 3.2). The K -index of F λ is the sum of the indices of each comp onen t. T o compute this index, we compare with the classical BGG complex. Let µ := w ⋆ λ b e in the shifted W eyl orbit of λ . The induced bundle E µ of our normalized B GG -complex a nd the holo morphic bundle L hol µ of the classica l B GG complex (1.1) are identical as K -ho mogeneous line bundles. The classica l BGG resolution is exact and K -equiv ariant, so exact in each K -type. It follows that the index of F λ is [ π λ ]. Thus the co mposition Res G K ◦ ι ∗ ◦ θ is the ident ity on R ( K ). By Theore m 1.1, Res G K : γ R ( G ) → R ( K ) is a ring iso morphism, so it s uﬃces to show that the imag e of ι ∗ θ is in γ R ( G ). Using [Ka s88, Theo rem 3.6(1 )], w e hav e γ · ( ι ∗ θ λ ) = ι ∗ ⊗ C ( G / B ) (1 C ( G / B ) ⊗ γ ) ⊗ C ( G / B ) θ λ = ι ∗ ⊗ C ( G / B ) (Ind G B Res G B γ ) ⊗ C ( G / B ) θ λ . Since B is amenable, γ restricts to the unit in R ( B ), so γ · ( ι ∗ θ λ ) = ι ∗ θ λ . Corollary 4.22. γ = [( H 0 , U, F 0 )] ∈ R ( G ) . 25 A Harmonic analysis of longitudinal pseudo dif- feren tial op erators This appendix desc r ibes the pro o f of T he o rem 3 .1 8 (ii). As in [Y un], the key computation will be made using Gelfand-Tsetlin bas es. The follo wing summary of Ge lfa nd-Tsetlin bases follows the exp ository pap er [Mol0 6] together with some rema rks of [Y un]. W e immedia tely sp ecialize to the case of sl (3 , C ). W eights for gl (3 , C ) corresp ond to triples o f int egers m = ( m 1 , m 2 , m 3 ) via m :   t 1 0 0 0 t 2 0 0 0 t 3   7→ X i m i t i . Dominant w eights corresp ond to descending tr iples , m 1 ≥ m 2 ≥ m 3 . A Gelfand-Tsetlin p attern is an ar ra y of integers Λ :=   λ 3 , 1 λ 3 , 2 λ 3 , 3 λ 2 , 1 λ 2 , 2 λ 1 , 1   satisfying the interlea ving conditions λ k +1 ,j ≥ λ k,j ≥ λ k +1 ,j +1 . (A.1) T o each Gelfand-Tsetlin pa ttern there is asso ciated a vector ξ Λ in the irreducible representation π m with highest w eight m = ( λ 31 , λ 32 , λ 33 ). Thes e vectors ξ Λ form an orthogo nal (not orthono rmal) basis for this r epresent ation. When dealing with sl (3 , C ), rather than gl (3 , C ), there is some redundancy here. Two triples des c ribe the same sl (3 , C )-weight if and o nly if they diﬀer by a m ultiple of (1 , 1 , 1). Two Gelfand-Tsetlin patterns describ e the sa me basis vector if and o nly if they diﬀer by a multiple of the constant pattern   1 1 1 1 1 1   . W e us e the following standa rd no tation: s k := P k j =1 λ k,j is the s um o f the ent ries of the k th row; l k,j = λ k,j − j + 1; and Λ ± δ k,j denotes the Gelfand-Tsetlin pattern obtained from Λ b y adding ± 1 to the ( k , j )-ent ry . Then  ξ Λ is a weigh t vector, with weight ( s 1 − s 0 , s 1 − s 2 , s 3 − s 2 ).  