Journal of Lie Theory Volume 9 (1999) 491–506 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

Spectra of self-gradients on spheres

Thomas Branson

Communicated by B. Ørsted

Abstract. We give a general formula for the spectral resolution of a class of first-order differential operators on the sphere S^{n} which includes, among the most elementary cases, the Dirac and Rarita-Schwinger operators.

1. Introduction

There is a long literature on spectra of natural second-order differential operators on vector bundles over homogeneous spaces; see, e.g., [11, 9, 10, 3]. Recently, there has been some interest in explicit spectra of natural first-order operators like the Dirac and Rarita-Schwinger operators. The purpose of this paper is to show, by theorem and example in the case of the sphere S^{n}, that such explicit spectra are computable using general results of [6] on the spectra of intertwining operators for group representations, and results like those of [4] which employ intertwinors to get spectral information on other operators which do not enjoy quite as much symmetry.

All spaces of eigensections on the sphere treated below are expressed as irreducible Spin(n + 1)-modules, or direct sums of such. As a result their dimensions, and thus the multiplicities of all eigenvalues, are computable via Weyl’s dimension formula.

2. Parameterization by dominant weights

Irreducible representations of Spin(n), and thus irreducible associated Spin(n)- bundles, are parameterized by dominant weights (λ_{1} , …, λ_{ℓ}) ∈ ℤ^{ℓ} ∪ (\frac{1}{2} + ℤ)^{ℓ}, ℓ = [n/2], satisfying the inequality constraint

\begin{matrix}λ_{1} ≥ … ≥ λ_{ℓ} ≥ 0, & n odd, \\ λ_{1} ≥ … ≥ λ_{ℓ−1} ≥ |λ_{ℓ}| , & n even. \end{matrix}

The dominant weight λ is the highest weight of the corresponding representation. The representations which factor through SO(n) are exactly those with λ ∈ ℤ^{ℓ}.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag We shall denote by V (λ) the representation with highest weight λ. If M is an ndimensional smooth manifold with Spin(n) structure and F is the bundle of spin frames, we denote by 𝕍(λ) the vector bundle F ×_{λ} V (λ). When spin structure is not involved (i.e. when λ is integral), we may use the orthonormal frame bundle in constructing 𝕍(λ). We shall denote by χ(n) the set of dominant Spin(n) weights.

3. The selection rule

We shall discuss several familiar examples of bundles and identify their highest weights below. One important highest weight is that of the defining representation V (1, 0, …, 0) of SO(n). The classical selection rule describes the Spin(n) decomposition of V (1, 0, …, 0) ⊗ V (λ) for an arbitrary dominant λ:

V (1, 0, …, 0) ⊗ V (λ) ≅_{Spin(n)} V (σ_{1}) ⊕ … ⊕ V (σ_{N(λ)}),  (1)

where the σ_{u} are distinct (σ_{u} ≅_{Spin(n)} σ_{v} ⇒ u = v), and a given σ appears if and only if σ is a dominant weight and

σ = λ ± e_{a} , some a ∈ \{1, …, ℓ\},  \underline{or}  (2)
n is odd, λ_{ℓ} ≠ 0, σ = λ.  (3)

Here e_{a} is the a\underline{th} standard basis vector in ℝ^{ℓ}. (1) implicitly defines a numerical invariant N(λ), the number of selection rule “targets” of V (λ). We shall use the notation

λ ↔ σ

for the selection rule: λ ↔ σ if and only if V (σ) appears as a summand in V (1, 0, …, 0) ⊗ V (λ). The notation ↔ is justified because the relation is symmetric. In fact, one can see a priori that the relation must be symmetric: the defining representation of SO(n) is real, and thus self-contragredient.

4. The branching rule

The classical branching rule governs the restriction of a Spin(n + 1) representation to a copy of Spin(n) which is embedded in the standard way, as follows. By changing n to n + 1 above we have a parameterization of the irreducible representations of Spin(n + 1). For a dominant Spin(n + 1)-weight α, let V(α) be the corresponding irreducible representation. The branching rule says that dim Hom_{Spin(n)}(V (λ), V(α)|_{Spin(n)}) ∈ {0, 1}, with dim Hom_{Spin(n)}(V (λ), V(α)|_{Spin(n)}) = 1 if and only if

α_{1} − λ_{1} ∈ ℤ and \{ \begin{matrix}α_{1} ≥ λ_{1} ≥ α_{2} ≥ ⋯ ≥ λ_{ℓ} ≥ |α_{ℓ+1}|, & n odd, \\ α_{1} ≥ λ_{1} ≥ α_{2} ≥ ⋯ ≥ λ_{ℓ−1} ≥ α_{ℓ} ≥ |λ_{ℓ}|, & n even. \end{matrix} (4)

We use α ↓ λ or λ ↑ α as an abbreviation for (4).

5. Stein-Weiss operators (gradients)

An interesting concept related to the selection rule is that of generalized gradients, or Stein-Weiss operators [13]. The covariant derivative ∇ carries sections of 𝕍(λ) to sections of

T^{∗}M ⊗ 𝕍(λ) ≅_{Spin(n)} 𝕍(1, 0 …, 0) ⊗ 𝕍(λ)
≅_{Spin(n)} 𝕍(σ_{1}) ⊕ … ⊕ 𝕍(σ_{N(λ)}).

Since the selection rule is multiplicity free, we may project onto the unique σ_{u} summand; the result is our gradient:

G_{u} = G_{λσ_{u}} = Proj_{u} ∘∇.

Up to normalization, some examples of gradients, or direct sums of gradients, are the exterior derivative d, its formal adjoint δ, the conformal Killing operator S, the Dirac operator, the twistor operator, and the Rarita-Schwinger operator. In fact, every first-order Spin(n)-equivariant differential operator is a direct sum of gradients [8].

