Sub-Laplacians of holomorphic \(L^p\)-type on exponential Lie groups (Q2568772)

It is known that every left-invariant Laplacian or sub-Laplacian \(L\) on a connected Lie group of polynomial growth admits a differentiable \(L^p\)-functional calculus for \(1\leq p <\infty\), i.e., there is an integer \(k\) such that \(C_0^k(\mathbf{R})\subset \mathcal{M}_p(L)\), where \(\mathcal{M}_p(L)\) stands for the space of all \(L^p\)-multipliers for \(L\). On a somewhat opposite direction, Laplace-Beltrami operators \(T\) on Riemannian symmetric spaces of non-compact type are operators of the so-called holomorphic \(L^p\)-type for every \(p\in [1,\infty)\), \(p \neq 2\), they have a non-isolated point \(\lambda_0\) in their \(L^2\)-spectrum, such that every \(m\in \mathcal{M}_p(L)\), \(p\neq 2\), that vanishes at infinity admits a holomorphic extension to a fixed neighbourhood \(W\) of \(\lambda_0\) in \(\mathbf{C}\). The present paper examines the situation for sub-Laplacians of Lie groups (with exponential volume growth). It should be remarked that despite the hint suggested by Laplace-Beltrami operators, some exponential Lie groups of exponential volume growth do admit differentiable \(L^p\)-functional calculi, see for instance \textit{M. Cowling} et al. [Studia Math. 111, 103--121 (1994; Zbl 0820.43001)] or \textit{W. Hebisch} [Colloq. Math. 73, 155--164 (1997; Zbl 0874.22005)] and other references in the paper. The paper contains no full proofs and concentrates on motivating and sketching the main ideas of the main theorem of \textit{W. Hebisch} and the author [J. Lond. Math. Soc. 72, 364--390 (2005)] that relates the existence of sub-Laplacians of holomorphic \(L^p\)-type on a Lie group \(G\) with representation-theoretic properties of \(G\), namely with the existence of coadjoint orbits that satisfy a technical condition, named \`\` Boidol condition\'\', that is equivalent to the non-symmetry of \(L^1(G)\).