2.1 Harmonic Poisson transforms

Throughout this section let $\mathfrak {g}$ be a non-compact real semisimple Lie algebra with Cartan involution $\theta $ and corresponding Cartan decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {q}$ so that $\mathfrak {q} \neq \{0\}$ . Choose a maximal abelian subspace $\mathfrak {a}_0 \subset \mathfrak {q}$ and let $\Delta _r \subset \mathfrak {a}_0^*$ be the set of restricted roots of $\mathfrak {g}$ with respect to $\mathfrak {a}_0$ . For $\lambda \in \Delta _r$ let $\mathfrak {g}_{\lambda }$ be its restricted root space and define $\mathfrak {m}_0$ as the centralizer of $\mathfrak {a}_0$ in $\mathfrak {k}$ . Choose subsets $\Delta _r^+$ and $\Sigma _0$ of positive and simple roots, respectively, and let $\mathfrak {n}_0 = \bigoplus _{\lambda \in \Delta _r^+} \mathfrak {g}_{\lambda }$ . Then $\mathfrak {p}_0 := \mathfrak {m}_0 \oplus \mathfrak {a}_0 \oplus \mathfrak {n}_0$ is a Borel subalgebra of $\mathfrak {g}$ . Let $\mathfrak {p}$ be a standard parabolic subalgebra of $\mathfrak {g}$ with respect to $\mathfrak {p}_0$ and let $\Sigma \subset \Sigma _0$ be its corresponding subset of simple roots. Let $\mathfrak {g} = \mathfrak {g}_{-k} \oplus \dotso \oplus \mathfrak {g}_k$ be the $|k|$ -grading of $\mathfrak {g}$ induced by $\Sigma $ -height. Define $\mathfrak {p}_+$ and $\mathfrak {g}_-$ to be the sum of all positive and negative grading components, respectively; then by construction, we have $\mathfrak {p} = \mathfrak {g}_0 \oplus \mathfrak {p}_+$ . Put $\mathfrak {m} := \mathfrak {k} \cap \mathfrak {p} = \mathfrak {k} \cap \mathfrak {g}_0$ as well as $\mathfrak {a} := \mathfrak {q} \cap \mathfrak {g}_0$ . Let G be a connected semisimple Lie group with Lie algebra $\mathfrak {g}$ and assume that the centre of G is finite. Let K be the maximal compact subgroup with Lie algebra $\mathfrak {k}$ , $P = N_G(\mathfrak {p})$ be the parabolic subgroup of G with Lie algebra $\mathfrak {p}$ and let $P_+ \subset P$ be its unipotent radical whose Lie algebra coincides with $\mathfrak {p}_+$ . We identify the quotient group $G_0 := P/P_+$ with a subgroup of G so that the Lie algebra of $G_0$ coincides with $\mathfrak {g}_0$ .

We briefly recall the construction of a G-equivariant operator mapping between vector-valued differential forms on $G/P$ and $G/K$ as presented in [Reference Harrach14], see also [Reference Harrach15]. The Iwasawa decomposition yields a diffeomorphism $G/M \cong G/K \times G/P$ of G-manifolds, where $M := K \cap P$ . Denoting the natural projections from $G/M$ to $G/K$ and $G/P$ by $\pi _K$ and $\pi _P$ , respectively, we obtain an induced decomposition of the tangent bundle of $G/M$ into the direct sum of $T_K := \ker (T\pi _P)$ and $T_P := \ker (T\pi _K)$ . Next, for an irreducible G-representation $\mathbb {V}$ and a closed subgroup H of G we denote by $V_H := G \times _H \mathbb {V}$ its associated vector bundle over $G/H$ . Now for another irreducible G-representation $\mathbb {W}$ we obtain a decomposition

$$ \begin{align*}\Lambda^{k}T^*(G/M, L(V_M, W_M)) = \bigoplus_{k_1+k_2=k} \Lambda^{k_1,k_2}T^*(G/M, L(V_M, W_M)), \end{align*} $$

where the elements in $\Lambda ^{k_1,k_2}$ vanish unless exactly $k_1$ of their entries come from $T_K$ and the remaining $k_2$ entries are from $T_P$ .

Assuming that $G/P$ is oriented and $\dim (G/P) = N$ , we fix a G-invariant $(0,N)$ -form $\operatorname {\mathrm {vol}}_M$ on $G/M$ , which is unique up to multiples, and normalize it so that the fibre $\pi _K^{-1}(eK)$ has unit volume. Then each G-equivariant bundle map

(1) $$ \begin{align} \phi \colon \Lambda^{0,k} T^*(G/M, V_M) \to \Lambda^{\ell,0} T^*(G/M, W_M) \end{align} $$

induces, via integration along fibers, a G-equivariant integral operator

called a Poisson transform. Equivalently, any such bundle map induces a G-invariant differential form $\varphi $ of bidegree $(\ell , N-k)$ on $G/M$ with values in $L(V_M, W_M)$ via the relation

(2) $$ \begin{align} \varphi \wedge \pi_P^*\alpha = \phi(\pi_P^*\alpha) \otimes \operatorname{\mathrm{vol}}_M. \end{align} $$

For each closed subgroup H of G the homogeneous bundle $V_H = G \times _H \mathbb {V}$ carries a canonical G-invariant flat connection $\nabla ^{V_H}$ , which induces a covariant exterior derivative $d^{V_H}$ on $\Omega ^*(G/H, V_H)$ in the usual way. Additionally, we can find an inner product on $\mathbb {V}$ so that $\langle X v_1, v_2 \rangle = \langle v_1, \theta (X) v_2\rangle $ for all $X \in \mathfrak {g}$ and $v_1$ , $v_2 \in \mathbb {V}$ . By construction, this inner product is K-invariant and thus yields a G-invariant bundle metric on $V_K \to G/K$ . Next, we define $\nabla ^{\theta }$ as the dual connection to $\nabla ^{V_K}$ with respect to this bundle metric, which is again flat. Moreover, we obtain an induced covariant exterior derivative $d^{\theta }$ on $\Omega ^*(G/K,V_K)$ and we define $\delta ^{V_K}$ as the formal adjoint of $d^{\theta }$ with respect to the induced $L^2$ -norm. Finally, define the Laplace operator $\Delta ^{V_K}$ via $\Delta ^{V_K} := \delta ^{V_K}d^{V_K} + d^{V_K}\delta ^{V_K}$ . In [Reference Harrach14, Theorem 6] the following was proven.

In the next step we recall the BGG-calculus on $G/P$ and afterwards consider Poisson transforms which are naturally linked to BGG complexes. Consider the standard complex $(C^*(\mathfrak {g}_-, \mathbb {V}), \mathfrak {d})$ for computing Lie algebra cohomology of the nilpotent Lie algebra $\mathfrak {g}_-$ with values in a $\mathfrak {g}$ -representation $\mathbb {V}$ . Each chain space is naturally a $G_0$ -module, and the differentials are $G_0$ -equivariant. On the other hand, under the identification $\mathfrak {g}_-^* \cong \mathfrak {p}_+$ the Kostant codifferential $\mathfrak {d}^* \colon C^k(\mathfrak {g}_-, \mathbb {V}) \to C^{k-1}(\mathfrak {g}_-, \mathbb {V})$ introduced originally in [Reference Kostant20] is defined via

(3) $$ \begin{align} \begin{aligned} \mathfrak{d}^*(Z_1 \wedge \dotso \wedge Z_k \otimes v) &:=\ \sum_{i=1}^k (-1)^i Z_1\wedge \dotso \wedge \widehat{Z_i} \wedge\dotso \wedge Z_k \otimes Z_i \cdot v \\ &+ \sum_{i<j} (-1)^{i+j} [Z_i, Z_j] \wedge Z_0 \wedge \dotso \wedge \widehat{Z_i} \wedge \dotso \wedge \widehat{Z_j} \wedge \dotso \wedge Z_k \otimes v \end{aligned} \end{align} $$

for all $Z_i \in \mathfrak {p}_+$ and $v \in \mathbb {V}$ , where the hat denotes omission. It is immediate that this map is P-equivariant and squares to zero. Therefore, the induced homology spaces $H_k := H_k(\mathfrak {p}_+, \mathbb {V}) := \ker (\mathfrak {d}^*)/\operatorname {\mathrm {im}}(\mathfrak {d}^*)$ are naturally P-modules which are completely reducible, c.f. [Reference Čap and Slovák8, Corollary 3.3.1]. Let $\Box = \mathfrak {d} \mathfrak {d}^* + \mathfrak {d}^* \mathfrak {d}$ be the Kostant Laplacian on $C^k(\mathfrak {g}_-, \mathbb {V})$ , which is $G_0$ -equivariant. We obtain a $G_0$ -invariant decomposition

(4) $$ \begin{align} C^k(\mathfrak{g}_-, \mathbb{V}) = \operatorname{\mathrm{im}}(\mathfrak{d}) \oplus \ker(\Box) \oplus \operatorname{\mathrm{im}}(\mathfrak{d}^*), \end{align} $$

and the sums of the first two and the last two summands coincide with $\ker (\mathfrak {d})$ and $\ker (\mathfrak {d}^*)$ , respectively. In particular, the homology spaces $H_k$ are isomorphic to $\ker (\Box )$ as $G_0$ -modules.