The repres e n tatio n π m acts on this basis inﬁnitesimally a s follows: π ( X 1 ) ξ Λ = − ( l 11 − l 21 )( l 11 − l 22 ) ξ Λ+ δ 11 π ( X ∗ 1 ) ξ Λ = ξ Λ − δ 11 π ( X 2 ) ξ Λ = − ( l 21 − l 31 )( l 21 − l 32 )( l 21 − l 33 ) ( l 21 − l 22 ) ξ Λ+ δ 21 − ( l 22 − l 31 )( l 22 − l 32 )( l 22 − l 33 ) ( l 22 − l 21 ) ξ Λ+ δ 22 π ( X ∗ 2 ) ξ Λ = ( l 21 − l 11 ) ( l 21 − l 22 ) ξ Λ − δ 21 + ( l 22 − l 11 ) ( l 22 − l 21 ) ξ Λ − δ 22 26  The norm o f ξ Λ is g iv e n b y k ξ Λ k 2 = 3 Y k =2 Y 1 ≤ i ≤ j 1 2 ( k + 1 )( m + 1) 2 , so that | a m,j,k +1 − b m,j,k +1 | ≤ [16 C ( k ) + 2 C ( k − 1) + 2( k + 1) − 1 C 2 ( k + 1)]( m + 1) − 2 for all 0 ≤ j ≤ m with m ≥ 2 k . (A.20) If C ( k + 1 ) is the maxim um of the constants C 1 ( k + 1) a nd [16 C ( k ) + 2 C ( k − 1) + 2( k + 1) − 1 C 2 ( k + 1)], we are done. R emark A.6 . W e observed that | b m,j,k | ≤ ( m + 1) − 1 for all 0 ≤ j, k ≤ m . In the light of the estimate (A.17) we also have | a m,j,k | ≤ (1 + C ( k ))( m + 1) − 1 . Now co nsider the actio n of X 1 ∈ k C on the zero weigh t space of π ( m, 0 , − m ) . Note that X 1 maps p 0 V π ( m, 0 , − m ) to p α 1 V π ( m, 0 , − m ) . F rom Equatio n (A.6), (Ph X 1 ) ξ m,j = X 1 . ( X ∗ 1 X 1 ) − 1 2 ξ m,j = 1 p j ( j + 1 ) X 1 ξ m,j . (A.21) for j > 0 , a nd (Ph X 1 ) ξ m, 0 = 0. W e next giv e an a ppro ximate formula for Ph X 1 with res p ect to the alter nativ e basis { η Λ } . Recall that { η m,j } m j =0 and { η ′ m,j } m j =1 are orthogonal ba s es for p 0 V ( m, 0 , − m ) and p α 1 V ( m, 0 , − m ) , re s pectively . W e let y m,j := η m,j / k η m,j k = 1 m !(2 m + 1)! 1 2 η m,j , (A.22) y ′ m,j := η ′ m,j / k η ′ m,j k = 1 m !(2 m + 1)! 1 2 s 2 j ( m + 1 ) 2 − j 2 η m,j (A.23) be the corresp onding orthono r mal bas es. Lemma A.7. F or e ach ﬁxe d k ∈ N , lim m →∞ |h (Ph X i )y m, 0 , y ′ m,k i| = √ 2 k  1 2 k − 1 − 1 2 k + 1  . Pr o of. Using , success iv ely , Equations (A.22) and (A.23), Lemma A.2, Equa tion 31 (A.21), E quation (A.10), and E q uation (A.14), we compute h (Ph X 1 )y m, 0 , y ′ m,k i = 1 m ! 2 (2 m + 1)! s 2 k ( m + 1 ) 2 − k 2 h (Ph X 1 ) η m, 0 , η ′ m,k i = ω m m ! 2 (2 m + 1)! s 2 k ( m + 1 ) 2 − k 2 m X j =0 ( − 1) j 2 j + 1 m + 1 h (Ph X 1 ) ξ m,j , η ′ m,k i = ω m m ! 2 (2 m + 1)! s 2 k ( m + 1 ) 2 − k 2 m X j =0 ( − 1) j m + 1 2 j + 1 p j ( j + 1 ) h ( X 1 ξ m,j , η ′ m,k i = ω m m ! 2 (2 m + 1)! 1 ( m + 1 ) s 2 k ( m + 1) 2 − k 2 m X j =0 ( − 1) j (2 j + 1) p j ( j + 1) h ( ξ m,j , X ∗ 1 η ′ m,k i = ω m m ! 2 (2 m + 1)! p 2 k (( m + 1) 2 − k 2 ) ( m + 1) × m X j =0 ( − 1) j 2 j + 1 2 p j ( j + 1) h ( ξ m,j , η m,k − 1 + η m,k i = ω m s 2 k  1 − k 2 ( m + 1 ) 2  m X j =0 j + 1 2 p j ( j + 1) ( a m,j,k − 1 + a m,j,k ) . (A.24) W rite the in the ﬁnal line as m X j =0 j + 1 2 p j ( j + 1) ( a m,j,k − 1 + a m,j,k ) (A.25) = m X j =0 j + 1 2 p j ( j + 1 ) − 1 ! ( a m,j,k − 1 + a m,j,k ) (A.26) + m X j =0 ( a m,j,k − 1 − b m,j,k − 1 ) + ( a m,j,k − b m,j,k ) (A.27) + m X j =0 ( b m,j,k − 1 + b m,j,k ) . (A.28) In the sum (A.26), j + 1 2 p j ( j + 1) − 1 ! = s j 2 + j + 1 4 j 2 + j − 1 ≤ 1 8 j 2 , which is summable, while a m,j,k − 1 and a m,j,k are bo th O ( m − 1 ) for ﬁxed k , by Remark A.6, so (A.26) tends to 0 as m → ∞ . By Lemma A.5 , the sum (A.27) is bo unded by ( m + 1) . [ C ( k − 1) + C ( k )]( m + 1) − 2 , so also v a nishes as m → ∞ . Finally , m X j =0 b m,j,k = 1 m + 1 m X j =0 P