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The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space I]ZP2‘“`L

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The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space I]ZP2‘“`L The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space i]ZP2‘“`l S. Seifarth, U. Semmelmann SFB 288 Preprint No. 95 Diese Arbeit ist mit Untersttitzung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 288 entstanden und als Manuskript vervielfiitigt worden. Berlin. Dezember 1993 OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON THE ODD DIMENSIONAL COMPLEX PROJECTIVE SPACE CP2"‘"l S. SEIFARTH AND U. SEMMELMANN ABSTRACT. We give explicitly the eigenvalues of the Dirac operator on the complex projective space CP" 0. INTRODUCTION In this paper we want to present a calculation of the eigenvalues of the Dirac operator on the complex projective space CP". This calculation has been done in [CFGKQ]. As their result differs from ours, we included two independent proofs of our result in a special case (for the subbundle S0). We prove the following theorem Theorem 0.1. On the complex projective space of complex dimension n, endowed with the F·u.bini—Stady metric, the spectrum of the Dirac operator consists of the series 1,/AU, and ;l;/y,;_k,, where AU, =l2 + %l(3n —- 2k — 1)+ %(n — k)(·n. —- 1) k€{1,...,n} andl2mam{1,k—%} (01) n pu, =l2 + §l(3n — 2k +1)+ %(n — +1) k€{0,...,n—1}andl2max{O,k— Though the method of calculating the spectrum of the Dirac operator using the Casimir operator is well known, we wrote down everything as explicitly as possible to make the computations more transparent. Our paper is organized as follows. We begin with fixing the notations and parame trisations. ln the second part we recall geometric aspects like the spin structure and the spinor bundle. Then we define the Dirac operator, recall the connection with the Casimir operator and the Peter—Weyl theorem. ln section four we cite the branching rules from [CFG89] and [lT78l. Then we have the necessary material for the explicit calculation (section tive). OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON <CP ”""`1 In the last section we included an independent proof of our assertion for the special case S0 and further remarks. 1. NOTATIONS We regard the Riemannian symmetric pair (cf. [Hel'78], ch.IV) (G, K) with G = S U (n + 1) a compact semi—simple Lie group and K : S(U(n) X U(1)) : {3( dc,<j_,-.) IA 6 uw} Q mn) a reductive subgroup. If go and E0 are the (real) Lie algebras of the groups G and K, we will denote by po the orthogonal complement of Eg in go with respect to the Killing form B(-, For G, K as above the Killing form is a multiple of the trace form: B(X, Y) : 2(n +1)tr(XY). With our choice of the parametrisation of K we get ,,0: gx gw}. The complexified Lie algebras of gc, E0 etc. we will denote by g, E etc. We fix a maximal torus n+1 (1.1) _ _ T = {diag(e"8‘, . .. ,e`p"*‘) |Hj E R, = 0}, 5:1 common to G and K, with (complexified) Lie algebra n+1 (1-2) *= {B ¤= d1¤g(U1 ,--- .%+1) I Bk € C, Z Bj = 0} j=1 Then 11+1 *' = {E 7161 I rj 6 C}, .i=1 where z—:;,(B) = Bk. The element rjsj we simply denote by (1-1,. , ·r,,+1) Because of the relation Bj = 0 we can set @,+1 : —— EQ2, Bj, i.e. the torus is in fact n—dimensional. Moreover, 11+1 n n Z Tieiipl Z" ZT5Bj ` 2:%+1% : Zi"} ‘ T¤+1 j=I j=1 j:1 j:]. By settting Z: Tj ·* 7},+1 OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2""1 for j : 1,... ,n we get a parametrisation of the elements of t' as ·n.~vectors, which we will denote by [t1,... ,¢,,]. The inverse formula is t,+t forj= 1,... ,n T- __ J `_ t for j = ·n. + 1 with t = —T—£-3 2;}:, tj. ln the sequel we will use freely both parametrisations without further comments. The root system <I> = <I>(g, t) of t in g is generated by the simple roots (1.4) :={O¢;:6;—E;+1, i=l,...,TL+l}. By <I>+ we will denote the positive system, corresponding to this set of simple roots Z and by EG the corresponding halfsum of the positive roots. Then 6;; = -1,... ,——§). Since { is a maximal torus for E too, we can also define the root system <I>K : = <I>K(l?, t), generated by the simple roots (1.5) ::{oz,=s;—e,+1, i:1,... ,·n.