6. Proper self-gradients

The exceptional case (3) in the selection rule, in which λ ↔ λ, together with the above concept of gradient, provides an interesting series of first-order operators. In this case, G_{λλ} carries sections of 𝕍(λ) to sections of a copy of 𝕍(λ) which lives as a subbundle in T^{∗}M ⊗ 𝕍(λ). Examples are, up to constant multiples, the Dirac and Rarita-Schwinger operators (the cases λ = (\frac{1}{2}, …, \frac{1}{2}) and (\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}) respectively), and the operator ⋆d on (n − 1)/2-forms, in odd dimensions n ≥ 3. Given any tensor-spinor realization 𝕍 of 𝕍(λ), the target of the corresponding realization of G_{λλ} is a subbundle 𝕎 of T^{∗}M ⊗ 𝕍, with 𝕎 ≅_{Spin(n)} 𝕍. If we would like to use the same realization 𝕍 as both source and target bundle for a realization D_{λ}^{self} of G_{λλ}, we need a choice of normalization. First, normalize the Hermitian inner product on T^{∗}M ⊗ 𝕍 so that

|ξ ⊗ v|^{2} = |ξ|^{2}|v|^{2};  (5)

then normalize D_{λ}^{self} so that (D_{λ}^{self})^{2} = G_{λλ}^{∗}G_{λλ}. This determines D_{λ}^{self} only up to multiplication by ±1. It is important to note that this ambiguity is in the nature of things: it is analogous to the ambiguity in the naming of the complex units ±\sqrt{−1}. Indeed, this is more than an analogy: gradients generalize the Cauchy-Riemann equations [13], which are sensitive to the renaming of ±\sqrt{−1}. In our examples, the ambiguity may be viewed as residing in a choice of fundamental tensor-spinor (or more generally, a Clifford structure) γ. The Clifford relations and spin connection (in particular the relation ∇γ = 0) are invariant under interchange of γ and −γ, but the Dirac operator γ^{a}∇_{a} undergoes a sign change. As is standard in complex variables and spinor theory, we shall implicitly make a choice in each example. When examples interact, as in the series of spinor-form bundles treated in Sec. 10., it is of course important to make a consistent choice.

7. Frobenius reciprocity

Let λ ∈ χ(n) and α ∈ χ(n + 1) be given. By Frobenius reciprocity (see, e.g., [7]),

Hom_{Spin(n+1)}(V(α), Ind_{Spin(n)}^{Spin(n+1)} V (λ)) ≅_{Spin(n+1)} Hom_{Spin(n)}(V(α)|_{Spin(n)}, V (λ)).

The induced representation Ind_{Spin(n)}^{Spin(n+1)} V (λ) is carried by E(λ), the space of Spin(n + 1)-finite sections of 𝕍(λ) on Spin(n + 1)/ Spin(n) = S^{n}. As a result,

E(λ) ≅_{Spin(n+1)} ⨁_{α↓λ} V(α).  (6)

For α ↓ λ, let V(α; λ) be the copy of V(α) appearing in this decomposition; then

E(λ) = ⨁_{α↓λ} V(α; λ).

Spin(n)-equivariant differential operators are Spin(n+1)-invariant in the induced representation. Thus by Schur’s Lemma, every Spin(n)-equivariant differential operator D takes on an eigenvalue on each V(α; λ):

D|_{V(α;λ)} = eig(D, α) id_{V(α;λ)} .

8. Eigenvalue formulas

In [6], formulas for eigenvalues of intertwining operators on G/P (G a semisimple Lie group and P a maximal parabolic) are developed. In the present case, these results apply with G = Spin_{0}(n+1, 1); then G/P = S^{n} (as a conformal manifold). This special case was developed earlier, in [2], Sec. 5. A technical condition for the validity of the eigenvalue formulas involves the effect of multiplication by the cocycle of the principal series. Luckily, in the present setting, this condition has an elementary statement:

α ↔ β, α ≠ β, α, β ↓ λ ⇒ Proj_{V(β;λ)}(xV(α; λ)) ≠ 0

for x any of the homogeneous coordinate functions on S^{n} (i.e. the coordinates in the ambient ℝ^{n+1}). This condition was established for general λ in [4], Sec. 6. For ease of notation, we write the result here only for operators of nonnegative order; in particular, this covers all differential intertwinors.

Let L := [(n + 1)/2]; this is the number of entries in a Spin(n + 1)-weight. Theorem 8.1. Suppose it is not the case that

n is even and λ_{ℓ} ≠ 0.  (7)

Let Re(r) ≥ 0. Then there exists a nonzero intertwinor Ind_{P}^{G} λ ⊗ (−rν_{0}) ⊗ 1 → Ind_{P}^{G} λ ⊗ rν_{0} ⊗ 1, and any such operator must have eigenvalue

c ∏_{a=1}^{L} \frac{Γ(α̃_{a} + \frac{1}{2} + r)}{Γ(α̃_{a} + \frac{1}{2} − r)}  (8)

on V(α; λ). In particular, for r = \frac{1}{2} , the eigenvalue is

c ∏_{a=1}^{L} α̃_{a}.  (9)

Here c is a nonzero constant which is independent of α, and α̃ is the rho-shift of α:

α̃ = α + ρ_{n+1}, (ρ_{n+1})_{a} = (n + 1 − 2a)/2.

ν_{0} is the unique positive (𝔤, 𝔞) root, MAN being the Langlands decomposition of the parabolic subgroup P; see [6] for details. Note that if poles of the Gamma function come into play in (8), the order of the pole of the denominator at r is at least the order of the pole of the numerator; thus the expression (8) is analytic in r in at least the half-plane Re(r) ≥ −1/2.

By the results of [8] on first-order differential intertwinors and the conformal covariance of gradients, (9) is not the spectrum of a differential operator unless we are in case (3). (If neither (7) nor (3) holds, the intertwinor for which (9) is the spectrum is nonlocal.) In case (3), by the uniqueness theorem of [4], Sec. 6, we have: Corollary 8.2. In case (3),

eig(D_{λ}^{self}, α) = c_{λ} ∏_{a=1}^{L} α̃_{a},

for some nonzero constant c_{λ} which is independent of α.

Using results of [4], we may determine the constant c_{λ} in Corollary 8.2 up to a factor of ±1 (Theorem 8.4 below). By Sec. 6. above, this specifies the normalization to the full extent possible.

