The subelliptic heat kernel of the octonionic anti-de Sitter Fibration

In this note, we study the sub-Laplacian of the $15$-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space $\mathbb{O}H^1$. We also obtain two integral representations for the corresponding subelliptic heat kernel.


Introduction and results
In this note we study the sub-Laplacian and the corresponding sub-Riemannian heat kernel of the octonionic anti-de Sitter fibration This study follows the previous works [2,3,8] which respectively concerned: (1) The complex anti-de Sitter fibrations: (2) The quaternionic anti-de Sitter fibrations: F.B is partially funded by the NSF grant DMS-1901315. The 15-dimensional anti-de Sitter fibration is the last model space that remained to be studied of a sub-Riemannian manifold arising from a H-type semi-Riemannian submersion over a rank-one symmetric space, see the Table 3 in [4].
Similarly to the complex and quaternionic case, the sub-Laplacian is defined as the lift on AdS 15 (O) of the Laplace-Beltrami operator of the octonionic hyperbolic space OH 1 . However, in the complex and quaternionic case the Lie group structure of the fiber played an important role that we can not use here, since the fiber S 7 is not a group. Instead, we make use of some algebraic properties of S 7 that were already pointed out and used by the authors in [1] for the study of the octonionic Hopf fibration: Let us briefly describe our main results. Due to the cylindrical symmetries of the fibration, the heat kernel of the sub-Laplacian only depends on two variables: the variable r which is the Riemannian distance on OH 1 (the starting point is specified with inhomogeneous coordinate in section 3) and the variable η which is the Riemannian distance starting at a pole on the fiber S 7 . We prove in Proposition 1 that in these coordinates, the radial part of the sub-LaplacianL writes L = ∂ 2 ∂r 2 + (7 coth r + 7 tanh r) As a consequence of this expression for the sub-Laplacian, we are able to derive two equivalent formulas for the heat kernel. The first formula, see Proposition 2, reads as follows: for r ≥ 0, η ∈ [0, π), t > 0 where s t is the heat kernel of the Jacobi operator with respect to the measure sin 6 η dη, and where q t,15 is the Riemannian heat kernel on the 15-dimensional real hyperbolic space H 15 given in (4.1). The second formula, see Proposition 3, writes as follows: where q t,9 is Riemannian heat kernel on the 9-dimensional hyperbolic space H 9 and G t (r, η, ϕ, u) is given in (4.3).

The octonionic anti-de Sitter fibration
be the division algebra of quaternions, see [1]. The octonionic norm is defined for x ∈ O by The octonionic anti-de Sitter space AdS 15 (O) is the quadric defined as the pseudohyperbolic space by: In real coordinates we have x = 7 j=0 x j e j , y = 7 j=0 y j e j , and the pseudo-norm can be written as Let OH 1 denote the octonionic hyperbolic space. The map π : AdS 15 (O) → OH 1 , given by (x, y) → [x : y] = y −1 x is a pseudo-Riemannian submersion with totally geodesic fibers isometric to the seven dimensional sphere S 7 . Notice that, as a topological manifold, OH 1 can therefore be identified with the unit open ball in O. The pseudo-Riemannian submersion π yields the octonionic anti-de Sitter fibration For further informations on semi-Riemannian submersions over rank-one symmetric spaces, we refer to [6].

