Achraf Ouald Chaib | Searcharxiv

arXiv2023

A characterization of the $L^2$-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

We study the $L^2$-boundedness of the Poisson transforms associated to the homogeneous vector bundles $ Sp(n,1)\times_{Sp(n)\times Sp(1)} V_τ$ over the quaternionic hyperbolic spaces $ Sp(n,1)/Sp(n)\times Sp(1)$ associated with irreducible representations $τ$ of $ Sp(n)\times Sp(1)$ which are trivial on $ Sp(n)$. As a consequence, we describe the image of the section space $L^2(Sp(n,1)\times_{Sp(n)\times Sp(1)} V_τ)$ under the generalized spectral projections associated to a family of eigensections of the Casimir operator.

math.RT↗

arXiv2021

The Strichartz conjecture for the Poisson transform on homogeneous line bundles over Noncompact Complex Grassmann manifolds

Let $X=G/K$ be a noncompact complex Grassmann manifold of rank $r$. Let $τ_l$ be a character of $K$, $G\times_P{\C}$ and $G\times_K{\C}$ the homogeneous line bundles associated with the representations $σ_{λ,l}=τ_l\otimes a^{ρ-iλ}\otimes 1$ of $P=MAN$ and $τ_l$ of $K$. We give an image characterization for the Poisson transform $P_{λ,l}$ of \\$L^2$-sections of the unitary principal series representations of $G$ parametrized by $σ_{λ,l}$. More precisely for real and regular parameter $λ$ in $\mathfrak{a}^\ast$ we prove that $P_{λ,l}$ is an isomorphism from $L^2(K\times_M{\C})$ onto the space of joint eigensections $F$ of the algebra of $G$-invariant differential operators on $G\times_K{\C}$ that satisfy the following growth condition \begin{eqnarray*} \sup_{R>1}\frac{1}{R^r}\int_{B(R)}\mid F(g)\mid^2\, {\rm d}g<\infty. \end{eqnarray*} This generalizes a conjecture by Strichartz which corresponds to $τ_l$ trivial.

math.RT↗