Canonical Bi-Differential Operators

• Canonical bi-differentials are uniquely defined bi-differential operators with strong conformal covariance, serving as intertwining operators between principal series representations.
• Their construction employs Riesz kernels, Knapp–Stein operators, and precise normalization to yield operators of specific differential orders.
• They generalize classical brackets like the Rankin–Cohen brackets, offering deep insights into symmetry breaking, spectral theory, and geometric analysis.

A canonical bi-differential is a uniquely determined bi-differential operator characterized by strong covariance properties under a symmetry group action. The paradigmatic setting is provided by conformally covariant bi-differential operators on differential forms, generalizing classical covariant brackets such as the Rankin–Cohen brackets. Canonical bi-differentials arise as the unique intertwining operators between principal series representations associated to the conformal group, constructed by explicit compositions of Knapp–Stein intertwining operators with specific normalization. Their construction makes essential use of representation theory, harmonic analysis, and the algebraic properties of exterior and interior products on forms (Saïd et al., 2018).

1. Representation-Theoretic Framework

Canonical bi-differentials are formulated within the context of noncompact principal-series representations of the conformal group $G = \mathrm{SO}_0(1,n+1)$ acting on differential forms $\Omega^p(\mathbb{R}^n)$. The group acts by rational conformal maps with conformal factor $\Omega(g,x)$ such that all $|g(x)-g(y)|^2 = \Omega(g,x)\Omega(g,y)|x-y|^2$, and this structure defines, for each $0\leq p\leq n$ and $\lambda\in\mathbb{C}$, a representation

$$\pi^{(p)}_{\lambda}(g)\omega(x) = \Omega(g^{-1},x)^{-\lambda}(g^{-1})^*\omega(g^{-1}(x)).$$

Intertwining properties under the diagonal $G$-action determine when such bi-differentials exist. Operators intertwine precisely when the parameters obey $\lambda_2 = \lambda_1 + \mu_1 + 2m$ and with the appropriate Cartan component in the tensor product of exterior powers $\Lambda^p\mathbb{R}^n\otimes\Lambda^q\mathbb{R}^n$ (Saïd et al., 2018).

2. Operator Construction: Riesz Kernels and Source Operators

Central to the construction is the use of Riesz kernels $$R^p_s \in \mathcal{S}'(\mathbb{R}^n,\mathrm{End}(\Lambda^p))$$ and associated Knapp–Stein integral operators $J^p_\lambda$. The Riesz kernel acts on $p$-forms by integrating $|x|^s$ against a combination of interior and exterior multiplication: $$\langle R^p_s, \omega\rangle = 2^{2-n-s}\Gamma\Big(\frac{s+2p}{2}\Big)\Gamma\Big(\frac{s+n-2p}{2}\Big) \int_{\mathbb{R}^n} |x|^s (\iota_x\varepsilon_x - \varepsilon_x\iota_x)\omega(x)\,dx,$$ with the convolution $J^p_\lambda\omega = R^p_{-2n+2\lambda} * \omega$ realizing the Knapp–Stein operator. These satisfy explicit intertwining relations with the principal series representations (Saïd et al., 2018).

The construction proceeds via composition:

• Two Knapp–Stein operators,
• A multiplication $M$ by $|x-y|^2$ on $\mathbb{R}^n\times\mathbb{R}^n$,
• Followed by further Knapp–Stein operators as needed for higher $m$.

A normalization factor $k_{p,q}(\lambda,\mu)$ ensures that the resulting operator $F^{(p,q)}_{\lambda,\mu}$ is purely differential of order $4$ in $(x,y)$, avoiding residual convolution terms. Iteration yields operators $F^{(p,q;m)}_{\lambda,\mu}$ of order $4m$.

3. Explicit Formulas and Canonical Property

The normal-ordered explicit expression for the source operator $F^{(p,q)}_{\lambda,\mu}$ involves the exterior differential $d$, its adjoint $\delta$, and first-order operators $V_{p,\lambda,j}, V'_{q,\mu,j}$ built from basis contractions and exterior products: $$\begin{aligned} F^{(p,q)}_{\lambda,\mu} &= -|x-y|^2 \left( a_p(\lambda)d_x + b_p(\lambda)\delta_x \right)\otimes \left( a_q(\mu)d_y + b_q(\mu)\delta_y \right)\ & \quad + 2\sum_{j=1}^n (x_j-y_j) \Big[ V_{p,\lambda,j}\otimes \left( a_q(\mu)d_y + b_q(\mu)\delta_y \right) \ & \hspace{70pt} + \left( a_p(\lambda)d_x + b_p(\lambda)\delta_x \right) \otimes V'_{q,\mu,j} \Big] \ & \quad - (\lambda - 2p)(\lambda + n - 2p) \mathrm{Id}\otimes \left( a_q(\mu)d_y + b_q(\mu)\delta_y \right) \ & \quad - (\mu - 2q)(\mu + n - 2q) \left( a_p(\lambda)d_x + b_p(\lambda)\delta_x \right)\otimes \mathrm{Id} \ & \quad - (\lambda - 2p)(\mu - 2q)(\lambda-p)(\mu-q) \mathrm{Id}\otimes \mathrm{Id}, \end{aligned}$$ with $a_p(\lambda) = \lambda + n - 2p - 2$, $b_p(\lambda) = \lambda - 2p$ (Saïd et al., 2018).