Denote the G-equivariant bundle map on $\Lambda ^* T^*(G/P)\otimes V_P$ induced by $\mathfrak {d}^*$ by the same symbol. Moreover, let $\mathcal {H}_k = \mathcal {H}_k(G/P, V_P)$ be the homogeneous vector bundle over $G/P$ associated to $H_k$ . Since the operator $\Box $ is $G_0$ -equivariant, we obtain an induced tensorial operator on the associated graded of $\Lambda ^k T^*(G/P)\otimes V_P$ , and the decomposition (4) globalizes only to this space. To remedy this fact, let $\Box ^R := d^{V_P}\mathfrak {d}^* + \mathfrak {d}^* d^{V_P}$ be the curved Box operator; then from [Reference Calderbank and Diemer6, Theorem 5.2] we know that $\Box ^R$ is invertible on sections of the subbundle induced by $\operatorname {\mathrm {im}}(\mathfrak {d}^*)$ , and the inverse $Q^R$ can be written as a universal polynomial in $\mathfrak {d}^*d^{V_P}$ by [Reference Čap and Souček11, Theorem 3.9]. Using this, it can be shown that

(5) $$ \begin{align} \Omega^k(G/P, V_P) = \operatorname{\mathrm{im}}(d^{V_P}\mathfrak{d}^*) \oplus \ker(\Box^R) \oplus \Gamma(\operatorname{\mathrm{im}}(\mathfrak{d}^*)), \end{align} $$

(c.f. [Reference Harrach14, p. 24]) and the projection $\Pi ^R \colon \Omega ^k(G/P, V_P) \to \ker (\Box ^R)$ onto the middle summand is given by

(6) $$ \begin{align} \Pi^R(\alpha) := \alpha - Q^R\mathfrak{d}^*d^{V_P}\alpha - d^{V_P}Q^R\mathfrak{d}^*\alpha \end{align} $$

for all $\alpha \in \Omega ^k(G/P, V_P)$ . From the definition of $Q^R$ , one easily concludes that (6) implies $\mathfrak {d}^* \circ \Pi ^R = 0$ . In particular, $\Pi ^R$ factors to an operator $L \colon \Gamma (\mathcal {H}_k) \to \Gamma (\ker (\mathfrak {d}^*))$ , which is a G-equivariant splitting operator for the canonical projection $\pi _{\mathcal {H}}:\Gamma (\ker (\mathfrak {d}^*))\to \Gamma (\mathcal {H}_k)$ and it is characterized by $\mathfrak {d}^*d^{V_P}L = 0$ . We define the k-th BGG-operator $D_k \colon \Gamma (\mathcal {H}_{k-1}) \to \Gamma (\mathcal {H}_k)$ as the composition $D_k := \pi _{\mathcal {H}} d^{V_P} L$ . Since the connection $\nabla ^{V_P}$ on $V_P$ is flat, it follows that the BGG-sequence

is a complex which computes the twisted de Rham cohomology of $G/P$ with values in the vector bundle $V_P$ , c.f. [Reference Čap, Slovák and Souček9, Corollary 4.15].

Our aim is to construct Poisson transforms which satisfy $\Phi \circ \mathfrak {d}^* = 0$ as well as $\Phi \circ d^{V_P} \circ \mathfrak {d}^* = 0$ . By the first property, $\Phi $ naturally factors to an operator $\underline {\Phi } \colon \Gamma (\mathcal {H}_k) \to \Omega ^\ell (G/K, W_K)$ defined by $\underline {\Phi }(\sigma ) := \Phi (\alpha )$ , where $\alpha \in \Gamma (\ker (\mathfrak {d}^*))$ is any representative of $\sigma \in \Gamma (\mathcal {H}_k)$ . The second property yields compatibility with the k-th BGG-operator, i.e. for all $\tau \in \Gamma (\mathcal {H}_{k-1})$ the value $\underline {\Phi }(D_k\tau )$ coincides with $\Phi (d^{V_P}\beta )$ for any representative $\beta \in \Gamma (\ker (\mathfrak {d}^*))$ of $\tau $ , c.f. [Reference Harrach14, Section 3.1].

In order to relate the BGG complex to the geometry of $G/K$ , recall that for each homogeneous vector bundle $E \to G/H$ we have a natural G-action on $\Gamma (E)$ . Infinitesimally, one obtains an action of the universal enveloping algebra $\mathcal {U}(\mathfrak {g})$ on sections of E, so in particular the Casimir element $\mathcal {C} \in \mathcal {U}(\mathfrak {g})$ induces a G-invariant differential operator $\Gamma (E) \to \Gamma (E)$ which we denote by the same symbol.

Part (i) of the previous theorem shows that for $\mathbb {W} = \mathbb {V}$ the condition $\Phi \circ \Box ^R = 0$ is equivalent to $\Delta ^{V_K} \circ \Phi = 0$ . In light of this we have the following definition.

2.2 BGG-compatibility and equivariant maps

A crucial feature of the approach to Poisson transforms introduced above is that, in view of the homogeneity of $G/M$ , G-equivariant bundle maps as in (1) are in bijective correspondence with M-equivariant maps between the representations inducing the bundles. We next describe harmonic Poisson transforms in this language.

We start with some general observations. Since $M=K\cap P$ , any representation $\mathbb {V}$ of K restricts to a representation of M and hence gives rise to a homogeneous bundle $G\times _M\mathbb {V}\to G/M$ , which can be identified with the pullback along $\pi _K$ of the homogeneous bundle $G\times _K\mathbb {V}\to G/K$ . Natural bundle maps between homogeneous bundles over $G/K$ come from K-equivariant maps between the inducing representations and hence lift to natural bundle maps between the pullback bundles. This also applies to the projection to $G/P$ , but in this case, more is true. Since $M\subset G_0\subset P$ , also representations of $G_0$ give rise to natural bundles over $G/M$ , and $G_0$ -equivariant maps induce natural bundle maps between such homogeneous bundles. In the case of representations of $G_0$ which are not restrictions of representations of P, one has to be careful with the pullback interpretation, though. This is available for associated bundles to the principal $G_0$ -bundle $G/P_+\to G/P$ which can be interpreted as homogeneous bundles corresponding to completely reducible representations of P.

The M-representation inducing $T(G/M)$ is $\mathfrak {g}/\mathfrak {m}$ . The subbundle $T_K$ corresponds to $(\mathfrak {g}/\mathfrak {m})^{1,0} := \mathfrak {p}/\mathfrak {m}$ , which is isomorphic to the restriction to M of the K-representation $\mathfrak {g}/\mathfrak {k}$ . Likewise, $T_P$ corresponds to $(\mathfrak {g}/\mathfrak {m})^{0,1} := \mathfrak {k}/\mathfrak {m}$ , which is isomorphic to the restriction of the P-representation $\mathfrak {g}/\mathfrak {p}$ . For the induced decomposition of the dual, there is a more natural point of view as $(\mathfrak {g}/\mathfrak {m})^*=(\mathfrak {g}/\mathfrak {k})^*\oplus (\mathfrak {g}/\mathfrak {p})^*$ . Viewing $(\mathfrak {g}/\mathfrak {m})^*$ as the annihilator of $\mathfrak {m}$ in $\mathfrak {g}^*$ , the summands are naturally contained as annihilators of bigger subalgebras. Hence $\Lambda ^{k_1,k_2} (\mathfrak {g}/\mathfrak {m})^* := \Lambda ^{k_1} (\mathfrak {g}/\mathfrak {k})^* \otimes \Lambda ^{k_2} (\mathfrak {g}/\mathfrak {p})^*$ is the M-module underlying $\Lambda ^{k_1,k_2} T^*(G/M)$ . The above observations on restrictions of $G_0$ -representations to M apply to the chain spaces $C^k(\mathfrak {g}_-, \mathbb {V})$ and the equivariant maps $\mathfrak {d}$ , $\mathfrak {d}^*$ and $\Box $ , for which we denote the induced natural bundle maps by $\mathfrak {d}_P$ , $\mathfrak {d}_P^*$ and $\Box _P$ . The splitting (4) then reads as

(7) $$ \begin{align} \Lambda^{0,k}T^*(G/M)\otimes V_M = \operatorname{\mathrm{im}}(\mathfrak{d}_P) \oplus \ker(\Box_P) \oplus \operatorname{\mathrm{im}}(\mathfrak{d}_P^*), \end{align} $$

and the sum of the first and last two modules coincide with $\ker (\mathfrak {d}_P)$ and $\ker (\mathfrak {d}_P^*)$ , respectively. Finally, $H_k = H_k(\mathfrak {p}_+, \mathbb {V})\cong _M\ker (\Box _P)$ , so over $G/M$ we can identify $\Gamma (\mathcal {H}_k)$ with $\Gamma (\ker (\Box _P))$ , and its elements are described by M-equivariant maps $G\to \ker (\Box _P)$ .