Given λ ∈ χ(n), let λ̃ be the rho-shift of λ:

λ̃ = λ + ρ_{n} , (ρ_{n})_{a} = n − 2a,

and let T (λ) be the set of component labels a ∈ {1, …, L} for which α̃_{a}^{2} is allowed, by the condition α ↓ λ, to take on more than one value. Theorem 8.3. ([4], Theorems 4.1 and 5.2.) Let λ ∈ χ(n) be arbitrary. The cardinality of T (λ) is t(λ) = [(N(λ) + 1)/2], and the eigenvalue of G_{u}^{∗}G_{u} on the V(α) summand of E(λ) is

μ(α; λ, σ_{u}) := eig(G_{u}^{∗}G_{u}, α) = c̃_{λσ_{u}} ∏_{a∈T (λ)} (α̃_{a}^{2} − s_{u}^{2}) ,  (10)

where

s_{u} = \frac{1}{2}(|λ̃|^{2} − |σ̃_{u}|^{2}),

and c̃_{λσ_{u}} =

\frac{(−1)^{t(λ)+1}}{∏_{v≠u^{1≤v≤N(λ)}}(s_{v} − s_{u})}  if N(λ) is odd;  (11)

\frac{(−1)^{t(λ)+1}}{2 ∏_{u=1}^{N(λ)−2}(s_{u} + \frac{1}{2})}  if n is even, λ_{ℓ} = 0 ≠ λ_{ℓ−1} , σ_{u} = λ ± e_{ℓ}; (12)

\frac{(−1)^{t(λ)} (s_{u} + \frac{1}{2})}{∏_{v≠u^{1≤v≤N(λ)}}(s_{v} − s_{u})}  otherwise.  (13)

This shows that in case (3),

eig(D_{λ}^{self}, α)^{2} = eig((D_{λ}^{self})^{2}, α) = μ(α; λ, λ) = c̃_{λλ} ∏_{a∈T_{λ}} α̃_{a}^{2}.

This determines the constant in Corollary 8.2 up to sign:

c_{λ}^{2} = c̃_{λλ} ∏_{a∉T_{λ}} α_{a}^{2}.

(This is independent of α because for a ∉ T_{λ}, α_{a}^{2} takes on only one value.) We have shown:

Theorem 8.4. In case (3),

eig(D_{λ}^{self}, α) = (sgn α_{L})\sqrt{μ(α; λ, λ)} = (sgn α_{L})\sqrt{c̃_{λλ}} ∏_{a∈T (λ)} |α̃_{a}|.  (14)

In particular, for each real number η, the multiplicities of η and −η as eigenvalues of D_{λ}^{self} are equal.

Remark 8.5. In the statement of the theorem, we have made a choice of sign in the definition of D_{λ}^{self} , making eig(D_{λ}^{self}, α) > 0 on the sphere when α_{L} > 0. In particular, we have chosen one of the two possible leading symbols for D_{λ}^{self} on the sphere. This choice may be transferred to arbitrary Spin(n)-manifolds, as it is simply a choice of a connected component in the punctured real line of Spin(n) maps V (1, 0, …, 0) ⊗ V (λ) → V (λ).

Note that we cannot write simply \sqrt{c̃_{λλ}} ∏_{a∈T (λ)} α̃_{a} on the right in (14), because of the case where L ∉ T (λ); this happens exactly when λ_{ℓ} = \frac{1}{2} . The statement about the multiplicities of ±η follows from the equality of dim V(α) and dim V(ᾱ), where

ᾱ = (α_{1}, …, α_{L−1}, −α_{L}).

Note that the eigenvalue formula (14) predicts that c̃_{λλ} should be positive. We can also obtain this sign directly, by a counting argument. Suppose (3) holds; then s_{self} = 0. If N(λ) is odd, then half of the other s_{v} , that is (N(λ)−1)/2 = t(λ)−1 of them, are negative; this and (11) show that c̃_{λλ} > 0. If N(λ) is even, then N(λ)/2 = t(λ) of the other s_{v} are negative; this and (13) show that c̃_{λλ} > 0.

Remark 8.6. Equation (14), together with Corollary 8.2, shows that

c_{λ} = \{ \begin{matrix}\begin{matrix}√ & \overline{c̃_{λλ}} \\ ∏ & a∉T (λ) & α̃_{a} \end{matrix} , & λ_{ℓ} ≠ \frac{1}{2}, \\ 2\begin{matrix}√ & \overline{c̃_{λλ}} \\ ∏ & a∉T (λ)∪\{L\} & α̃_{a} \end{matrix} , & λ_{ℓ} = \frac{1}{2}, \end{matrix}

since sgn α_{L} = 2α_{L} in case λ_{ℓ} = \frac{1}{2} . Note that all terms in the denominator on the right are necessarily nonzero: if α̃_{a} appears and vanishes, then a = L, and the condition α ↓ λ constrains α_{L} to vanish; this can only happen if λ_{ℓ} = 0.

9. Reducible self-gradients

It is sometimes useful to expand the notion of self-gradient to reducible bundles. For example, in even dimensions, the Dirac operator interchanges the bundles Σ_{±} of positive and negative spinors. Thus to speak of the spectrum of the Dirac operator, one should work on Σ_{+} ⊕ Σ_{−}.

If λ ↔ σ ∈ χ(n), let G_{λσ} be the operator on sections of 𝕍(λ) ⊕ 𝕍(σ) given in block form by

G_{λσ} = ( \begin{matrix}0 & G_{λσ}^{∗} \\ G_{λσ} & 0 \end{matrix} ) .

This operator is formally self-adjoint, with square diag(G_{λσ}^{∗}G_{λσ}, G_{λσ}G_{λσ}^{∗}).

Now specialize to the sphere S^{n}, where the Spin(n + 1)-finite section space of 𝕍(λ) ⊕ 𝕍(σ) is

E(λ) ⊕ E(σ) = (⨁_{α↓λ} V(α; λ)) ⊕ (⨁_{β↓σ} V(β; σ)) .  (15)

Given α ↓ λ, either G_{λσ}|_{V(α;λ)} = 0, or μ(α; λ, σ) > 0. In the latter case, for each φ ∈ V(α; λ), we get G_{λσ} -eigensections with eigenvalues ±\sqrt{μ(α; λ, σ)}, namely

( \begin{matrix}φ \\ ±μ(α; λ, σ)^{−1/2}G_{λσ}φ \end{matrix} )  (16)

In fact, by [4], Lemma 4.7, or inspection of the formula (10), G_{λσ}|_{V(α;λ)} = 0 if and only if α ↓̸ σ or

n is odd, α_{L} = 0, λ is integral, σ = λ, and λ_{ℓ} ≠ 0.