k  2  j m +1  2 − 1  (A.29) 32 is a Riemann sum for the integral ( − 1) j R 1 0 P k (2 t 2 − 1 ) dt Thus, using the sub- stitution u = 1 − 2 t 2 , (A.29) conv er g es to 2 − 3 2 Z 1 − 1 (1 − u ) − 1 2 P k ( − u ) du = ( − 1) k 2 k + 1 (see e.g., [GR6 5, 7 .225(3)]). W e obtain that the sum (A.28), a nd hence (A.25) conv er ges to ( − 1) k − 1  1 2 k − 1 − 1 2 k +1  as m → ∞ . Putting this into Equation (A.24) pr o ves the result. In the following lemmas, σ 0 will denote the trivia l representation of K 2 . Lemma A.8. On a ny un itary K -r epr esentation H , (P h X 1 ) p σ 0 ∈ K α 2 ( H ) . Pr o of. W e will prove the equiv alent condition of Prop osition 3.5(ii). Note that if F ⊂ ˆ K 2 contains σ 0 then (Ph X 1 ) p σ 0 p ⊥ F = 0, so we only need that for an y ǫ > 0 there is a ﬁnite s et F ⊂ ˆ K 2 such that k p ⊥ F (Ph X 1 ) p σ 0 k < ǫ. (A.30) W e beg in with the cas e o f H an ir reducible K -repr esen tation, say of highest weigh t ( m 1 , m 2 , m 3 ). If η ∈ H is of trivial K 2 -type, then it has trivial s 2 -type and weigh t 0. By the de ﬁning prop erties of the Gelfa nd- Tsetlin basis , η m ust b e a sca la r m ultiple of η m, 0 for some m ∈ N . Therefore p σ 0 = 0 on an y irreducible representation other than the r e presen tations π ( m, 0 , − m ) considered ab ov e. On V π ( m, 0 , − m ) , p σ 0 is the pro jection onto the span of y m, 0 . Let σ ′ k denote the K 2 -type in the α 1 -weigh t space with hig he s t w eight ( k − 1 , − k ) for s 2 , s o that p σ ′ k is the pro jection onto y ′ m,k . Then, k p ′ σ k (Ph X 1 ) p σ 0 k 2 = |h y ′ m,k , (Ph X 1 )y m, 0 i| 2 . Cho ose l with (2 l + 1) − 2 < 1 2 ǫ . Applying Lemma A.7 to k = 1 , . . . , l , we can ﬁnd M suﬃciently lar ge that for all k ≤ l , |h y ′ m,k , (Ph X 1 )y m, 0 i| 2 ≥ 2 k  1 2 k − 1 − 1 2 k + 1  2 − 1 2 l ǫ for all m ≥ M . Putting F l := { σ ′ 1 , . . . , σ ′ l } , we get k p ⊥ F l (Ph X 1 ) p σ 0 k 2 = k (Ph X 1 ) p σ 0 k 2 − l X k =1 k p σ ′ k (Ph X 1 ) p σ 0 k 2 ≤ 1 − l X k =1 2 k  1 2 k − 1 − 1 2 k + 1  2 − 1 2 l ǫ ! ≤ 1 + 1 2 ǫ + l X k =1 1 (2 k − 1) 2 − 1 (2 k + 1) 2 = 1 + 1 2 ǫ −  1 − 1 (2 l + 1) 2  < ǫ, 33 so that (A.30) holds on all π ( m, 0 , − m ) with m ≥ M . In the ﬁnitely man y K -represe ntations π ( m, 0 , − m ) with m < M , there can only a ppear ﬁnitely many K 2 -types. Let F ⊂ ˆ K 2 be the ﬁnite set co n ta ining all of these K 2 -types as well as σ ′ 1 . . . , σ ′ l and σ 0 . Then p ⊥ F (Ph X 1 ) p σ 0 = 0 on V ( m, 0 , − m ) for all m < M . Hence k p ⊥ F (Ph X 1 ) p σ 0 k < ǫ on all ir reducible representations. F or a general unitary K -r epresen tation H , p ⊥ F (Ph X 1 ) p σ 0 decomp oses as a sum of ope r ators on each irreducible co mponent, so the s ame estimate holds . Lemma A.9. On any unitary K -r epr esentation H the op er ators (P h X 1 ∗ ) p σ 0 , and ther efor e p σ 0 (Ph X 1 ) , ar e in K α 2 ( H ) . Pr o of. Le t U b e a unitary r epresentation of K on H . The a n tilinea r map J : H → H † ; ξ 7→ h ξ , · i intert wines the repres en ta tio ns U and U † . One can c heck that for an y X in the