{'fraction_non_alphanumeric': 0.06430786758655611, 'fraction_numerical': 0.016520032913475535, 'mean_word_length': 3.449450858912982, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Symplectic Dirac Operators on Hermitian Symmetric Spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.


Introduction
Symplectic spinor fields were introduced by B. Kostant in the early 1970th [15]. This was done in the context of geometric quantization, where he was interested in constructing the so-called half-form bundle as well as the half-form pairing in order to establish the appropriate Hilbert space. Kostants construction was quite an essential step in geometric quantization, but afterwards no further work has been done to seek out interior aspects of symplectic spin geometry for two decades.
In 1995, the second author introduced symplectic Dirac operators and initialized a systematical investigation [7]. Some progress has been made in understanding basic properties of these operators [8]. Moreover, symplectic Dirac operators have been studied in the setting of parabolic geometries (cf. e.g. [13,16]) and received attention in mathematical physics (cf. e.g. [19]). A systematic and thorough introduction to the subject can be found in [11].
There are several topics of their own interest that are relevant to get a broader understanding of symplectic Dirac operators. For example, a deeper knowledge of the moduli space of symplectic structures may provide a way of analyzing how the symplectic Dirac operators are related to the choice of the symplectic structure of the underlying symplectic manifold. For considerations on moduli spaces of symplectic structures, see [5,6,22]. Furthermore, the symplectic Dirac operators are defined by means of a symplectic connection. It is well known that in symplectic geometry there is no analog of the Levi-Civita connection. With the aim to get a deeper insight into the structure of the space of symplectic connections, in [12] an approach of a purely symplectic Yang-Mills theory is given.
In order to develop the symplectic Dirac operator picture, it is significant to have more explicit examples. In the present paper, we study symplectic Dirac operators in the case where the underlying symplectic manifold is a Hermitian symmetric space. This gives a class of examples which were not figured out so far. In the particular case of odd-dimensional complex projective spaces, the spectrum of an associated second order operator has been already computed [9,23]. However, no calculations are performed for the first order symplectic Dirac operators there. Here, we are going to study the shape of these operators itself. Our strategy is based on the idea of considering symplectic Dirac operators as families of operators acting on more suitable spaces than the section space of an infinite rank vector bundle.
The paper is organized as follows. The necessary concepts of symplectic Dirac operators in the setting of symmetric symplectic spaces are introduced in Section 2. For the general situation, we refer to [11]. In Section 3 we describe a splitting of the symplectic spinor bundle into subbundles of finite rank and apply the Frobenius reciprocity to decompose the corresponding section spaces in the case of a Hermitian symmetric space. Moreover, we explain how the symplectic Dirac operators built with respect to the canonical Hermitian connection behave in relation to these decompositions. The last section is devoted to a detailed discussion of the case CP 1 .