The zero eigenvalue when α ↓̸ σ is required by Schur’s Lemma, as is a zero eigenvalue on V(β; σ) when β ↓̸ λ. With this collection of remarks, we have proved: Theorem 9.1. Let λ ↔ σ in χ(n). A complete spectral resolution of G_{λσ} is given by:

• Eigenvalue \sqrt{μ(α; λ, σ)} on one summand Spin(n + 1)-isomorphic to V(α) when α ↓ λ and α ↓ σ;

• Eigenvalue −\sqrt{μ(α; λ, σ)} on the V(α) summand complementary to that above when α ↓ λ and α ↓ σ;

• Eigenvalue 0 on summands V(α; λ) when α ↓̸ σ and on summands V(β; σ) when β ↓̸ λ.

In particular, for each real number η, the multiplicities of η and −η as eigenvalues of G_{λσ} are equal.

Note that the first two cases cover some of the zero eigenvalue situations. The Spin(n + 1)-types α with α ↓ λ and α ↓ σ occur with multiplicity 2 in the decomposition (15); there are infinitely many splittings of this isotypic submodule as V(α) ⊕ V(α); (16) is the unique choice of such a splitting with projections that commute with G_{λσ}. The nonzero spectrum of G_{σλ} is a constant multiple of the nonzero spectrum of G_{λσ}; in fact, the constant is \sqrt{c̃_{σλ}/c̃_{λσ}}. (c̃_{λσ} may be positive or negative, but (10) shows that sgn c̃_{σλ} = sgn c̃_{λσ}.)

An aspect of Theorem 9.1 that is inconvenient in some applications is that it calls for a realization of 𝕍(σ) as a subbundle of T^{∗}M ⊗ 𝕍(λ). If some other realization is more desirable, and if λ ↮ λ, σ ↮ σ, we may remedy this situation as follows. Choose realizations for 𝕍(λ) and 𝕍(σ), and choose an isometric, Spin(n)-equivariant bundle injection

ι : 𝕍(λ) ⊕ 𝕍(σ) → T^{∗}M ⊗ (𝕍(λ) ⊕ 𝕍(σ)).

By Schur’s Lemma, the range of ι is the (unique, by λ ↮ λ and σ ↮ σ) (𝕍(λ) ⊕ 𝕍(σ))-isomorphic summand of the target. The operator

D_{𝕍(λ)⊕𝕍(σ)}^{self} := ( \begin{matrix}0 & G_{λσ}^{∗}∘ι \\ ι^{∗}∘G_{λσ} & 0 \end{matrix} )  (17)

is then a realization of G_{λσ} which acts on the “original” realization of 𝕍(λ)⊕𝕍(σ). If we choose the same isometric injection ι for the construction of D_{𝕍(λ)⊕𝕍(σ)}^{self} as for D_{𝕍(σ)⊕𝕍(λ)}^{self}, we get the same operator, after the appropriate permutation of blocks in (17); this justifies the notation. We have: Theorem 9.2. Suppose λ ↮ λ ↔ σ ↮ σ in χ(n). A complete spectral resolution of D_{𝕍(λ)⊕𝕍(σ)}^{self} is given by:

• Eigenvalue \sqrt{μ(α; λ, σ)} on one summand Spin(n + 1)-isomorphic to V(α) when α ↓ λ and α ↓ σ;

• Eigenvalue −\sqrt{μ(α; λ, σ)} on the V(α) summand complementary to that above when α ↓ λ and α ↓ σ;

• Eigenvalue 0 on V(α) summands when α ↓ λ and α ↓̸ σ, or α ↓ σ and α ↓̸ λ.

In particular, for each real number η, the multiplicities of η and −η as eigenvalues of G_{λσ} are equal.

10. Examples

Example 10.1. Let n ≥ 3 be odd, and consider the self-gradient on Λ^{(n−1)/2} ≅_{Spin(n)} 𝕍(1, …, 1). By the branching rule, on the sphere S^{n},

E(Λ^{(n−1)/2}) ≅_{Spin(n+1)} ⨁_{q∈{0,±1}^{j∈ℕ}} V(α_{j,q}),

where

α_{j,q} := (1 + j, 1, …, 1, q).

There are three gradient target bundles with source bundle Λ^{(n−1)/2}, namely Λ^{(n−1)/2} itself (≅_{Spin(n)} Λ^{(n+1)/2}), Λ^{(n−3)/2}, and F ≅_{Spin(n)} 𝕍(2, 1, …, 1). We have

s_{Λ^{(n−1)/2}Λ^{(n−1)/2}} = 0,  s_{Λ^{(n−1)/2}Λ^{(n−3)/2}} = 1,  s_{Λ^{(n−1)/2}F} = −(n + 1)/2,

from which

c̃_{Λ^{(n−1)/2}Λ^{(n−1)/2}} = \frac{2}{n + 1}.

By Theorem 8.4,

eig(D_{Λ^{(n−1)/2}}^{self}, α_{j,q}) = q\sqrt{\frac{2}{n + 1}} (j + \frac{n + 1}{2}) .  (18)

The operator (⋆d)^{2} on Λ^{(n−1)/2} agrees with (−1)^{n(n+1)/2+1}δd, where d is the exterior derivative, and δ is the formal adjoint of d in the usual form metric

(φ, ψ)_{Λ^{k}} = \frac{1}{k!}g^{a_{1}b_{1}} … g^{a_{k}b_{k}}φ_{a_{1}...a_{k}}ψ_{b_{1}...b_{k}}.  (19)

Thus \sqrt{−1}^{n(n+1)/2+1} ⋆ d is a real nonzero multiple of D_{Λ^{k}}^{self} . Comparing the normalizations (19) and (5), we find that

d|_{Λ^{k}} = (k + 1)G_{Λ^{k}Λ^{k+1}}, δ|_{Λ^{k+1}} = G_{Λ^{k}Λ^{k+1}}^{∗},

so that

(δd)_{Λ^{(n−1)/2}} = \frac{n + 1}{2}(D_{Λ^{(n−1)/2}}^{self})^{2}.