complexiﬁcation k C , J − 1 U † ( X ) J = − U ( X ) ∗ . Since J is anti-unitary , J − 1 Ph ( U † ( X )) J = − Ph ( U ( X ∗ )) If ξ ∈ H has K 2 -type σ , then J ξ has K 2 -type σ † , so J − 1 p † σ J = p σ . Thus, by conjuga ting by J , the estimate A.30 implies k p ⊥ F † (Ph X 1 ∗ ) p σ 0 k < ǫ , where F † := { σ † | σ ∈ F } . This completes a base case in the pro of of Ph X 1 ∈ A α 2 . W e now need to replace σ 0 by an arbitrar y K 2 -type in the preceding tw o le mma s. T o do so, we use a trick based on the following fact. Lemma A.10. F or any σ ∈ ˆ K 2 , ther e exists a ﬁnite c ol le ction of c ontinuous functions ψ 1 , . . . , ψ n ∈ C ( K ) such that p σ = P n j =1 M ψ j p σ 0 M ψ j as an op er ator on L 2 ( K ) . Pr o of. B y the P eter -W eyl Theorem, the righ t regular repres en ta tion of K 2 has a ﬁnite dimensional σ -iso t ypica l subspace with a bas is consisting of contin uo us functions b 1 , . . . , b m ∈ C ( K 2 ). Thus, for any f ∈ L 2 ( K 2 ), p σ f = n X j =1 b j h b j , f i = n X j =1 M b j p σ 0 M b j f . (A.31) Let Y ⊂ K / K 2 be open and ζ : Y → K be a contin uous local sectio n of the principal K 2 -bundle q : K → K / K 2 . W e now deﬁne functions on K which e qual b 1 , . . . , b m on each ﬁbre ov er Y . T o be precise, deﬁne b Y 1 , . . . , b Y m on K by b Y j ( k ) := ( b j ( h ) , if k = ζ ( y ) h fo r some y ∈ Y , h ∈ K 2 , 0 if q ( k ) / ∈ Y . By applying (A.31) ﬁbrewise we get p σ f = P n j =1 M b Y j p σ 0 M b Y j f for any f ∈ L 2 ( K ) supp orted on q − 1 ( Y ). Now let U = { ( Y l , ζ l ) } be a ﬁnite atlas of suc h gauges. Let a l ∈ C ( K / K 2 ) be such that { a 2 l } is a par tition o f unity sub ordinate to this a tlas, and pull back to ˜ a l := a l ◦ q ∈ C ( K ). Then b Y l j ˜ a l is contin uous on K for each j, l , and for any f ∈ L 2 ( K ), p σ f = X l a 2 l f = X l X j M ( b Y l j ˜ a l ) p σ 0 M ( b Y l j ˜ a l ) f . 34 Lemma A.11. L et ν b e a weigh t of K . F or a ny f ∈ C ( K ) , [Ph X 1 , M f ] p ν and [Ph Y 1 , M f ] p ν ar e in K α 1 ( L 2 ( K )) . Pr o of. Supp ose ﬁrst that f is a weigh t vector for the right regular repres en ta tion, i.e. , f ∈ C ( X ; E − µ ) for some µ . Then Lemma 3 .19 says that [Ph X 1 , M f ] : p ν L 2 ( K ) → p ν + µ + α 1 L 2 ( K ) is in K α 1 , which implies the r e s ult. The subspace spanned by these weight vectors contains all matrix units, so is uniformly dense in C ( K ). A density argument completes the pr oof. Similar ly , [Ph Y 1 , M f ] p ν ∈ K α 1 . Theorem A.12. O n any un itary K -r epr esentation H , Ph X i and Ph Y i ar e in A ( H ) for i = 1 , 2 . Pr o of. W e b egin with H = L 2 ( K ) with the rig h t regula r re presen tation, a nd consider Ph X 1 . As in Lemma 3.7, the ﬁnite multiplicit y of K -t y pes in L 2 ( K ) implies that A Σ ( L 2 ( K )) = L ( L 2 ( K )), so Ph X 1 ∈ A Σ trivially . Since Ph X 1 maps the µ -weigh t space in to the ( µ + α 1 )-weigh t s pace for each w eight µ , it is M -harmonica lly proper , so in A ∅ . Since X 1 ∈ ( k 1 ) C , Ph X 1 preserves K 1 -types, so is in A α 1 . It remains to