Recall that for any G-representation $\mathbb {V}$ , the homology space $H_k = H_k(\mathfrak {p}_+, \mathbb {V})$ decomposes into a direct sum of irreducible P-representations $\mathbb {E}_i^k$ and we denote by $\pi ^k_i$ the projection onto the component $\mathbb {E}_i^k$ . In the same way, the M-representation $\ker (\Box _P)$ decomposes into irreducible components that we also denote by $\mathbb {E}_i^k$ . Correspondingly $\mathcal {H}_k = G \times _P H_k=\oplus _iE_i^k$ , where $E_i^k := G \times _P \mathbb {E}_i^k$ , and the projection onto one summand is a G-equivariant bundle map. We also denote this bundle map and the induced tensorial operator on sections by $\pi _i^k$ . This leads to a splitting of the kth BGG-operator into components $D^{j,i}_k$ defined by $D^{j,i}_k := \pi _i^k \circ D_k\circ \pi _j^{k-1}$ .

Now suppose that we are given a harmonic Poisson transform $\Phi :\Omega ^k(G/P,V_P)\to \Omega ^\ell (G/K,W_K)$ induced by $\phi \in \operatorname {\mathrm {Hom}}_M(\ker (\Box _P), \Lambda ^{\ell ,0} (\mathfrak {g}/\mathfrak {m})^*\otimes \mathbb {W})$ . Then we define $\phi _i:=\phi \circ \pi _i^k$ and this induces a harmonic Poisson transform $\Phi _i$ acting between the same spaces as $\Phi $ . Part (ii) of Theorem 3 readily shows that for the induced transform we get $\underline {\Phi }_i=\underline {\Phi }\circ \pi _i^k$ . Using this, we can now show that harmonic Poisson transforms are well behaved with respect to compositions with components of BGG-operators.

2.3 A specific family of transforms

We now specialize to a family of transforms that will be used in what follows. We do this only in the case that $\mathbb {V}$ and $\mathbb {W}$ are $\mathbb {R}$ or $\mathbb {C}$ and that $\mathfrak {g}$ is of real rank one. This implies that $\mathfrak {p}\subset \mathfrak {g}$ is a minimal parabolic subalgebra and that $\mathfrak {a}=\mathfrak {q}\cap \mathfrak {g}_0$ is a one-dimensional abelian subalgebra spanned by the grading element E. This element is characterized by $[E, X] = jX$ for all $X \in \mathfrak {g}_j$ , which implies that M acts trivially on $\mathfrak {a}$ .

It is well known that $\mathfrak {p}_+$ is the annihilator of $\mathfrak {p}$ under the Killing form B, so one obtains an isomorphism $(\mathfrak {g}/\mathfrak {p})^*\cong \mathfrak {p}_+$ . This is P-equivariant and hence M-equivariant and thus can also be viewed as an isomorphism $\Lambda ^{0,1}(\mathfrak {g}/\mathfrak {m})^*\cong _M\mathfrak {p}_+$ . Similarly, $\mathfrak {q}$ is the annihilator of $\mathfrak {k}$ under the Killing form, so we get an isomorphism $(\mathfrak {g}/\mathfrak {k})^*\cong \mathfrak {q}$ . This is K-equivariant and hence M-equivariant, so $\Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})^*\cong _M\mathfrak {q}$ . But now $\mathfrak {q}\cap (\mathfrak {g}_-\oplus \mathfrak {p}_+)$ is an M-invariant complement to $\mathfrak {a}$ in $\mathfrak {q}$ , and any element in there can be uniquely written as $\theta (Z)-Z$ for some $Z\in \mathfrak {p}_+$ . Thus, we conclude that $\Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})^*\cong \mathfrak {a}\oplus \mathfrak {p}_+$ as a representation of M. Otherwise put, sending $Z\in \mathfrak {p}_+$ to the map $Y\mapsto B(Y,\theta (Z)-Z)$ defines an M-equivariant injection $\mathfrak {p}_+\to \Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})^*$ . Together with the above, this defines an M-equivariant injection $\iota _1:\Lambda ^{0,1}(\mathfrak {g}/\mathfrak {m})^*\to \Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})^*$ . Taking exterior powers, we obtain M-equivariant maps $\iota _k:\Lambda ^{0,k}(\mathfrak {g}/\mathfrak {m})^*\to \Lambda ^{k,0}(\mathfrak {g}/\mathfrak {m})^*$ .

It is easy to make this map explicit. An element $\alpha \in \Lambda ^{0,k}(\mathfrak {g}/\mathfrak {m})^*$ is a k-linear alternating map $\mathfrak {g}^k\to \mathbb {K}$ , where $\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$ , which vanishes if one of its entries is from $\mathfrak {p}$ . Invariance of the Killing form readily implies that for $Z\in \mathfrak {p}_+$ and $Y\in \mathfrak {g}$ we get $B(Y,\theta (Z)-Z)=B(\theta (Y)-Y,Z)$ . Passing to exterior powers, we readily conclude that

(8) $$ \begin{align} \iota_k(\alpha)(Y_1,\dots,Y_k)=\alpha(\theta(Y_1)-Y_1,\dots,\theta(Y_k)-Y_k). \end{align} $$

Observe that, as expected, this vanishes if one of its entries is in $\mathfrak {k}$ . As an element of $\Lambda ^{k,0}(\mathfrak {g}/\mathfrak {m})^*$ , $\iota _k(\alpha )$ is uniquely determined by its value on entries from $\mathfrak {p}$ , and for $Y_i\in \mathfrak {p}$ , the formula simplifies to

(9) $$ \begin{align} \iota_k(\alpha)(Y_1,\dots,Y_k)=\alpha(\theta(Y_1),\dots,\theta(Y_k)), \end{align} $$

which explains our choice of sign for the map $\iota _1$ .

From Section 2.2 we know that the transform $\Phi ^k_i$ defines a map $\Gamma (E^k_i)\to \Omega ^k(G/K,\mathbb {K})$ and we next compute $d\circ \Phi ^k_i$ and $\delta \circ \Phi ^k_i$ . By Theorem 6 of [Reference Harrach14] these compositions are Poisson transforms induced by the kernels $d_K\varphi ^k_i$ and $\delta _K\varphi ^k_i$ , respectively, where $\varphi ^k_i$ is the kernel of $\Phi ^k_i$ . Here $d_K$ is the component of the exterior derivative which raises the bidegree by $(1,0)$ , and $\delta _K$ corresponds to the codifferential on $G/K$ , see [Reference Harrach14] for details. Since $\varphi ^k_i$ is a G-invariant differential form, these operations can be computed purely algebraically from the inducing M-invariant elements.

To do this, we use the Lie algebra cohomology differential $\mathfrak {d}_{\mathfrak {g}}:\Lambda ^k\mathfrak {g}^*\to \Lambda ^{k+1}\mathfrak {g}^*$ for $\mathfrak {g}$ with values in the trivial representation $\mathbb {K}$ . For $\alpha \in \Lambda ^k\mathfrak {g}^*$ this is explicitly given by

(10) $$ \begin{align} \mathfrak{d}_{\mathfrak{g}}\alpha(Y_0,\dots,Y_k)=\textstyle\sum_{i<j}(-1)^{i+j}\alpha([Y_i,Y_j],Y_0,\dots,\widehat{Y_i}, \dots, \widehat{Y_j},\dots, Y_k), \end{align} $$

where the hats denote omission. Now we first prove a technical lemma.

Lemma 5. Suppose that $\alpha \in \Lambda ^{0,k}(\mathfrak {g}/\mathfrak {m})^*$ satisfies $\mathfrak {d}_P\alpha =0$ and $\beta \in \Lambda ^{0,\ell }(\mathfrak {g}/\mathfrak {m})^*$ satisfies $E\cdot \beta =\nu \beta $ for $\nu \in \mathbb {Z}$ , and consider $\gamma :=\mathfrak {d}_{\mathfrak {g}}(i_k(\alpha )\wedge \beta )\in \Lambda ^{k+\ell +1}\mathfrak {g}^*$ . Take elements $Y_0,\dots , Y_{k+\ell }\in \mathfrak {p}_+$ such that $Y_j\in \mathfrak {g}_{s_j}$ for $j>k$ and put $\tilde Y_i:=\theta (Y_i)+Y_i$ .

Then $\gamma (Y_0,\dots ,Y_k,\tilde Y_{k+1},\dots ,\tilde Y_{k+\ell })=0$ provided that $s_{j+1}+\dots +s_{k+\ell }\leq \nu $ . Moreover, we get

(11) $$ \begin{align} \gamma(E,Y_1,\dots,Y_k,\tilde Y_{k+1},\dots ,\tilde Y_{k+\ell})=\big(-i_k(E\cdot\alpha)\wedge\beta+\nu i_k(\alpha)\wedge \beta\big) (Y_1,\dots ,\tilde Y_{k+\ell}). \end{align} $$

Proof. Computing $\gamma (Y_0,\dots ,Y_k,\tilde Y_{k+1},\dots ,\tilde Y_{k+\ell })$ via formula (10), we have to distinguish cases according to which types of elements go into the Lie bracket. We know that $i_k(\alpha )$ vanishes if one of its entries is from $\mathfrak {k}$ while $\beta $ vanishes if one of its entries is from $\mathfrak {p}$ . Now $[\tilde Y_i,\tilde Y_j]\in \mathfrak {k}$ , so in the corresponding terms we feed $k+1$ entries from $\mathfrak {p}_+$ and $\ell -1$ entries from $\mathfrak {k}$ into $i_k(\alpha )\wedge \beta $ , which automatically gives zero. Similarly, $[Y_i,Y_j]\in \mathfrak {p}$ , and hence for the corresponding summands, all the $\tilde Y$ ’s have to be inserted into $\beta $ . Thus, for these summands, we have to consider

$$ \begin{align*}\left(\textstyle\sum_{i<j\leq k}(-1)^{i+j}i_k(\alpha)([Y_i,Y_j],Y_0,\dots, \hat i,\dots, \hat j,\dots, Y_k)\right)\beta(\tilde Y_{k+1},\dots,\tilde Y_{k+\ell}). \end{align*} $$

Using formula (9) and that $\theta ([Y_i,Y_j])=[\theta (Y_i),\theta (Y_j)]$ , we conclude that these terms add up to zero since $\mathfrak {d}_P\alpha =0$ . Thus, we are left with the summands that contain $[Y_i,\tilde Y_j]$ , for which there are k more entries from $\mathfrak {p}_+$ . Hence all the remaining Y’s have to go into $i_k(\alpha )$ while the bracket term and the remaining $\tilde Y$ ’s have to be inserted into $\beta $ . Now $[Y_i,\tilde Y_j]=[Y_i,\theta (Y_j)]+[Y_i,Y_j]$ and the second summand lies in $\mathfrak {p}_+$ and hence inserts trivially into $\beta $ . For the same reason we can also replace the remaining entries of $\beta $ by $\theta (Y_r)$ instead of $\tilde Y_r$ .