}. The corresponding halfsum of the positive roots is 6K : ($,§— 1 ,... ,-%,0) A weight (1*1,... ,*5,+1) is dominant with respect to <I>+ if Tl 2 . .. 2 *5,+1, and integral, if Tg -—·rj is integral for all ·i, j. For the n—vecto1·s that means that [tl, . , tn] is dominant, if tl 2 . 2 tn 2 O and integral, if all ti are integers. 2. THE COMPLEX PROJECTIVE SPACE The complex projective space CP" can be realised as a compact symmetric space G/ K with G and K like in section 1. If we fix the point 1:0 = eK in G/K we may identify po with the tangent space in :1:0, pg : T:OG/K. We use the Riemannian metric induced from the negative of the Killing form B. Note that this is exactly (n + 1) times the Fubini—Study metric. Homogenous spin structures on symmetric spaces G/ K correspond to lifts 54 of the isotropy representation at K ——» SO(T,0G/K) 2 SO(2·n.), i.e. 54 is a homomorphism K —> Spin(2n) with cx : A 0 5, where A : S pin(2n) —> S O(2n) is the non-trivial twofold covering of S O(2·n.). OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP 2""1 The complex projective space CP" admits a spin structure if and only if n is odd. This spin structure is unique and homogenous. Let fi : C li f f2,, —> HomC(S, S) be an irreducible representation of the Cliilord algebra (unique up to equivalence). We will denote the restriction of fi to Spin(2n) as well by Then the spinor bundle S of CP" is given as the associated vector bundle 8 = G XR; S. Since the dimension of CP" is even, we have S = 8+ 6) S`, the decomposition into half—spinors. The representation ( ,5 0 54, S) of K = S(U X U( 1)) now decomposes into irreducible components. Lemma 2.1. [Par72], [CFG89]} The representation (fic Z2, S) of the group K = S(U(n) X U(1)) decomposes in (n + 1) irreducible components (2.1) S : QQ Sk k=O with the ha,U spinor modules S+ = @ Sk _ k;O mod.2 _ S : EB Sk. k£1 mod 2 Here Sk is the representation with highest weight uk:(&%il—k,..., #——k, %—k ,...,%—k) (vi.-Io) (n-k+x) Remark. It is easy to see that the highest weights Vk of the irreducible components Sk of S all are of the form 205;; — 5K for some element w in the Weyl group WG (cf. [Par72]). According to the decomposition (2.1) there is a decomposition of the spinor bundle in a sum of holomorphic subbundles (2.3) S : @98;,. k:O 3. THE Dimxc OPERATOR Our description of the action of the Dirac operator is similar to [CFG89]. We use the identification (3-1) U°°(G/K»$*) 2 (U°°(G)®$*)K» where ·· )K refers to the subspace of K —invariants with K acting on C'°°(G) by right translation and on sby sz:i 5*d: o Et. OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2 '""1 Let {X,} be an orthonormal basis of po. Each X, determines a left invariant vector field r(X,) by infinitesimal right translation. In terms of the identification (3.1) the Dirac operator on 8* is given as (3.2) 0* Z Zr(X,) rg) p*(x,) ; (0~=·(o) ® s*)K Z» (0°¤(G) ® sr) It is easy to see that the Dirac operator preserves the K ·-invariants in C°°(G) ® 5*, and that it commutes whith the action of G. Sections in the spinor bundle may also be viewed as functions on G with values in se which are equivariant, i.e. functions f : G ——+ 5* with f(gk) = .s*(k"1)f(g) for all g G G and all k E K. The space of these functions we will denote by C'°°(G, 5*)K. Using a hermitian scalar product on the spinor module 5*, we can define a scalar product on C°°(G, se )K and consider the Hilbert space completion LI’(G, 5*) The Dirac operator extends to a self adjoint operator on .C2(G, 5*) Denote by G the unitary dual of G and let (lQ,7r,) for ry 6 G be a representative of the equivalence class ry. The L2 sections of the subbundle Sk are described by the Frobenius reciprocity and the Peter—Weyl theorem: (3.3) r’(s,,) Z Q; iq ® HOmK(v,, sk), ·v€G where ® is a direct, orthogonal sum and HomK(l@,5k) denotes the space of K invariant homomorphisms V, ——> Sk. All these summands are finite dimensional and invariant under the action of the Dirac operator D. The elements v ® A of KQ ® HomK(V,, Sk) define smooth sections ofthe bundle Sk by (11 ® A)(g) : Ao 1r.,(g’1)·v and are therefore in the domain of the Dirac operator D. The spectrum of D is completely determined by the spectra of the operators D., for *7 E G, where D, is defined as follows.
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