This explains the factor of \sqrt{2/(n + 1)} in (18), and gives

eig(\sqrt{−1}^{n(n+1)/2+1}(⋆d)_{Λ^{(n−1)/2}}, α_{j,q}) = q (j + \frac{n + 1}{2}) .

Theorem 9.2 is of special interest in the case in which n is even, λ_{ℓ} = \frac{1}{2} , σ_{ℓ} = −\frac{1}{2} , and σ_{a} = λ_{a} for a < ℓ. Examples of G_{λσ} in this case are the evendimensional Dirac and Rarita-Schwinger operators, and generalizations to certain bundles of spinor-forms first studied in [1]. When n is odd, the Dirac, Rarita-Schwinger, and spinor-form self-gradients are covered by Theorem 8.4. Example 10.2. Let n ≥ 3 be odd. The spinor bundle Σ is a copy of 𝕍(\frac{1}{2}, …, \frac{1}{2}). On the sphere, by (6),

E(Σ) ≅_{Spin(n+1)} ⨁_{j=0}^{∞} (V(\frac{1}{2} + j, \frac{1}{2}, …, \frac{1}{2}, \frac{1}{2}) ⊕ V(\frac{1}{2} + j, \frac{1}{2}, …, \frac{1}{2}, −\frac{1}{2})) .

The Dirac operator ∇ / must be, up to a constant factor, the realization of the self-gradient D_{Σ}^{self}, which, by Theorem 8.4, takes the eigenvalue

\sqrt{c̃_{ΣΣ}}α̃_{1}

on V(α; Σ). (We abuse notation in the obvious way by writing Σ instead of its highest weight in expressions like c̃_{ΣΣ}, V(α; Σ), etc.) There two gradient targets, Σ itself, and the twistor bundle 𝕋 ≅_{Spin(n)} 𝕍(\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}). Since c̃_{ΣΣ} = 1/n by (13), we have

eig(D_{Σ}^{self}, (\frac{1}{2} + j, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2})) = ±\frac{1}{\sqrt{n}} (\frac{n}{2} + j) .

By [3], Sec. 5, the eigenvalues of the classical Dirac operator ∇ / = γ^{a}∇_{a} (in abstract index notation) on the sphere are ±(\frac{n}{2} + j). The factor 1/\sqrt{n} which emerges above from the general results can also be seen classically, as follows. The 𝕋 subbundle of T^{∗}M ⊗ Σ consists of the spinor one-forms φ_{a} for which γ^{a}φ_{a} = 0. For ψ a spinor, ∇_{a}ψ decomposes as

∇_{a}ψ = (∇_{a}ψ + \frac{1}{n}γ_{a}γ^{b}∇_{b}ψ) + (−\frac{1}{n}γ_{a}γ^{b}∇_{b}ψ)  (20)

into 𝕋 and Σ parts; thus

(G_{ΣΣ}ψ)_{a} = −\frac{1}{n}γ_{a}γ^{b}∇_{b}ψ.

This gives

G_{ΣΣ}^{∗}φ = \frac{1}{n}γ^{b}γ^{a}∇_{b}φ_{a}, G_{ΣΣ}^{∗}G_{ΣΣ} = \frac{1}{n} ∇ / ^{2},

since γ^{a}γ_{a} = −n. Thus

D_{Σ}^{self} = ∇ / /\sqrt{n}.  (21)

Note that both sides of (21) are subject to a sign ambiguity (recall Sec. 6.). Example 10.3. In even dimensions, the Dirac operator carries sections of

Σ_{+} ≅_{Spin(n)} V (\frac{1}{2}, …, \frac{1}{2})

to sections of

Σ_{−} ≅_{Spin(n)} V (\frac{1}{2}, …, \frac{1}{2}, −\frac{1}{2})

and vice versa. Let Σ = Σ_{+} ⊕ Σ_{−}. To get the spectrum of D_{Σ}^{self} on S^{n} using Theorem 9.2, note that

E(Σ_{+}) ≅_{Spin(n+1)} E(Σ_{−}) ≅_{Spin(n+1)} ⨁_{j=0}^{∞} V(\frac{1}{2} + j, \frac{1}{2}, …, \frac{1}{2}).

There are two gradient target bundles with source bundle Σ_{+}, namely Σ_{−} and the positive twistor bundle 𝕋_{+} ≅_{Spin(n)} 𝕍(\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}). We have

s_{Σ_{−}} = 0, s_{𝕋_{+}} = −n/2,

and by (13), c̃_{Σ_{+}Σ_{−}} = 1/n. Setting

α_{j} := (\frac{1}{2} + j, \frac{1}{2}, …, \frac{1}{2}),

G_{Σ_{+}Σ_{−}} takes the eigenvalues

±\sqrt{μ(α_{j}; Σ_{+}, Σ_{−})} = ±\frac{1}{\sqrt{n}} (\frac{n}{2} + j)

on two complementary summands in the isotypic submodule V(α_{j}; Σ_{+})⊕V(α_{j}; Σ_{−}). Switching the roles of Σ_{+} and Σ_{−}, the calculation is the same. (20) gives the decomposition of a spinor (i.e. a section of Σ_{+} ⊕ Σ_{−}) into 𝕋_{+} ⊕ 𝕋_{−} and Σ_{+} ⊕ Σ_{−} parts, and a calculation similar to the one in Example 10.2 explains the factor of 1/\sqrt{n}.

Example 10.4. Let n ≥ 3 be odd, and consider the twistor bundle

𝕋 ≅_{Spin(n)} 𝕍(\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2})

defined above. If n ≥ 5, there are four gradient targets: 𝕋 itself, Σ, Z := 𝕍(\frac{5}{2}, \frac{1}{2}, …, \frac{1}{2}), and 𝕋^{2} := 𝕍(\frac{3}{2}, \frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}). (See Example 10.6 below for more on the series 𝕋^{k}.) If n = 3, the last of these targets is missing. As mentioned in Example 10.2, in abstract index notation, 𝕋 is realized in spinor-one-forms φ annihilated by interior multiplication by γ:

γ^{a}φ_{a} = 0.