By assumption, $\theta (Y_r)\in \mathfrak {g}_{-s_r}$ , and since $Y_i\in \mathfrak {p}_+$ , the bracket $[Y_i,\theta (Y_j)]$ is a sum of elements of $\mathfrak {g}_i$ with $i>-s_j$ . Hence the degrees of the elements inserted into $\beta $ add up to a number which is $>-s_{k+1}-\dots -s_{k+\ell }\geq -\nu $ . But since we have assumed that $E\cdot \beta =\nu \beta $ , $\beta $ can only be non-zero on a family of elements if these degrees add up exactly to $-\nu $ . Hence these terms do not contribute either, and the proof of the first part is complete.

For the second part, we first observe that E inserts trivially into $i_k(\alpha )\wedge \beta $ , so in the expansion according to (10), we only have to consider the terms in which E gets inserted into the Lie bracket. Moreover, $[E,Y_i]\in \mathfrak {p}$ , so this bracket has to go into $i_k(\alpha )$ together with the other Y’s, while all the $\tilde Y$ ’s have to go into $\beta $ . So we have to consider

$$ \begin{align*}-\textstyle\sum_{i=1}^ki_k(\alpha)(Y_1,\dots,[E,Y_i],\dots,Y_k)= -\textstyle\sum_{i=1}^k\alpha(\theta(Y_1),\dots,\theta([E,Y_i]),\dots,\theta(Y_k)). \end{align*} $$

Using that $\theta ([E,Y_i])=-[E,\theta (Y_i)]$ , we see that this sum equals $-(E\cdot \alpha )(\theta (Y_1),\dots ,\theta (Y_k))$ . So the first k summands in (10) add up to $-(i_k(E\cdot \alpha )\wedge \beta )(Y_1,\dots ,\tilde Y_{k+\ell })$ . For the remaining terms, $[E,\tilde Y_j]=[E,\theta (Y_j)]+[E,Y_j]$ has to be inserted into $\beta $ , so the second summand does not contribute, while all the $Y_i$ have to go into $i_k(\alpha )$ . Thus, we have to consider the sum

$$ \begin{align*}-\textstyle\sum_{j=k+1}^{k+\ell}\beta(\theta(Y_{k+1}),\dots,[E,\theta(Y_j)],\dots, \theta(Y_{k+\ell}))=(E\cdot\beta)(\tilde Y_{k+1},\dots,\tilde Y_{k+\ell}), \end{align*} $$

so these summands provide the rest of the claimed formula.

Viewed as an element of $\mathfrak {a}\subset \Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})$ , the grading element E defines an M-invariant element $E^*\in \Lambda ^{1,0}(\mathfrak {g}/\mathfrak {m})^*$ and hence an element of $\Omega ^{1,0}(G/M)^G$ that we denote by the same symbol. Using this, we can now formulate the following theorem.

3.1 Geometric setup

We realize $G=\operatorname {\mathrm {SU}}(n+1,1)$ as the stabilizer in $\operatorname {\mathrm {GL}}(n+2, \mathbb {C})$ of the sesquilinear form $(z, w) \mapsto z_0\overline {w_{n+1}} + z_{n+1}\overline {w_0} + \sum _{j=1}^n z_j \overline {w_j}$ for z, $w \in \mathbb {C}^{n+2}$ . The maximal compact subgroup K is isomorphic to $S(\operatorname {\mathrm {U}}(n+1)\times \operatorname {\mathrm {U}}(1))$ , i.e. all pairs $(B, b) \in \operatorname {\mathrm {U}}(n+1) \times \operatorname {\mathrm {U}}(1)$ with $b\det (B)= 1$ . The parabolic subgroup P is isomorphic to $\operatorname {\mathrm {CSU}}(n+1) \rtimes (\mathbb {C}^{n} \oplus i\mathbb {R})$ , and M is given by $S(\operatorname {\mathrm {U}}(n) \times \operatorname {\mathrm {U}}(1))$ , i.e. all pairs $(B, b) \in \operatorname {\mathrm {U}}(n) \times \operatorname {\mathrm {U}}(1)$ so that $b^2\det (B)=1$ . The Lie algebra $\mathfrak {g} = \mathfrak {su}(n+1,1)$ is naturally $|2|$ -graded with grading components $\mathfrak {g}_{\pm 1} \cong \mathbb {C}^n$ and $\mathfrak {g}_{\pm 2} \cong i\mathbb {R}$ , whereas the component $\mathfrak {g}_0$ splits into the direct sum of $\mathfrak {a} \cong \mathbb {R}$ and $\mathfrak {m} \cong i\mathbb {R} \oplus \mathfrak {su}(n)$ . Under these identifications the M-action on $\mathfrak {g}_{\pm 1}$ is given by $(B, b) \cdot X = b^2BX$ for all $X \in \mathfrak {g}_{\pm 1} \cong \mathbb {C}^n$ , whereas the components $\mathfrak {g}_{\pm 2}$ are M-invariant.

The space $\mathbf {G/K.}$ The symmetric space $G/K$ is the complex hyperbolic space of (complex) dimension $n+1$ , which is a Kähler manifold. The complex structure on $G/K$ induces a decomposition of $\Omega ^k(G/K, \mathbb {C}) = \bigoplus _{p+q=k} \Omega ^{p,q}(G/K)$ into $(p,q)$ -types, and the exterior derivative splits accordingly as $d = \partial + \overline {\partial }$ . Next, the Kähler metric induces an $L^2$ -inner product on the space of differential forms and we define $\delta $ , $\partial ^*$ and $\overline {\partial }^*$ to be the formal adjoints of d, $\partial $ and $\overline {\partial }$ , respectively. Consequently, we have the Laplacian $\Delta = d\delta +\delta d$ and similarly $\Box = \partial \overline {\partial } + \overline {\partial }\partial $ and its complex conjugate $\overline {\vphantom{1^{.}}\Box }$ , which are all formally self-adjoint with respect to the $L^2$ -inner product and satisfy $\Delta = 2\Box = 2\overline {\vphantom{1^{.}}\Box }$ , c.f. [Reference Ballmann2, Corollary 5.26]. Additionally, the wedge product with the Kähler form $\omega \in \Omega ^{1,1}(G/K)$ yields the Lefschetz map L and we denote by $L^*$ its adjoint with respect to the induced inner product on $\Lambda ^k T^*(G/K)$ and call it the Co-Lefschetz map. These operators are linked via the Kähler relations $[L^*, \partial ] = i\overline {\partial }^*$ and $[L^*, \overline {\partial }] = - i\partial ^*$ , c.f. [Reference Ballmann2, Proposition 5.22]. A differential form $\alpha \in \Omega ^k(G/K)$ is said to be primitive if $L^*\alpha = 0$ , and primitive forms only exist for $0 \le k \le n+1$ .

We have seen that the images of the transforms from Definition 2 consist of coclosed and harmonic differential forms. Using the Kähler relations, we can prove an equivalent description of these two properties, which will significantly simplify the necessary computations.

The space $\mathbf {K/M.}$ In the next step we describe the geometry of the homogeneous space $G/P$ in terms of its underlying pseudo-Hermitian structure induced by the isomorphism $K/M \cong G/P$ . The group M acts trivially on the grading component $\mathfrak {g}_2 \cong i\mathbb {R}$ , so each element induces a K-invariant $1$ -form $\vartheta $ on $K/M$ , which is a contact form and unique up to multiples. We define $\zeta \in \mathfrak {X}(K/M)$ to be the Reeb vector field corresponding to $\vartheta $ which is characterized by $\vartheta (\zeta ) = 1$ and $\iota _{\zeta }d\vartheta = 0$ . Then $T(K/M)$ is the direct sum of the contact subbundle $H := \ker (\vartheta )$ and the line bundle spanned by $\zeta $ , which is isomorphic to the quotient $Q := T(K/M)/H$ . Since H is associated to the M-module $\mathfrak {g}_{-1} \cong \mathbb {C}^n$ , it carries a canonical K-invariant complex structure J, which induces a decomposition $H \otimes \mathbb {C} = H^{1,0} \oplus H^{0,1}$ into holomorphic and antiholomorphic parts and accordingly a decomposition $\Lambda ^k H^* \otimes \mathbb {C} = \bigoplus _{p+q=k} \Lambda ^{p,q} H^*$ into $(p,q)$ -types. In particular, we obtain a splitting

(12) $$ \begin{align} \Lambda^k T^*(K/M) \otimes \mathbb{C} = \bigoplus_{p+q=k} \Lambda^{p,q} H^* \oplus \bigoplus_{p+q=k-1} Q^* \otimes \Lambda^{p,q} H^* \end{align} $$

for all $0 \le k \le n$ , and since $[\mathcal {L}_{\zeta }, J] = 0$ , this decomposition is invariant under the Lie derivative with respect to the Reeb vector field.