Preliminaries
Let G be a simply connected real Lie group, let H be a closed and connected subgroup of G and set M to be the homogeneous space G/H. Let g and h denote the Lie algebras of G and H, respectively. We suppose that there exists a subspace m ⊂ g such that Moreover, let ω 0 be an ad(h)-invariant non-degenerate skew-symmetric bilinear form on m. Identifying the tangent space T o M to M at the point o = eH with m, ω 0 induces a G-invariant almost symplectic structure ω on M . Here and in the sequel, e denotes the unit element of G. Since the ad(h)-invariance of ω 0 is equivalent to the trivial extension of ω 0 to g being a Chevalley 2-cocycle, ω is closed. Thus ω is even a symplectic structure on M . In this way, we have described (M, ω) as a simply connected symplectic symmetric space (cf. [2]).
The canonical projection π : G → M and the right action of H on G give G the structure of an H-principal fiber bundle. Let κ : H → GL(m) be the isotropy representation of the homogeneous space M , i.e. the restriction of the adjoint representation of G to H acting on m. By assumption, κ maps into the symplectic group Sp(m) of the symplectic vector space (m, ω 0 ). We fix a symplectic basis (X 1 , . . . , X 2n ) of (m, ω 0 ), i.e. a basis of m such that ω 0 (X j , X k ) = ω 0 (X n+j , X n+k ) = 0 and ω 0 (X j , X n+k ) = δ jk for j, k = 1, . . . , n, and identify the symplectic frame bundle R of (M, ω) with the Sp(m)-principal fiber bundle G × κ Sp(m) associated to G with respect to κ : Theorem 2.1 The mappingκ → (P, F P ) described above induces a 1 : 1 correspondence between the liftsκ of κ and the isomorphism classes of metaplectic structures (P, F P ) of the simply connected symplectic symmetric space (M, ω).
According to the above, we identify the tangent bundle T M of M with the vector bundle G × κ m associated to G with respect to κ via The space of smooth vector fields on M , i.e. the space Γ(T M ) of smooth sections of T M is then the space of smooth maps ξ : G → m such that for g ∈ G and h ∈ H and the symplectic structure ω is given by Consequently, the function ω(ξ 1 , ξ 2 ) ∈ C ∞ (M ) for ξ 1 , ξ 2 ∈ Γ(T M ) satisfies ω(ξ 1 , ξ 2 )(π(g)) = ω 0 (ξ 1 (g), ξ 2 (g)) . (
Thus ∇ is a torsion-free symplectic connection on the almost symplectic manifold (M, ω). By a classical result of Tondeur (cf. [20]), this also implies that (M, ω) is symplectic.
Using the fixed symplectic basis (X 1 , . . . , X 2n ) of (m, ω 0 ), we identify Sp(m) and its double cover Mp(m) with the symplectic group Sp(n, R) and the metaplectic group Mp(n, R), respectively. Let m : Mp(m) → U(L 2 (R n )) be the metaplectic representation. Letκ be a lift of κ and let (P, F P ) be the metaplectic structure of (M, ω) constructed fromκ as described above. The symplectic spinor bundle is then the . Accordingly, a symplectic spinor field, i.e. a smooth section of Q is a smooth map ϕ : for g ∈ G and h ∈ H. Since the symplectic connection ∇ on M is induced by the G-invariant connection on the H-principal fiber bundle G that corresponds to the decomposition (2.1), the same holds true for the spinor derivative on Q, also denoted by ∇. Therefore, for ϕ ∈ Γ(Q) and ξ ∈ Γ(T M ).
According to our conventions, the symplectic Clifford multiplication is the ho- . , x n ) ∈ R n and j = 1, . . . , n, where the multiplication by X ∈ m has to be seen as an unbounded operator on L 2 (R n ). As usual, we shall write X ·f for µ 0 (X ⊗f ). The multiplication µ 0 induces a multiplication µ : The first symplectic Dirac operator is now defined as whereω means the isomorphismω : T * M → T M generated by the symplectic structure ω on M .

Proposition 2.4
The symplectic Dirac operator D can be written as for ϕ ∈ Γ(Q), where X(ϕ) for X ∈ m denotes the derivative of ϕ in the direction of the left-invariant vector field determined by X, i.e. X(ϕ)(g) = dϕ(dL g X) .
To construct the second Dirac operatorD, we need an ω-compatible almost complex structure J on M . We assume that J is G-invariant. That is, we suppose that the endomorphism J : T M → T M is given by Then for any X, Y ∈ m. Let g denote the G-invariant Riemannian metric on M induced by g 0 . This means that By Equation (2.7), for all ξ 1 , ξ 2 ∈ Γ(T M ). Furthermore, it follows that (M, g, J) is a Hermitian symmetric space and that ∇ is its Levi-Civita connection.
The second symplectic Dirac operator is defined as whereḡ : T * M → T M is the identification by means of the Riemannian metric g on M . Analogously to the proof of Proposition 2.4, one can show Proposition 2.5 Suppose that the basis (X 1 , . . . , X 2n ) of m is unitary, i.e., in addition to symplecticity, it satisfies J 0 X j = X n+j for j = 1, . . . , n. Then the symplectic Dirac operatorD takes the form for ϕ ∈ Γ(Q).