After decomposing ∇φ into its components in these bundles, one gets

\begin{aligned}(G_{𝕋𝕋}φ)_{ab} & = \frac{1}{(n + 2)(n − 2)}\{ − nγ_{a}γ^{c}∇_{c}φ_{b} + 2γ_{b}γ^{c}∇_{c}φ_{a} \\ & +2γ_{a}γ_{b}∇^{c}φ_{c} − \frac{4}{n}γ_{b}γ_{a}∇^{c}φ_{c}\}. \end{aligned}  (22)

(See [5] for details of the calculation.)

The Rarita-Schwinger operator is the self-gradient D_{𝕋}^{self} corresponding to the gradient G_{𝕋𝕋}. To get a formula for D_{𝕋}^{self}, we need an explicit identification of the 𝕋 summand in T^{∗} ⊗𝕋 with the “original” realization of 𝕋 as spinor-one-forms φ_{a} with γ^{a}φ_{a} = 0. For this, consider the spinor-two-tensor γ_{a}φ_{b}, and project to the 𝕋 summand in T^{∗}M ⊗ 𝕋:

Proj_{𝕋}(γ_{a}φ_{b}) = \frac{n}{(n + 2)(n − 2)}(nγ_{a}φ_{b} − 2γ_{b}φ_{a}) =: −\frac{n}{(n + 2)(n − 2)}(ι̃φ)_{ab}.

Note that ι̃ is injective, with adjoint

(ι̃^{∗}φ)_{a} = nγ^{b}φ_{ba}.

Since ι̃^{∗}ι̃ = n(n + 2)(n − 2), the maps

ι = {n(n + 2)(n − 2)}^{−1/2}ι̃, ι^{∗} = {n(n + 2)(n − 2)}^{−1/2}ι̃^{∗}

are isometric bundle isomorphisms of the original 𝕋 realization and the 𝕋-isomorphic summand in T^{∗}M ⊗𝕋. Thus we get a concrete realization of the Rarita-Schwinger operator as

D_{𝕋}^{self} = ι^{∗}∘G_{𝕋𝕋}.

This and the above formulas give

(D_{𝕋}^{self}φ)_{a} = {n(n + 2)(n − 2)}^{−1/2} (nγ^{b}∇_{b}φ_{a} − 2γ_{a}∇^{b}φ_{b}) .  (23)

For the spectrum of D_{𝕋}^{self} on the sphere S^{n}, note first that the branching rule gives

E(𝕋) ≅_{Spin(n+1)} ⨁_{q∈{0,1}^{j∈ℕ}} (V(α_{j,q,+}) ⊕ V(α_{j,q,−})) ,

where

α_{j,q±} := \{ \begin{matrix}(\frac{3}{2} + j, \frac{1}{2} + q, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}), & n > 3, \\ (\frac{3}{2} + j, ±(\frac{1}{2} + q)), & n = 3. \end{matrix}

We have

s_{Σ} = \frac{n}{2}, s_{𝕋} = 0, s_{𝕋^{2}} = −\frac{n − 2}{2} (n > 3), s_{Z} = −\frac{n + 2}{2}, c̃_{𝕋𝕋} = \frac{4}{n(n + 2)(n − 2)}.

By Theorem 8.4,

eig(D_{𝕋}^{self}, α_{j,q,±}) = \frac{2}{\sqrt{n(n + 2)(n − 2)}} (j + \frac{n}{2} + 1) (q + \frac{n}{2} − 1) .

Thus if S is the non-normalized Rarita-Schwinger operator

(Sφ)_{a} := nγ^{b}∇_{b}φ_{a} − 2γ_{a}∇^{b}φ_{b},  (24)

we have

eig(S, α_{j,q,±}) = ±2 (j + \frac{n}{2} + 1) (q + \frac{n}{2} − 1) .

Example 10.5. Let n ≥ 4 be even. The twistor bundle splits as 𝕋 = 𝕋_{+} ⊕𝕋_{−}, where

𝕋_{±} ≅_{Spin(n)} 𝕍(\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}).

The right side of (22) gives a formula for G_{𝕋_{+}𝕋_{−}} (when φ is a section of 𝕋_{+}) and for G_{𝕋_{−}𝕋_{+}} (when φ is a section of 𝕋_{−}). Equation (23) holds (with the meaning of D_{𝕋}^{self} given in (17)).

For the spectrum of D_{𝕋}^{self}, note that on the sphere S^{n}, the branching rule gives

E(𝕋_{+}) ≅_{Spin(n+1)} E(𝕋_{−}) ≅_{Spin(n+1)} ⨁_{q∈{0,1}^{j∈ℕ}} V(α_{j,q}),

where

α_{j,q} := (\frac{3}{2} + j, \frac{1}{2} + q, \frac{1}{2}, …, \frac{1}{2}).

Let I_{j,q} be the multiplicity 2 isotypic component V(α_{j,q}; 𝕋_{+})⊕V(α_{j,q}; 𝕋_{−}) of E(𝕋). Theorem 9.2 applies, and gives

D_{𝕋}^{self}|_{I_{j,q}} = \frac{2}{\sqrt{n(n + 2)(n − 2)}} (j + \frac{n}{2} + 1) (q + \frac{n}{2} − 1) ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

relative to some Spin(n + 1)-invariant block decomposition of I_{j,q} as V(α_{j,q}) ⊕ V(α_{j,q}). (Note that this is not the decomposition V(α_{j,q}; 𝕋_{+}) ⊕ V(α_{j,q}; 𝕋_{−}), but rather the one described in the argument leading to Theorem 9.1, as realized in the argument leading to Theorem 9.2.) For the non-normalized Rarita-Schwinger operator S of (24),

S|_{I_{j,q}} = 2 (j + \frac{n}{2} + 1) (q + \frac{n}{2} − 1) ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

Example 10.6. This example contains Examples 10.2 and 10.4 as special cases. Let n ≥ 3 be odd, and consider the bundle S^{k} of spinor-k-forms as in [1]. If k < n/2, this is a copy of

𝕍(\frac{1}{2}, …, \frac{1}{2}) ⊗ 𝕍(\underbrace{1, …, 1}_{k}, 0, …, 0).