Since $\vartheta $ is a contact form, the restriction $\omega _H\in \Gamma (\Lambda ^{1,1} H^*)$ of $d\vartheta $ is nondegenerate and thus $g_H(\ , \ ) := \omega _H(\ , J\ )$ defines a bundle metric on H. Denote by $\langle \ , \ \rangle _H$ the induced pointwise inner product on $\Lambda ^{*} H^*$ and by $\langle \!\langle \ , \ \rangle \!\rangle _H$ its induced $L^2$ -inner product on $\Gamma (\Lambda ^* H^*)$ . The $(n,n)$ -form $\operatorname {\mathrm {vol}}_H := \frac {1}{n!} \omega _H^n$ is normed with respect to $\langle \ , \ \rangle _H$ and thus we define the H-Hodge star operator $\ast _H \colon \Lambda ^{p,q} H^* \to \Lambda ^{n-q,n-p} H^*$ via the relation $\alpha \wedge \ast _H\beta = \langle \alpha , \overline {\beta }\rangle _H \operatorname {\mathrm {vol}}_H$ for all $\alpha \in \Lambda ^{q,p} H^*$ .

The inclusion of H into $T(K/M)$ dualizes to the projection $\pi _H \colon \Lambda ^k T^*(K/M) \to \Lambda ^k H^*$ and we define $d_H \colon \Gamma (\Lambda ^k H^*) \to \Gamma (\Lambda ^{k+1} H^*)$ via $d_H := \pi _H \circ d$ . Since both $H^{1,0}$ and $H^{0,1}$ are involutive subbundles of $T(K/M) \otimes \mathbb {C}$ , we can split the complex linear extension of $d_H$ into the sum of $\partial _H$ and $\overline {\partial }_H$ , where the first and second operators map sections of $\Lambda ^{p,q} H^*$ to sections of $\Lambda ^{p+1,q} H^*$ and $\Lambda ^{p,q+1} H^*$ , respectively. Next, define $\delta _H$ as the (formal) $L^2$ -adjoint of $d_H$ , which again splits into $\delta _H = \partial _H^* + \overline {\partial }_H^*$ according to $(p,q)$ -types. Finally, we obtain the H-Laplacian $\Delta _H := d_H\delta _H + \delta _H d_H$ as well as the operator $\Box _H := \partial _H\partial _H^* + \partial _H^*\partial _H$ and its complex conjugate.

On sections of $\Lambda ^{p,q} H^*$ , define the map $L_H$ as the wedge product with $\omega _H$ and $L_H^*$ as its adjoint with respect to $\langle \ , \ \rangle _H$ . It is immediate that $L_H$ and $L_H^*$ raise and lower, respectively, the bidegree by $(1,1)$ , and a chase through the definitions shows that $L_H^*$ coincides with the Levi-bracket under the identification of Q with the line subbundle of $T(K/M)$ spanned by the Reeb vector field. Next, by nondegeneracy of $\omega _H$ we see that $L_H$ is injective on $\Lambda ^{p,q}H^*$ for all $0 \le p + q < n$ , which by duality also implies that $L_H^*$ is surjective on this bundle. Moreover, via the duality given by the H-Hodge star we deduce that $L_H$ is surjective and $L_H^*$ is injective on $\Lambda ^{p,q}H^*$ for $p+q \ge n+1$ . In the remaining case $p+q = n$ the kernels of $L_H$ and $L_H^*$ coincide.

In the next step we describe the BGG complex on $G/P$ in terms of the operators induced by the pseudo-Hermitian structure. The group P acts on $\mathfrak {g}_2$ by scalars, so the subbundle $H = \ker (\vartheta )$ is G-invariant. Furthermore, by naturality of the exterior derivative the group G acts on the restriction $\omega _H$ of $d\vartheta $ by multiplication with positive smooth functions. In particular, for all $0 \le k \le 2n$ the kernel of $L_H^*|_{\Lambda ^k H^*}$ is an invariant subbundle which we denote by $\Lambda ^k_0 H^*$ . The definition of the Kostant codifferential easily implies that $\mathfrak {d}^*\alpha $ coincides with $\vartheta \wedge L_H^*\alpha $ up to a multiple. Therefore, the homology bundle $\mathcal {H}_k$ is isomorphic to $\Lambda ^k_0 H^*$ for $k \le n$ and to a quotient of $\Lambda ^{k-1}H^*\otimes Q^*$ for $k \ge n+1$ , cf. [Reference Čap, Harrach and Julg7, Proposition 3.1]. Additionally, on the fibre of H over the origin of $G/P$ the group $P_+$ acts trivially, whereas $G_0$ acts by complex linear maps, implying that the complex structure J on H and thus the decomposition of $\Lambda ^k H^*$ into $(p,q)$ -types is G-invariant. Thus, for $k \le n$ we obtain an induced decomposition of $\mathcal {H}_k \otimes \mathbb {C} = \bigoplus _{p+q=k} \mathcal {H}_{p,q}$ into $(p,q)$ -types, and it turns out that each of the G-invariant bundles $\mathcal {H}_{p,q}$ is irreducible. Accordingly, for $1 \le k \le n$ the k-th BGG-operator $D = D_k \colon \Gamma (\mathcal {H}_{k-1}) \to \Gamma (\mathcal {H}_{k})$ splits into the sum $D = \mathcal {D} + \overline {\mathcal {D}}$ , where $\mathcal {D}$ and $\overline {\mathcal {D}}$ map sections of $\mathcal {H}_{p,q}$ to sections of $\mathcal {H}_{p+1,q}$ and $\mathcal {H}_{p,q+1}$ , respectively.

Via the identification of $\mathcal {H}_{p,q}$ as a subbundle of $\Lambda ^{p+q}_0 H^* \otimes \mathbb {C}$ , we obtain a pointwise Hermitian product on the holonomy bundles as well as an induced $L^2$ -Hermitian product, which are both K-invariant. Denote by $D^* = D^*_k \colon \Gamma (\mathcal {H}_k) \to \Gamma (\mathcal {H}_{k-1})$ the $L^2$ -adjoint of the k-th BGG-operator. For $k \le n-1$ we can split $D^*$ into $\mathcal {D}^* + \overline {\mathcal {D}}^*$ , where the first and second operators lower the first and second degree, respectively. Using the formula for the BGG-operator in Theorem 9, we immediately deduce that $\mathcal {D}^*$ corresponds to $\partial _H^*$ under the identification $\mathcal {H}_{p,q} \cong \Lambda ^{p,q}_0 H^*$ , and similarly for their complex conjugates. In particular, Proposition 8(iii) implies that $\mathcal {D}^*$ and $\overline {\mathcal {D}}^*$ anticommute.

Corollary 10. For $k \le n-1$ the BGG-operators and their adjoints satisfy on $\Gamma (\Lambda ^{k}_0 H^*)$ the relations

$$ \begin{align*} \lambda_k\overline{\mathcal{D}}^*_k \mathcal{D}_k + (\lambda_k -1) \mathcal{D}_{k-1}\overline{\mathcal{D}}^*_{k-1} &= 0, & \lambda_k \mathcal{D}^*_{k}\overline{\mathcal{D}}_{k} + (\lambda_k -1) \overline{\mathcal{D}}_{k-1} \mathcal{D}^*_{k-1} &= 0, \end{align*} $$

where $\lambda _k = n+1-k$ .

3.2 The distinguished harmonic Poisson transform

The invariant structures on the tangent bundles of $G/K$ and $G/P$ induce corresponding structures on the subbundles $T_K$ and $T_P$ of $T(G/M)$ , which yield further properties of the G-equivariant operators from Definition 2.

The bundle $T_P = \ker (\pi _K)$ is associated to the same M-representation as $T(K/M)$ , implying that each tensorial object on the latter space induces analogous objects on $T_P$ , which we denote by the same symbol. Explicitly, $T_P$ contains a rank- $2n$ bundle H, which is endowed with an invariant complex structure J as well as a bundle metric. The fibre $H_{eM}$ of H at the origin $eM \in G/M$ is isomorphic to $\mathbb {C}^n$ , and the M-invariant inner product on $H_{eM}$ is a multiple of the standard Hermitian inner product. Moreover, there is an invariant section $\omega _H \in \Gamma (\Lambda ^2 T_P^*)$ , and wedging with $\omega _H$ defines a tensorial map $L_H$ with adjoint $L_H^*$ . The kernel $\Lambda ^k_0 H^*$ of the restriction of $L_H^*$ to $\Lambda ^k H^*$ is a G-invariant subbundle, whose complexification decomposes into $(p,q)$ -types. From the description of the homology bundles on $G/P$ it follows that the subbundle $\ker (\Box _P) \subset \Lambda ^k T^*_P$ coincides with $\Lambda ^k_0 H^*$ for $0 \le k \le n$ .