s ho w Ph X 1 ∈ A α 2 . Let σ ∈ ˆ K 2 and let ψ 1 , . . . , ψ n ∈ C ( K ) be as in Lemma A.10. Then (Ph X 1 ) p σ = n X j =1 (Ph X 1 ) M ψ j p σ 0 M ψ j = n X j =1 M ψ j (Ph X 1 ) p σ 0 M ψ j + n X j =1 [(Ph X 1 ) , M ψ j ] p σ 0 M ψ j . Since p σ 0 pro jects int o the 0-weigh t spa ce, Lemmas A.8, A.11 and 3.11, im- ply (Ph X 1 ) p σ ∈ K α 2 . A s imilar computation using Lemma A.9 shows that (Ph Y 1 ) p σ = (Ph X 1 ∗ ) p σ ∈ K α 2 , so p σ (Ph X 1 ) ∈ K α 2 . By Pr opos ition 3.6, Ph X 1 ∈ A α 2 . Therefore Ph X 1 ∈ A . By taking adjoints, Ph Y 1 ∈ A . Conjugation by the long est W eyl group element interc hanges A α 1 and A α 2 and ﬁxes A ∅ and A Σ , so ﬁxes A . It also sends X 1 and Y 1 to Y 2 and X 2 , resp ectiv ely . W e obtain Ph Y 2 , Ph X 2 ∈ A . The theorem remains true if H is a dir ect sum of arbitra rily ma ny copies of the regula r repre s en ta tion. Since every unitar y K - representation can be equiv- ariantly em b edded into such a direct sum, we are done. Corollary A.13. L et H b e any unitary K -r epr esen t ation. F or i = 1 , 2 and any weight µ , Ph X i : p µ H → p µ + α i H is in A . In p articular, P h X i : L 2 ( X ; E − µ ) → L 2 ( X ; E − µ − α i ) ∈ A . The ab ov e r esult is suﬃcient for the applications of this pap er. But the generaliza tion to a rbitrary order zero longitudinal pseudo diﬀeren tial op erators (Theorem 3.18) is easily deduced from it, a nd p erhaps useful for future appli- cations. 35 Pr o of of The or em 3.18. Start with the case E = E ′ = E 0 , the trivial line bundle ov er X . Recall Co nnes’ short exac t sequence 0 / / Ψ − 1 F i ( E 0 ) / / Ψ 0 F i ( E 0 ) Symb F i / / C ( S ∗ F i ) / / 0 . W e know from [Y un] that Ψ − 1 F i ( E 0 ) ⊆ A . Let C ⊆ C ( S ∗ F i ) b e the ima ge of Ψ 0 F i ( E 0 ) ∩ A under the long itudinal pr inc ipa l s ym b ol map. W e prov e that C = C ( S ∗ F i ) b y sho wing that it sepa rates the po in ts of S ∗ F i , in the sense o f the Stone-W eierstrass Theo rem. F or an y f ∈ C ( X ), the m ultiplication oper ator M f is in Ψ 0 F i ( E 0 ) ∩ A , so the function algebra C separates p oints in diﬀerent ﬁbres of S ∗ F i . The longitudinal principal sym b ol o f Ph X i separates points in the ﬁbre at the identit y coset (see Lemma 3.17). Let ϕ ∈ C ( X ; E α ) be a n y smo oth section of E α which is nonzer o at the iden tity coset. Then M ϕ Ph X i ∈ Ψ 0 F i ( E 0 ) ∩ A and its principa l s ym b ol separates p oin ts of the ﬁbre a t the identit y coset. Conjugating by translations by k ∈ K , C separates p oin ts in any ﬁbre. Now suppose E = E µ , E ′ = E ν are gener al K -homogeneo us line bundles. Find par titions of unit y ϕ 1 , . . . , ϕ n ∈ C ( X ; E µ ), ϕ ′ 1 , . . . , ϕ ′ m ∈ C ( X ; E ν ) in the sense of Lemma 2.1. If A ∈ Ψ 0 F i ( E , E ′ ), then M ϕ ′ j AM ϕ k ∈ Ψ 0 F i ( E 0 ) ⊆ A for each j, k . Hence A = P j,k M ϕ ′ j M ϕ ′ j AM ϕ k M ϕ k ∈ A . 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The Bernstein-Gelfand-Gelfand complex and Kasparov theory for SL(3,C)

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