Decompositions and invariant subspaces
First we decompose the symplectic spinor bundle Q into subbundles of finite rank. For this, let U(m) denote the unitary group of the Hermitian vector space (m, g 0 , J 0 ) and setÛ(m) = ρ −1 (U(m)), which is a connected double cover of U(m). The irreducible components of the restriction u :Û(m) → U(L 2 (R n )) of the metaplectic representation m toÛ(m) can be described as follows. Let N 0 denote the set of non-negative integers and let h α ∈ L 2 (R n ) for a multi-index α = (α 1 , . . . , α n ) ∈ N n 0 be the Hermite function on R n defined by Here, h l for l ∈ N 0 are the classical Hermite functions on R, which are given by As is well known, the Hermite functions h α form a complete orthogonal system in L 2 (R n ). Furthermore, for any l ∈ N 0 , the span W l of the functions h α with α 1 + · · · + α n = l is an irreducible u-invariant subspace of L 2 (R n ) (cf. [3]).
Since the complex structure J 0 on m is assumed to be ad(h)-invariant, the homomorphism κ maps into U(m) and, therefore, the liftκ maps intoÛ(m). Hence the unitary representation λ : H → U(L 2 (R n )) can be written as λ = u •κ. Let u l :Û(m) → U(W l ) be the restriction of u to the subspace W l and set λ l = u l •κ.

The above yields
Proposition 3.1 The symplectic spinor bundle Q splits into the orthogonal sum of the finite rank subbundles Q l = G × λ l W l , l ∈ N 0 .
In the following considerations, we want to make use of the Frobenius reciprocity.
To be able to do this, we assume from now on that G is compact. Let τ a : G → GL(V a ), a ∈ A, form a complete system of representatives of isomorphism classes of irreducible complex representations of G. Let ν : H → GL(W) be a finite-dimensional representation of H and let Hom H (V a , W) be the space of all H-equivariant homomorphisms L : V a → W. We embed V a ⊗ Hom H (V a , W) into the space Γ(G × ν W) of smooth sections of the associated vector bundle G × ν W by assigning to v ⊗ L ∈ V a ⊗ Hom H (V a , W) the H-equivariant map For a proof of the theorem below, we refer to [21].
Applying Theorem 3.2 to the representation u l , we obtain a decomposition of Γ(Q l ) into the sum a∈A V a ⊗ Hom H (V a , W l ) .
Using this and Proposition 3.1, we now want to find invariant subspaces for the symplectic Dirac operators D andD. We start with expressing the symplectic Clifford multiplication by means of the Hermite functions h α . Lemma 3.3 For any α ∈ N n 0 and j = 1, . . . , n, where j = (δ 1j , . . . , δ nj ).

Proof. This follows from Equations (2.5) and (3.1) and the relations
for the classical Hermite functions h l . Then with the convention that W −1 = 0.
for a ∈ Mp(m) and f ∈ L 2 (R n ), we furthermore have

Equations (3.3) and (3.4) imply
Since the definition of D a (L) does not depend on the choice of the symplectic basis (X 1 , . . . , X 2n ) and since (κ(h)X 1 , . . . , κ(h)X 2n ) for any h ∈ H is again a symplectic basis, this proves the lemma.
For a ∈ A, set Hom H (V a , W l ) .
By Lemma 3.4, Equation (3.2) defines a homomorphism D a : U a → U a . The symplectic Dirac operator D and the homomorphisms D a are related by Proof. We deduce that for X ∈ m and g ∈ G. This together with Proposition 2.4 gives the assertion.
The decompositions obtained above and Lemma 3.5 allow us to consider the symplectic Dirac operator D as a map D : Moreover, we have Proposition 3.6 For any a ∈ A, the space V a ⊗ U a is invariant under D.
Proof. This is a consequence of Lemmas 3.4 and 3.5.
The same can be done for the symplectic Dirac operatorD and the second order operator P = i D , D .
where here the basis (X 1 , . . . , X 2n ) of m is assumed to be unitary. Theñ Proof. Using that κ(h) is unitary for all h ∈ H, the proof is the same as the proof of Lemma 3.4.
Consequently, Equation (3.5) gives rise to a homomorphismD a : U a → U a .
where P a = i D a , D a .
Proof. This follows from Proposition 2.5 and Lemma 3.5.