On the other hand, we have seen in Section 2.3 that the grading element $E \in \mathfrak {a}$ induces an invariant vector field $\zeta _E \in \Gamma (T_K)\subset \mathfrak {X}(G/M)$ as well as an invariant $1$ -form $E^* \in \Omega ^1(G/M)$ of bidegree $(1,0)$ . Denoting by $E^*_{\mathbb {C}}:T_K\to \mathbb {C}$ the complex linear extension of $E^*:T_K\to \mathbb {R}$ , $\ker (E^*_{\mathbb {C}}) \subseteq T_K$ is a G-invariant subbundle whose underlying representation is isomorphic to $\mathbb {C}^n$ . Furthermore, the pullback of the metric and the complex structure on $G/K$ along the projection $\pi _K \colon G/M \to G/K$ yields an invariant bundle metric as well as an invariant complex structure on $T_K$ , respectively, and the restriction of the metric to $\ker (E^*_{\mathbb {C}})$ corresponds to the standard Hermitian inner product on $\mathbb {C}^n$ . A vector field $\xi \in \Gamma (T_K)$ determines sections $\xi ^{1,0}$ and $\xi ^{0,1}$ of the holomorphic and the antiholomorphic part of $T_K\otimes \mathbb {C}$ . The Co-Lefschetz map on $G/K$ is tensorial and induces a G-invariant complex linear map $L_K^*$ on $\Lambda ^* T^*_K\otimes \mathbb {C}$ . Explicitly, if $\{\xi _{\ell }\}$ is a complex orthonormal frame of $\ker (E^*_{\mathbb {C}})\subset T_K$ , then we can write $L_K^*$ as a linear combination of $\iota _{\zeta _E^{1,0}}\iota _{\zeta _E^{0,1}}$ and $\sum _{\ell }\iota _{\xi _{\ell }^{1,0}}\iota _{\xi _{\ell }^{0,1}}$ . Observe that after complexification, $E^*_{\mathbb {C}}$ induces two sections $E^{*(1,0)}$ and $E^{*(0,1)}$ of $(T_K\otimes \mathbb {C})^*$ , which vanish upon insertion of antiholomorphic and holomorphic vector fields, respectively.

For $0 \le p, q \le n$ with $0 \le p+q=k \le n$ we denote the Poisson transform as in Definition 2 corresponding to the irreducible subbundle $\mathcal {H}_{p,q}$ of $\mathcal {H}_k \otimes \mathbb {C}$ by $\Phi _{p,q} \colon \Omega ^k(G/P, \mathbb {C}) \to \Omega ^k(G/K, \mathbb {C})$ with the induced operator $\underline {\Phi }_{p,q} \colon \Gamma (\mathcal {H}_{p,q}) \to \Omega ^k(G/K, \mathbb {C})$ . From the construction of the underlying map $\phi _{p,q} \in \operatorname {\mathrm {Hom}}_G(\Lambda ^{p,q}_0 H^*, \Lambda ^{p+q} T_K^* \otimes \mathbb {C})$ it is obvious that $\phi _{p,q}$ preserves the $(p,q)$ -type and is conformal with respect to the bundle metrics on H and $\ker (E^*_{\mathbb {C}})$ .

Now we can construct a family of real operators $\underline {\Phi }_k \colon \Gamma (\mathcal {H}_k) \to \Omega ^k(G/K)$ for $0 \le k \le n$ via suitable linear combinations of the operators $\underline {\Phi }_{p,q}$ in such a way that we obtain a chain map between the BGG complex on $G/P$ and the de Rham complex on $G/K$ .

Theorem 12. For all $0 \le p, q \le n$ with $0 \le p+q \le n$ define $\lambda _{p,q} \in \mathbb {C}$ via $\lambda _{0,0} := \frac {1}{\operatorname {\mathrm {vol}}(K/M)}$ and

$$ \begin{align*} \lambda_{p,q} := \lambda_{0,0}\frac{n!}{(n+1-(p+q))!} \prod_{i=1}^p c_{i,q}^{-1} \prod_{j=1}^q \overline{c_{j,0}}^{-1}, \end{align*} $$

where $c_{p,q} \neq 0$ are the constants appearing in Proposition 11(iii). Then for all $0 \le k \le n$ the images of the operators $\underline {\Phi }_k \colon \Gamma (\mathcal {H}_k) \to \Omega ^k(G/K)$ defined by $\underline {\Phi }_k := \sum _{p+q=k} \lambda _{p,q} \underline {\Phi }_{p,q}$ consist of primitive, coclosed and harmonic differential forms. Moreover, for all $1 \le k \le n$ they satisfy the relation $d \circ \underline {\Phi }_{k-1} = \underline {\Phi }_k \circ D_{k}$ .

Proof. Using the relations $c_{p,q}\overline {c_{q,p-1}} = c_{p,q-1} \overline {c_{q,p}}$ , it is easy to see that the coefficients satisfy

(13) $$ \begin{align} (n+1-(p+q))\lambda_{p,q} = \lambda_{p+1,q}c_{p+1,q} = \lambda_{p,q+1} \overline{c_{q+1,p}}, \end{align} $$

which by part (iii) of the previous proposition imply that these operators are chain maps.

The above construction of real Poisson transforms provides a geometric connection between the differential calculi on $G/P$ and $G/K$ in each degree. For the following developments, the cases of main interest are the middle degrees n and $n+1$ . In these degrees, we have the diagram

where we denote by $\Omega ^k_{\Delta }(G/K)$ the space of harmonic k-forms on $G/K$ . Now it is well known that the space $L^2\Omega ^{n+1}_{\Delta }(G/K)$ of $L^2$ -harmonic $(n+1)$ -forms on $G/K$ is a realization of the direct sum of all discrete series representations of G with trivial infinitesimal character (see [Reference Pedon26] and Section 6). Our aim for the rest of the paper is to show that the Poisson transform $d\underline {\Phi }_n$ maps into the space of $L^2$ -harmonic forms and has dense image there. On the one hand, this leads to a realization of (a dense subspace in) the sum of discrete series representations with trivial infinitesimal character on a quotient of the space $\Gamma (\mathcal {H}_n)$ on the boundary sphere. On the other hand, as we will prove in Theorem 28, we obtain an induced exact sequence

for which the last operator has dense image. So one can ‘cut off’ (a completion of) the BGG complex in the middle degree by the sum of discrete series representations with trivial infinitesimal character, which is important for applications towards the Baum-Connes conjecture.

4.1 Polar decomposition of complex hyperbolic space

In the next step we relate differential operators on $K/M$ and $G/K$ in the Poincaré ball model via the polar decomposition $\Psi = (r,\chi ) \colon B \setminus \{ 0\} \to (0,1) \times K/M$ given by $r(z) := |z|$ and $\chi (z) := r(z)^{-1}z$ . For fixed $r \in (0,1)$ the map $\Psi ^{-1}_r := \Psi ^{-1}(r,-) \colon K/M \to B \setminus \{0\}$ is a K-equivariant isomorphism, implying that the image $S_r := \Psi ^{-1}_r(K/M)$ is a K-orbit of B and coincides with the sphere of radius r. Define the distribution $H_r \subset TS_r$ as the image of the CR-distribution $H \subset T(K/M)$ under $T\Psi ^{-1}_r$ .

The (complex) standard basis of $\mathbb {C}^{n+1}$ induces complex coordinates $z = (z_0, \dotsc , z_n)$ of $z \in \mathbb {C}^{n+1}$ . Then $\{z_j, \overline {z}_j\}$ are real coordinates on B and we define the partial derivatives $\partial _{z_j}$ and $\partial _{\overline {z}_j}$ with respect to these coordinates. Define the holomorphic vector field $T := \sum _{j=0}^n z_j \partial _{z_j}$ as well as the $(1,0)$ -form $\tau := \sum _{j=0}^n \overline {z}_j dz_j$ . We obtain the relations $\tau (T) = \overline {\tau }(\overline {T}) = r^2$ and therefore $\partial _r = \frac {1}{r}(T + \overline {T})$ and $dr = \frac {1}{2r}(\tau + \overline {\tau })$ . Furthermore, we fix the pseudo-Hermitian structure $\vartheta \in \Omega ^1(K/M)$ so that its corresponding Reeb vector field $\zeta \in \mathfrak {X}(K/M)$ is given by $\zeta (u) = i(T(u) - \overline {T}(u))$ for all $u \in S^{2n+1}$ . Using the definition of $\Psi ^{-1}_r$ , a direct computation shows that $\zeta $ is pushed forward to the vector field $J\partial _r = \frac {i}{r}(T-\overline {T}) \in \mathfrak {X}(B\setminus \{0\})$ .