These operators are canonical: for generic parameters, the space of $G$-intertwiners

$$\mathrm{Hom}_G\left( \pi^{(p)}_\lambda \widehat\otimes \pi^{(q)}_\mu, \pi^{(r)}_{\lambda+\mu+2m} \right)$$

is at most one-dimensional. Any such operator must factor through the same composition of source operator and diagonal restriction, with normalization fixed by a minimal vanishing property. This determines "canonical" as uniqueness up to explicit normalization (Saïd et al., 2018).

4. Diagonal Restriction and Covariance

Applying the restriction map $\mathrm{res}$ to the diagonal $x=y$ in $$\Omega^p(\mathbb{R}^n)\,\widehat\otimes\,\Omega^q(\mathbb{R}^n)$$ yields the bi-differential operator: $$D^{(p,q;m)}_{\lambda,\mu} = \mathrm{res} \circ F^{(p,q;m)}_{\lambda,\mu}$$ mapping $\Omega^p(\mathbb{R}^n)\times\Omega^q(\mathbb{R}^n)$ to sections of $\Lambda^p \otimes \Lambda^q$. Projection onto an irreducible Cartan component $\Lambda^r$ yields the operator

$$D_{\lambda,\mu}^{(p,q; m)}: \Omega^p(\mathbb{R}^n)\times\Omega^q(\mathbb{R}^n)\rightarrow \Omega^r(\mathbb{R}^n)$$

which is exactly covariant: $$D_{\lambda,\mu}^{(p,q;m)}\circ [\pi^{(p)}_{\lambda}\otimes\pi^{(q)}_{\mu}](g) = \pi^{(r)}_{\lambda+\mu+2m}(g)\circ D_{\lambda,\mu}^{(p,q;m)},\qquad g\in G.$$ This intertwining property identifies $D_{\lambda,\mu}^{(p,q; m)}$ as the canonical bi-differential for the prescribed symmetry and representation data (Saïd et al., 2018).

5. Low-Dimensional Examples and Classical Limits

For $n=1$ and $p=q=0$, the canonical bi-differential recovers the classical Rankin–Cohen brackets: $$D_{\lambda,\mu}^{(0,0;1)}(f,g) = \mu f'(x)g(x) - \lambda f(x)g'(x).$$ For higher $m$, all Rankin-Cohen brackets are obtained, demonstrating the generalization from scalar functions to higher forms and from $\mathrm{SL}(2,\mathbb{R})$ to the conformal group in higher dimensions. Similarly, in $n=2$ with $p=q$ either $0$ or $1$, one obtains higher-dimensional analogs of the classical brackets, with explicit actions on differential forms (Saïd et al., 2018).

6. Significance and Connections

Canonical bi-differentials provide the unique conformally covariant differential operators intertwining prescribed representation-theoretic data. They play a fundamental role in symmetry breaking, spectral theory, and geometric analysis on symmetric spaces. Their explicit construction via Riesz kernels and Knapp–Stein operators ties harmonic analysis on homogeneous spaces with invariant theory and modern approaches to symmetry-breaking differential operators, as in the work of Kobayashi, Speh, and Juhl (Saïd et al., 2018).

The general methodology and algebraic structure of canonical bi-differentials resonate with the broader theory of (strongly) bicovariant differential graded algebras on quantum groups and Hopf algebra structures, though the constructions and normalization conditions in the quantum algebraic setting (as in (Aziz et al., 2019)) require separate machinery and extend beyond the strictly classical or conformal group context.

• S. Ben Saïd, J.-L. Clerc, K. Koufany, "Conformally covariant bi-differential operators for differential forms" (Saïd et al., 2018).
• A. Kobayashi, B. Speh, "Symmetry breaking for rank-one orthogonal groups," Mem. AMS 238 (2015).
• A. Juhl, "Families of conformally covariant differential operators, Q-curvature and holography," Birkhäuser (2009).