This bundle is generally reducible, but the Cartan summand, i.e. the summand

𝕍(\underbrace{\frac{3}{2}, …, \frac{3}{2}}_{k}, \frac{1}{2}, …, \frac{1}{2})

containing the highest weight vector (which necessarily occurs in the tensor product with multiplicity one) consists of spinor-forms φ_{a_{1}...a_{k}} with γ^{a_{1}}φ_{a_{1}...a_{k}} = 0. We denote this bundle by 𝕋^{k}; note that 𝕋_{0} = Σ and 𝕋^{1} = 𝕋.

The following operators, defined in [1], are notationally convenient. Let φ be a section of S^{k}, and put

\begin{aligned}(d̃φ)_{a_{0}...a_{k}} & = ∑_{i=0}^{k}(−1)^{i}∇_{a_{i}}φ_{a_{0}...a_{i−1}a_{i+1}...a_{k}}, \\ (δ̃φ)_{a_{2}...a_{k}} & = −∇^{b}φ_{ba_{2}...a_{k}}, \\ (ε(γ)φ)_{a_{0}...a_{k}} & = ∑_{i=0}^{k}(−1)^{i}γ_{a_{i}}φ_{a_{0}...a_{i−1}a_{i+1}...a_{k}}, \\ (ι(γ)φ)_{a_{2}...a_{k}} & = γ^{b}φ_{ba_{2}...a_{k}}. \end{aligned}

Note that 𝕋^{k} as defined above is the null space of ι(γ). (The lower weight summands in S^{k} may be distinguished by computing the null spaces of powers ι(γ)^{h}; see [12].) [1] shows that the operator

P_{k} := \frac{n − 2k + 4}{2}ι(γ)d̃+ \frac{n − 2k}{2} (d̃ι(γ) − δ̃ε(γ)) − \frac{n − 2k − 4}{2}ε(γ)δ̃  (25)

is conformally covariant: if we change the metric conformally, ḡ = Ω^{2}g for 0 < Ω ∈ C^{∞}(M), and make the compatible change in the fundamental tensor-spinor, γ̄ = Ω^{−1}γ, then

P_{k}(Ω^{(n−2k−1)/2}φ) = Ω^{(n−2k+2)/2}P̄_{k}φ.  (26)

By uniqueness of the first-order conformal covariant on irreducible bundles [8], the compression Proj_{𝕋^{k}} P_{k}|_{𝕋^{k}} of P_{k} to an operator on sections of 𝕋^{k} is a constant multiple of the self-gradient D_{𝕋_{k}}^{self}. (This might a priori be the zero multiple, but it is not, as we shall show presently.) Moreover, since the conformal biweights of gradients from 𝕋^{k} to other bundles differ from that in (26) (see [8]), this compression is equal to the restriction: P_{k}|_{𝕋^{k}} = a_{k}D_{𝕋_{k}}^{self}. The constant a_{k} is universal; i.e. independent of the particular n-dimensional Riemannian spin manifold on which we realize these operators.

On the sphere S^{n},

E(𝕋^{k}) ≅_{Spin(n+1)} ⨁_{q∈{0,1}^{j∈ℕ}}(V(α_{k,j,q,+}) ⊕ V(α_{k,j,q,−}))

where

α_{k,j,q,±} = \{ \begin{matrix}(\frac{3}{2} + j, \frac{3}{2}, …, \frac{3}{2}, \underbrace{\frac{1}{2} + q}_{(k+1)\underline{st}}, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}), & k < (n − 1)/2, \\ (\frac{3}{2} + j, \frac{3}{2}, …, \frac{3}{2}, ±(\frac{1}{2} + q)), & k = (n − 1)/2. \end{matrix}

Let

(Dφ)_{a_{1}...a_{k}} = γ^{b}∇_{b}φ_{a_{1}...a_{k}}.

Some short calculations with the definitions give

ι(γ)d̃+ d̃ι(γ) = D  (27)
δ̃ι(γ) + ι(γ)δ̃ = 0  (28)

Since

d̃^{∗} = δ̃,  ε(γ)^{∗} = −ι(γ),  D^{∗} = D,

we also have, by taking formal adjoints in (27),

δ̃ε(γ) + ε(γ)δ̃ = −D  (29)

As a result,

a_{k}D_{𝕋^{k}}^{self} = P_{k}|_{𝕋^{k}} = (n − 2k + 2)D + 2ε(γ)δ̃.  (30)

Compare this to (21) and (23); this shows in particular that

a_{0} = (n + 2)\sqrt{n},  a_{1} = \sqrt{n(n + 2)(n − 2)}.  (31)

Since δ̃D − Dδ̃ and δ̃δ̃ are 0\underline{th} order curvature terms, (29) and (28) (in particular, the fact that δ̃ carries 𝕋^{k} to 𝕋^{k−1}) give

δ̃P_{k} = (n − 2k + 2)δ̃Dφ + 2δ̃ε(γ)δ̃
= (n − 2k)Dδ̃ + (0\underline{th} order curvature terms)
= \frac{n − 2k}{n − 2k + 4}P_{k−1}δ̃ + (0\underline{th} order curvature terms)

on 𝕋^{k}. But δ̃, being Spin(n + 1)-invariant, carries V(α_{k,j,0,+}; 𝕋^{k}) to V(α_{k−1,j,1,+}; 𝕋^{k−1}), since α_{k,j,0,+} = α_{k−1,j,1,+}. By [4], Lemma 4.7 the restriction of δ̃ to V(α_{k,j,0,+}; 𝕋^{k}) is nonzero. Thus

eig(a_{k}D_{𝕋^{k}}^{self}, α_{k,j,0,+}) = \frac{n − 2k}{n − 2k + 4} eig(a_{k−1}D_{𝕋^{k−1}}^{self}, α_{k−1,j,1,+}) + O(j^{0}).  (32)

By (14), the O(j^{0}) term must vanish.