It is immediate that for $z \in S_r$ the tangent space $T_zB\otimes \mathbb {C}$ is the direct sum of $H_r\otimes \mathbb {C}$ and the complex subspace spanned by $T(z)$ and $\overline {T}(z)$ . Using this and the description (14) for the Bergman metric on B, a direct computation yields that the Kähler metric g and the Kähler form $\omega $ are given in terms of the polar decomposition by

$$ \begin{align*} g &= \frac{1}{(1-r^2)^2r^2} \frac{1}{2}\tau \odot \overline{\tau} + \frac{r^2}{1-r^2} \frac{1}{2} \chi^*g_H, & \omega &= \frac{1}{(1-r^2)^2r^2}\frac{i}{2} \tau \wedge \overline{\tau} + \frac{r^2}{1-r^2} \frac{1}{2}\chi^*\omega_H. \end{align*} $$

Furthermore, since the distribution $H_r$ was defined as the pullback of $H \subset T(G/P)$ along the angular map, every $(p,q)$ -form $\alpha $ on $B\setminus \{0\}$ can be decomposed as

$$ \begin{align*} \alpha = \chi^*\sigma_{p,q} + \tau \wedge \chi^*\sigma_{p-1,q} + \overline{\tau} \wedge \chi^*\sigma_{p,q-1} + \tau \wedge \overline{\tau} \wedge \chi^*\sigma_{p-1,q-1}, \end{align*} $$

where $\sigma _{s,t}$ is a smooth map $(0,1) \to \Gamma (\Lambda ^{s,t} H^*)$ . In the sequel we determine an explicit formula for the image of the operator $\underline {\Phi }_{p,q}$ in terms of this decomposition, where we will use that this image consists of primitive, coclosed and harmonic differential forms. In view of Lemma 7 we need to determine formulae for $\partial $ , $\overline {\partial }$ and the Co-Lefschetz map $L^*$ .

Proof. (i) Follows from a direct computation using the definition of $\tau $ .

(ii) Since $\iota _{\partial _r}d\chi ^*\sigma = 0$ by naturality of the exterior derivative, it follows that $\iota _{T}\chi ^*\sigma = -\iota _{\overline {T}}\chi ^*\sigma $ . On the other hand, the vector field $i(T - \overline {T})$ is $\chi $ -related to the Reeb vector field $\zeta $ , implying the first formula. For the second formula we use naturality of the exterior derivative to write

(15) $$ \begin{align} d\chi^*\sigma = \chi^*\partial_H\sigma + \chi^*\overline{\partial}_H\sigma + \chi^*\vartheta \wedge \chi^*\mathcal{L}_{\zeta}\sigma. \end{align} $$

A direct computation shows that $\chi ^*\vartheta = \frac {1}{2ir^2}(\tau - \overline {\tau })$ , and after inserting this into (15) and using (i), we decompose the resulting equation according to $(p,q)$ -types.

(iii) Using the expression for the Kähler form yields explicit formulae for the Lefschetz map and thus for $L^*$ by adjointness.

4.2 Boundary values of Poisson transforms

As a first step towards understanding the boundary asymptotics of the values of $\underline {\Phi }_{p,q}$ , we will consider the case of zeros of the initial section in the case $p+q\leq n$ . Given $x \in \partial B$ and a section $\alpha $ of the homology bundle with $\alpha (x) = 0$ , we prove that the differential form $\underline {\Phi }_{p,q}(\alpha )$ also tends to $0$ along every smooth curve on $\overline {B}$ that ends in x. This will be done via a norm estimate argument, for which we need to analyze the behaviour of the G-equivariant map underlying $\Phi _{p,q}$ along the fibres of the projection $\pi _K \colon G/M \to G/K$ .

In order to do so, recall that for every $w \in \mathbb {R}$ there is a real line bundle $\mathcal {E}(w) = G \times _P \mathbb {R}(w) \to G/P$ whose sections are densities of weight w, normalized so that $\mathcal {E}(-2n-1)$ is isomorphic to the bundle of volume densities on the oriented manifold $G/P$ . Given such a density $\sigma \in \Gamma (\mathcal {E}(w))$ with the corresponding P-equivariant map $f \in C^{\infty }(G, \mathbb {R}(w))^P$ , the pullback $\pi _P^*\sigma \in C^{\infty }(G/M)$ is the smooth function which is induced by viewing f as an M-equivariant map.

In the next step we need to analyze the asymptotics of powers of $\mathcal {P}$ as these govern the change of the differential form $|\pi _P^*\alpha |_M \operatorname {\mathrm {vol}}_M$ on $G/M$ along the fibres over $G/P=S^{2n+1}$ for all $\alpha \in \Gamma (\Lambda ^k H^*)$ . After expressing the induced G-actions on the Poincaré ball model and its boundary sphere an explicit computation yields for $z = gK \in B$ and $u = hP \in G/P$ with $h \in K$ that

$$ \begin{align*} \mathcal{P}(z,u) = \frac{1-|z|^2}{|1-\langle z, u \rangle|^2}. \end{align*} $$

In particular, by Lemma 14 the change of the volume form $\operatorname {\mathrm {vol}}_M$ is governed by $\mathcal {P}(z,u)^{n+1}$ , which is (a multiple of) the Poisson-Bergman kernel on the unit ball in $\mathbb {C}^{n+1}$ , c.f. [Reference Krantz21, Section 3.2.2]. It turns out that the asymptotic behaviour of $\mathcal {P}$ can be described via Gaussian hypergeometric functions with specific parameters, and since these will appear again later, we collect important information in the next proposition.

Proposition 15. Let a, b, c with $c \not \in \mathbb {Z}_{\le 0}$ and for $z \in \mathbb {C}$ with $|z| < 1$ let

$$ \begin{align*} {}_2F_1(a,b,c;z) := \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!} \end{align*} $$

be the (Gaussian) hypergeometric function, where $(q)_n$ is the rising Pochhammer symbol. Then

1. (i) The function ${}_2F_1(a,b,c;z)$ is the unique solution up to multiples of the hypergeometric differential equation

   $$ \begin{align*} z(1-z) F"(z) + (c - (a+b+1)z) F'(z) - ab F(z) &= 0 \end{align*} $$
   which is bounded at $z = 0$ .

2. (ii) $\frac {d}{dz} {}_2F_1(a,b,c;z) = \frac {ab}{c} {}_2F_1(a+1,b+1,c+1;z)$ .

3. (iii) If $\operatorname {\mathrm {re}}(c-a-b)> 0$ , then the limit of ${}_2F_1(a,b,c;z)$ for $z \to 1$ exists and is given by

   $$ \begin{align*} {}_2F_1(a,b,c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}. \end{align*} $$

4. (iv) For $\operatorname {\mathrm {re}}(c-a-b) = 0$ we have ${}_2F_1(a,b,c;z) \in L^1([0,1])$ .

Using this information on hypergeometric functions, we are ready to prove the asymptotic behaviour of specific powers of the function $\mathcal {P}$ .

Proposition 16. For all $0 \le k \le n+1$ we have

$$ \begin{align*} I_k(z) := \int_{S^{2n+1}} \mathcal{P}(z,u)^{n+1-\frac{k}{2}} du = \operatorname{\mathrm{vol}}(S^{2n+1}) \cdot {}_2F_1\left(\frac{k}{2}, \frac{k}{2}, n+1;|z|^2\right). \end{align*} $$

In particular, $I_k(z)$ is globally bounded for $0 \le k \le n$ .

Proof. First, we show that $I_k$ is an eigenfunction of the Laplacian, which can be done via a tedious computation of $\Delta \mathcal {P}(z,u)^{n+1-\frac {k}{2}}$ . However, we can avoid this by use of representation theory. Explicitly, combining parts (ii) and (iii) of Lemma 14 the integrals $I_{-k}(z)$ coincide with a multiple of the Poisson transform $\Phi _0(\sigma _k)$ of the k-density $\sigma _{k}$ , which is defined by the P-equivariant function $f_{k} \colon G \to \mathbb {R}$ given by $f_{k}(g\exp (tE)n):= e^{-kt}$ for any $g\in K$ and $n\in N$ . In particular, since $\Delta $ is the negative of the differential operator $\mathcal {C}$ induced by the Casimir element, G-equivariance of the Poisson transform implies that $\Delta I_{k}(z) = -\Phi _0(\mathcal {C} \cdot \sigma _{-k})$ . Now $\sigma _{-k}$ is a section of a $1$ -dimensional representation with highest weight $\lambda := kE^*$ on which $\mathcal {C}$ acts by the scalar $\langle \lambda , \lambda - 2 \rho \rangle $ , c.f. [Reference Knapp18, Proposition 5.28]. Here the inner product $\langle \ , \ \rangle $ on $\mathfrak {a}^*$ is induced by a multiple of the Killing form and normed so that the grading element is of unit length. Since the lowest form $\rho $ is given by $(n+1)E^*$ , we thus obtain that $I_k(z)$ is an eigenfunction of the Laplace operator on $G/K$ with eigenvalue $k(2(n+1)-k)$ .