Cylindric coordinates and radial part of the sub-Laplacian
The sub-Laplacian L on AdS 15 (O) we are interested in is the horizontal Laplacian of the Riemannian submersion π : AdS 15 (O) → OH 1 , i.e the horizontal lift of the Laplace-Beltrami operator of OH 1 . It can be written as where AdS 15 (O) is the d'Alembertian, i.e., the Laplace-Beltrami operator of the pseudo-Riemannian metric and △ V is the vertical Laplacian. Since the fibers of π are totally geodesic and isometric to S 7 ⊂ AdS 15 (O), we note that AdS 15 (O) and △ V are commuting operators, and we can identify The sub-Laplacian L is associated with a canonical sub-Riemannian structure on AdS 15 (O) which is of H-type, see [4].
To study L, we introduce a set of coordinates that reflect the cylindrical symmetries of the octonionic unit sphere which provides an explicit local trivialization of the octonionic anti-de Sitter fibration. Consider the coordinates w ∈ OH 1 , where w is the inhomogeneous coordinate on OH 1 given by w = y −1 x, with x, y ∈ AdS 15 (O). Consider the pole p = (1, 0, · · · , 0) ∈ S 7 and take Y 1 , ..., Y 7 to be an orthonormal frame of T p S 7 . Let us denote exp p the Riemannian exponential map at p on S 7 . Then the cylindrical coordinates we work with are given by i . More precisely f is radial cylindric if there exists a function g so that We denote by D the space and smooth and compactly supported functions on [0, 1) × [0, π). Then the radial part of L is defined as the operator L such that for any f ∈ D, we have We now compute L in cylindric coordinates.
where r = tanh −1 ρ is the Riemannian distance on OH 1 from the origin.
Proof. Note that the radial part of the Laplace-Beltrami operator on the octonionic hyperbolic space OH 1 is △ OH 1 = ∂ 2 ∂r 2 + (7 coth r + 7 tanh r) ∂ ∂r , and the radial part of the Laplace-Beltrami operator on S 7 is Since the octonionic anti-de Sitter fibration defines a totally geodesic submersions with base space OH 1 and fiber S 7 , the semi-Riemannian metric on AdS 15 (O) is locally given by a warped product between the Riemannian metric of OH 1 and the Riemannian metric on S 7 . Hence the radial part of the d'Alembertian becomes  For the point with coordinates one has We therefore deduce that AdS 15 (O) (cosh r cos η) = 15 cosh r cos η.
Using the formula (3.4), after a straightforward computation, this yields g(r) = − 1 cosh 2 r and therefore AdS 15 (O) = ∂ 2 ∂r 2 + (7 coth r + 7 tanh r) Finally, to conclude, one notes that the sub-Laplacian L is given by the difference between the Laplace-Beltrami operator of AdS 15 (O) and the vertical Laplacian.

Integral representations of the subelliptic heat kernel
In this section, we give two integral representations of the subelliptic heat kernel associated withL. We denote by p t (r, η) the heat kernel ofL issued from the point r = η = 0 with respect to the measure (3.5).

First integral representation.
We denote by s t the heat kernel of the operator with respect to the reference measure sin 6 η dη. The operator△ S 7 belongs to the family of Jacobi operators which have been extensively studied in the literature, see for instance the Appendix in [5] and references therein. In particular, the spectrum of△ S 7 is given by and the eigenfunction corresponding to the eigenvalue m(m + 6) is P 5/2,5/2 m (cos η) where P 5/2,5/2 m is the Jacobi polynomial As a consequence, one has the following spectral decomposition for the heat kernel: Proposition 2. For r ≥ 0, η ∈ [0, π), and t > 0 we have: is the Riemannian heat kernel on the 15-dimensional real hyperbolic space H 15 .
Proof. Since π : AdS 15 (O) → OH 1 is a (semi-Riemannian) totally geodesic submersion, the operators AdS 15 (O) and △ S 7 commute. Thus We deduce that the heat kernel ofL can be written as where s t is the heat kernel of (3.3) with respect to the measure sin 6 ηdη, η ∈ [0, π), In order to write (4.2) more precisely, let us consider the analytic change of variables τ : (r, η) → (r, iη) that will be applied on functions of the type f (r, η) = h(r)e −iλη , with h smooth and compactly supported on [0, ∞) and λ > 0. Then as we saw in the proof of Proposition 1 one can see that Then, one deduces Therefore, coming back to (4.2), one infers that using the analytic extension of s t one must havê where q t,15 is the Riemannian heat kernel on the real hyperbolic space H 15 given in (4.1).

Second integral representation.
Proposition 3. For r ≥ 0, η ∈ [0, π), and t > 0 we have: where q t,9 is the 9-dimensional Riemannian heat kernel on the hyperbolic space: Proof. The strategy of the following method appeals to some results proved in [7]. Firstly, we decompose the subelliptic heat kernel in the basis which are the normalized eigenfunctions in (4.4) of△ S 7 = ∂ 2 ∂η 2 + 6 cot η ∂ ∂η which is associated to the eigenvalue −m(m + 6). Accordingly, where for each m, h m is given by and f m (t, .) solves the following heat equation: We consider then the operator L m := ∂ 2 ∂r 2 + (7 coth r + 7 tanh r) ∂ ∂r + m(m + 6) cosh 2 r + 7 2 which was studied in [7]; see page 229 with From Theorem 2 in [7], we deduce that the solution to the wave Cauchy problem associated with the subelliptic Laplacian is given f ∈ C ∞ 0 (OH 1 ) by d(cosh s)K m (s, 0, y)q t,9 (cosh s).