If k < (n − 1)/2, there are four gradient target bundles with source bundle 𝕋^{k}, namely

𝕋^{k−1}, 𝕋^{k}, 𝕋^{k+1}, Z^{k} ≅_{Spin(n)} 𝕍(\frac{5}{2}, \underbrace{\frac{3}{2}, …, \frac{3}{2}}_{k−1}, \frac{1}{2}, …, \frac{1}{2}).

If k = (n − 1)/2, the summand 𝕋^{k+1} is missing. (If we define 𝕋^{k+1} as the spinor- (k+1)-forms φ with ι(γ)φ = 0, we get another Spin(n)-isomorphic copy of 𝕋^{k−1}.) We have

\begin{matrix}s_{𝕋^{k−1}} = (n − 2k + 2)/2, & s_{𝕋^{k}} = 0, \\ s_{𝕋^{k+1}} = −(n − 2k)/2 (k < (n − 1)/2), & s_{Z^{k}} = −(n + 2)/2. \end{matrix}

By (13) (or (11) if k = (n − 1)/2),

c̃_{𝕋^{k}𝕋^{k}} = \frac{4}{(n − 2k + 2)(n + 2)(n − 2k)}.

By (14),

\begin{matrix}eig(D_{𝕋^{k}}^{self}, α_{k,j,q,±}) = \\ ±\frac{2}{\sqrt{(n − 2k + 2)(n + 2)(n − 2k)}} (j + \frac{n}{2} + 1) (q + \frac{n}{2} − k) . \end{matrix}  (33)

By (33) and (32),

a_{k} = \sqrt{\frac{n − 2k}{n − 2k + 4}}a_{k−1}.

(Compare (31).) Computing inductively, we have

a_{k} = \sqrt{\frac{(n − 2k + 2)(n − 2k)}{(n + 2)n}} a_{0} = \sqrt{(n − 2k + 2)(n + 2)(n − 2k)}.

This and (33) give

eig(P_{k}|_{𝕋^{k}}, α_{k,j,q,±}) = ±2 (j + \frac{n}{2} + 1) (q + \frac{n}{2} − k)

Example 10.7. This example contains Examples 10.3 and 10.5 as special cases. Let n ≥ 4 be even. For k < n/2, the bundle of spinor-k-forms φ with ι(γ)φ = 0 splits as 𝕋^{k} = 𝕋_{+}^{k} ⊕ 𝕋_{−}^{k}, where

𝕋_{±}^{k} ≅_{Spin(n)} 𝕍(\underbrace{\frac{3}{2}, …, \frac{3}{2}}_{k}, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}).

(30), or (25) restricted to 𝕋^{k}, gives a formula for a conformally covariant operator 𝕋_{±}^{k} → 𝕋_{∓}^{k}. By a calculation similar to that in Example 10.6,

P_{k}|_{𝕋^{k}} = \sqrt{(n − 2k + 2)(n + 2)(n − 2k)}D_{𝕋^{k}}^{self},

where in this case, D_{𝕋^{k}}^{self} is defined by (17).

On the sphere S^{n},

E(𝕋_{+}^{k}) ≅_{Spin(n+1)} E(𝕋_{−}^{k}) ≅_{Spin(n+1)} ⨁_{q∈{0,1}^{j∈ℕ}} V(α_{k,j,q}),

where

α_{k,j,q} := (\frac{3}{2} + j, \frac{3}{2}, …, \frac{3}{2}, \underbrace{\frac{1}{2} + q}_{(k+1)\underline{st}}, \frac{1}{2}, …, \frac{1}{2}).

Let I_{k,j,q} be the multiplicity 2 isotypic component V(α_{k,j,q}; 𝕋_{+}) ⊕ V(α_{k,j,q}; 𝕋_{−}) of E(𝕋^{k}). By Theorem 9.2,

D_{𝕋^{k}}^{self}|_{I_{k,j,q}} = \frac{2}{\sqrt{(n − 2k + 2)(n + 2)(n − 2k)}} (j + \frac{n}{2} + 1) (q + \frac{n}{2} − k) ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

relative to some Spin(n + 1)-invariant block decomposition of I_{k,j,q} as V(α_{k,j,q}) ⊕ V(α_{k,j,q}). As a result, relative to the same decomposition,

(P_{k}|_{𝕋^{k}})|_{I_{k,j,q}} = 2 (j + \frac{n}{2} + 1) (q + \frac{n}{2} − k) ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

References

[1] Branson, T., Intertwining differential operators for spinor-form representations of the conformal group, Advances in Math. 54 (1984), 1–21.

[2] —, Second order conformal covariants, University of Copenhagen Mathematical Institute preprints #2,3 (1989).

[3] —, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. 106 (1992), 314–328.

[4] —, Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334– 383.

[5] Branson, T. and O. Hijazi, Eigenvalue estimates for the Rarita-Schwinger and related operators, In preparation.

[6] Branson, T., G. Ólafsson and B. Ørsted, Spectrum generating operators, and intertwining operators for representations induced from a maximal parabolic subgroup, J. Funct. Anal. 135 (1996), 163–205.

[7] Bröcker, T. and T. tom Dieck, “Representations of Compact Lie Groups, ” Springer-Verlag, Berlin, 1985.

[8] Fegan, H., Conformally invariant first order differential operators, Quart. J. Math. Oxford 27 (1976), 371–378.

[9] Gallot, S. et D. Meyer, Opérateur de courbure et laplacian des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. 54 (1975), 259–284.

[10] Ikeda, A. and Y. Taniguchi, Spectra and eigenforms of the Laplacian on S^{n} and P^{n}(ℂ), Osaka J. Math. 15 (1978), 515–546.

[11] D. Ray, Reidemeister torsion and the Laplacian on lens spaces, Advances in Math. 4 (1970), 109–126.

[12] Delanghe, R., F. Sommen, and V. Soucek, “Clifford Algebra and Spinor-Valued Functions,” Kluwer Academic Publishers, Dordrecht, 1992.

[13] Stein, E. and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163– 196.

Department of Mathematics University of Iowa Iowa City IA 52242 USA

Received August 4, 1998 and in final form November 24, 1998