The transformation formula for integrals shows that $I_k(z)$ is K-invariant and thus only depends on the norm of z. Define the smooth function $f \colon (0,1) \to \mathbb {R}$ via $f(r(z)) := I_k(z)$ ; then f is coclosed and primitive. Therefore, we can apply Lemma 7, which shows that $I_k$ being an eigenfunction of $\Delta $ is equivalent to $L^*\overline {\partial }\partial I_k(z) = \frac {k(2(n+1)-k)}{2i}I_k(z)$ . Expanding the left hand side using the formulae from Proposition 13 a few lines of computation yields that f has to satisfy the ordinary differential equation

(16) $$ \begin{align} f"(r) + \left(\frac{2n+1}{r} + \frac{2n r}{1-r^2}\right)f'(r) + \frac{k(2(n+1)-k)}{(1-r^2)^2} f(r) = 0. \end{align} $$

After a change of variables to $s(r) = r^2$ this can be transformed into a hypergeometric differential equation in s with parameters $a = b = \frac {k}{2}$ and $c = n+1$ . Applying Proposition 15(i), it follows that the bounded solution of (16) is given by $f(r) = C \cdot {}_2F_1\left (\frac {k}{2}, \frac {k}{2}, n+1;r^2\right )$ with a constant $C \in \mathbb {C}$ . Forming the limit $r \to 0$ yields that $C = \operatorname {\mathrm {vol}}(S^{2n+1})$ and thus the claimed formula. Since $c - a - b = n+1-k> 0$ , the boundedness claim follows from Proposition 15(iii).

Using the previous result, we can prove a boundary value theorem for the operators $\underline {\Phi }_{p,q}$ in the case of a trivial initial value. This will also show that the boundary value of a Poisson transform at a boundary point only depends on the value of the initial data at this point.

Theorem 17. For $0\le p, q \le n$ with $0 \le p+q \le n$ let $\alpha \in \Gamma (\mathcal {H}_{p,q})$ with $\alpha (u) = 0$ for $u \in S^{2n+1}$ . Let $\gamma \colon [0, a) \to \overline {B}$ be a continuous curve with $\gamma (0) = u$ and $\gamma (t) \in B$ for $t> 0$ . Then

(17) $$ \begin{align} \lim_{t \to 0} \underline{\Phi}_{p,q}(\alpha)(\gamma(t)) = 0. \end{align} $$

Proof. Using the identification of $\mathcal {H}_{p,q}$ with $\Lambda ^{p,q}_0H^*$ , we can find a section $\beta $ of the latter bundle so that $\underline {\Phi }_{p,q}(\alpha ) = \Phi _{p,q}(\beta )$ . Moreover, we have seen that the G-equivariant map $\phi _{p,q}$ underlying the Poisson transform $\Phi _{p,q}$ is a conformal map with respect to the inner products on $\Gamma (\Lambda ^k_0 H^*)$ and $\Gamma (\Lambda ^k T_K^*)$ whose conformal factor $c_{p,q} \in \mathbb {R}$ is constant on $G/M$ . Thus, if we denote by $\| \ \|$ the norm on $\Omega ^{p,q}(G/K)$ induced by the Kähler metric, we obtain for all $z \in B$ that

(18) $$ \begin{align} \left\|\underline{\Phi}_{p,q}(\alpha)(z)\right\| \le c_{p,q} \int_{S^{2n+1}} \|\pi_P^*\beta\|_M(z,u) \ \operatorname{\mathrm{vol}}_M(z,u). \end{align} $$

Next, we use Lemma 14 to move the integrand to the fibre over the origin $\mathbf {0} \in B$ , which produces the factor $\mathcal {P}(z,u)^{n+1-\frac {p+q}{2}}$ . Furthermore, along the fibre $\pi _K^{-1}(\mathbf {0})$ the norm of $\pi _P^*\beta $ coincides with the norm of $\alpha $ with respect to the inner product induced by the K-invariant Riemannian metric on $S^{2n+1} \cong K/M$ . Inserting this into (18) yields

(19) $$ \begin{align} \left\|\underline{\Phi}_{p,q}(\alpha)(z)\right\| \le c_{p,q} \int_{S^{2n+1}} \mathcal{P}(z,u)^{n+1-\frac{p+q}{2}} \|\alpha(u)\| \ du. \end{align} $$

Now let $\varepsilon> 0$ be arbitrary. Since $\alpha (u) = 0$ , we can find $\delta> 0$ so that for all $v \in S^{2n+1}$ in the $\delta $ -ball $B_{\delta }(u)$ around u we have $\|\alpha (v)\| < \varepsilon $ . Furthermore, since $n+1 - \frac {p+q}{2}> 0$ , $\mathcal {P}^{n+1-\frac {p+q}{2}}$ tends to $0$ uniformly on every compact subset of $S^{2n+1}$ away from u. Therefore, since the curve $\gamma $ is continuous, we can find $T> 0$ so that for all $t < T$ and all $v \in S^{2n+1}\setminus B_{\delta }(u)$ we have $\mathcal {P}(\gamma (t),v)^{n+1-\frac {p+q}{2}} < \varepsilon $ . Furthermore, we have seen in Proposition 16 that the integral over $\mathcal {P}(z,u)^{n+1-\frac {p+q}{2}}$ is globally bounded by a constant $C \in \mathbb {R}$ . Therefore, by (19) we have for all $t < T$ that

$$ \begin{align*} \|\underline{\Phi}_{p,q}(\alpha)(\gamma(t))\| &\le c_{p,q} \left(\int_{B_{\delta}(u)} + \int_{S^{2n+1}\setminus B_{\delta}(u)}\right) \mathcal{P}(\gamma(t),v)^{n+1-\frac{p+q}{2}} \|\alpha(v)\| \ dv \\ &< \varepsilon c_{p,q}\int_{B_{\delta}(u)} \mathcal{P}(\gamma(t),v)^{n+1-\frac{p+q}{2}} \ dv + \varepsilon \int_{S^{2n+1}} \|\alpha(v)\| \ dv \\ &\le \varepsilon c_{p,q}\left(C + \|\alpha\|_{\infty} \operatorname{\mathrm{vol}}\left(S^{2n+1}\right)\right), \end{align*} $$

where $\| \alpha \|_{\infty }$ denotes the maximum norm of $\alpha $ on the compact manifold $S^{2n+1}$ . Since $\varepsilon $ was arbitrary, the claim follows.

5.1 Frobenius forms and differential operators

For $0 \le p+q\le n$ and $\mathbb {V} \in \hat {K}$ let $\sigma \in \operatorname {\mathrm {Hom}}_K(\mathbb {V}, \Gamma (\mathcal {H}_{p,q}))$ with $\sigma \neq 0$ . Since the partial BGG-operators as well as their adjoints are both K-equivariant differential operators, their compositions with $\sigma $ are again Frobenius forms. Moreover, the kernel of $\mathcal {D}^*$ is closed, so since $\mathbb {V}$ is irreducible, we either have $\sigma (\mathbb {V}) \subseteq \ker (\mathcal {D}^*)$ or $\sigma (\mathbb {V}) \cap \ker (\mathcal {D}^*) = \{0\}$ , and similarly for $\overline {\mathcal {D}}^*$ . This shows that for each irreducible K-representation the subspace $\sigma (\mathbb {V}) = \Gamma (\mathcal {H}_{p,q})_{\mathbb {V}}$ of $\Gamma (\mathcal {H}_{p,q})$ falls into one of four categories, which will influence the properties of its image under $\underline {\Phi }_{p,q}$ .

Note that by adjointness the condition $\sigma (\mathbb {V}) \cap \ker (\mathcal {D}^*) = \{0\}$ implies that $\sigma = \mathcal {D} \hat {\sigma }$ for an element $\hat {\sigma } \in \operatorname {\mathrm {Hom}}_K(\mathbb {V}, \Gamma (\mathcal {H}_{p-1,q}))$ , which has to be a multiple of $\mathcal {D}^*\sigma \neq 0$ due to Proposition 18. If in addition $\sigma (\mathbb {V}) \subseteq \ker (\overline {\mathcal {D}}^*)$ , then $\hat {\sigma }(\mathbb {V}) \subseteq \ker (\mathcal {D}^*) \cap \ker (\overline {\mathcal {D}}^*)$ since the adjoint operators anticommute. On the other hand, if additionally $\sigma (\mathbb {V}) \cap \ker (\overline {\mathcal {D}}^*) = \{0\}$ , then adjointness implies that $\hat {\sigma }$ is a multiple of $\mathcal {D}^*\overline {\mathcal {D}}\overline {\mathcal {D}}^*\sigma \neq 0$ , which in turn is a nontrivial multiple of $\overline {\mathcal {D}} \mathcal {D}^*\overline {\mathcal {D}}^*\sigma $ due to Corollary 10. In particular, we can write $\sigma = \mathcal {D}\overline {\mathcal {D}}\tilde {\sigma }$ with $\tilde {\sigma } \in \operatorname {\mathrm {Hom}}_K(\mathbb {V}, \Gamma (\mathcal {H}_{p-1,q-1}))$ satisfying $\tilde {\sigma }(\mathbb {V}) \subseteq \ker (\mathcal {D}^*) \cap \ker (\overline {\mathcal {D}}^*)$ . By exchanging the roles of $\mathcal {D}^*$ and $\overline {\mathcal {D}}^*$ we obtain analogous results, implying that the case of $\sigma (\mathbb {V})$ being contained in the intersection of the kernels of $\mathcal {D}^*$ and $\overline {\mathcal {D}}^*$ induces all other cases by taking images of $\sigma $ under partial BGG-operators.