Proposition 3.9
The spaces V a ⊗ U a are invariant underD and P.
Proof. This is immediate from Lemmas 3.7 and 3.8 and Proposition 3.6.
Example 3. 10 We consider the complex projective space CP n = SU(n + 1)/U(n), where the group U(n) is embedded into SU(n + 1) by means of Let su(n + 1) and u(n) denote the Lie algebras of SU(n + 1) and U(n), respectively, and let m be the image of the homomorphism Then m is an ad(u(n))-invariant complement of u(n) ⊂ su(n + 1) satisfying [m, m] ⊂ u(n). We identify m with C n via Φ. With this, the isotropy representation κ takes the form κ(B) = det(B)B for B ∈ U(n). Moreover, ω 0 , J 0 and g 0 are the standard symplectic form, complex structure and inner product on C n , i.e.
ω 0 (z, w) = Im z T w , J 0 z = iz and g 0 (z, w) = Re z T w for z, w ∈ C n . There exists a liftκ of κ iff n is odd and in this case it is unique (for more details, see [23]). Since κ maps onto U(n) and since the representations u l are irreducible and pairwise inequivalent, by Schur's lemma, Hom H (V a , W l ) is non-trivial for only finitely many a ∈ A. Therefore, the spaces V a ⊗ U a have finite dimension.
In the next section, we examine the case of the complex projective line CP 1 = SU(2)/U(1) in more detail.
Let V k for k ∈ N 0 be the vector space of homogeneous polynomials of degree k in two variables z 1 and z 2 with complex coefficients. The polynomials p k,0 , . . . , p k,k defined by p k,j (z 1 , z 2 ) = z k−j 1 z j 2 form a basis of V k . Let τ k : SU(2) → GL(V k ) be the representation of SU (2) induced by the canonical action of SU(2) on C 2 , i.e.
Proof. Let H 0 : L 2 (R) → L 2 (R) be the Hamilton operator of the one-dimensional harmonic oscillator, i.e.
As shown in the previous section, the spaces V k ⊗ U k with are invariant under the symplectic Dirac operators D andD.

Corollary 4.3 If k is odd, then
and dim(V k ⊗ U k ) = (k + 1) 2 /2. If k is even, then U k is trivial.
Proof. This is a consequence of Lemma 4.2 and dim V k = k + 1.
According to Lemmas 3.5 and 3.8, in order to calculate the symplectic Dirac operators D andD on CP 1 , we have to compute the linear maps D k ,D k : U k → U k given by . Thereby, we may assume that k is odd. Then the maps L k,0 , . . . , L k,(k−1)/2 defined in Lemma 4.2 form a basis of U k .
Proof. Assume that j = (k + 1)/2 + l. Then, by Lemmas 3.3 and 4.1, we get Analogously, we seeD This together with Lemmas 3.4 and 3.7 yields the claim.
We draw the following conclusions from Corollary 4.5. First, for the operators P k = i D k , D k , we obtain  (ii) The spectrum of D k andD k is symmetric, i.e. if λ is an eigenvalue, so is −λ.
(iii) The kernels of D k andD k are one-dimensional, if (k + 1)/2 is odd, and trivial otherwise.
Proof. One easily verifies that the characteristic polynomials of D k andD k are the same and of the form λp k λ 2 , if (k + 1)/2 is odd, and p k λ 2 otherwise, for some polynomial p k . Since a k,1 , . . . , a k,(k−1)/2 are non-zero, the dimension of the kernel of D k is at most 1. Moreover, if (k + 1)/2 is even, then det(D k ) = (k+1)/4 r=1 a 2 k,2r−1 .
Corollary 4.8 The spectrum of the symplectic Dirac operators D andD on CP 1 is unbounded above and below.
Proof. Let U k be endowed with the norm defined by

Remark 4.9
The spectrum of the second order operator P on CP 1 was already computed in [9]. For the general case, the coincidence of the spectra of D andD and their symmetry was proven in [10] (cf. also [11]) by means of a Fourier transform for symplectic spinor fields. For an explicit calculation of the eigenvalues and eigenvectors of D k andD k for small k, see [18].