Fields of Toeplitz algebras for the principal symbol of regular 2-step nilpotent groups
Clément Cren
TL;DR
This work develops a unified C*-algebraic framework to recover the order-0 principal-symbol algebra in the filtered calculus for regular 2-step nilpotent groups, by representing it as a crossed product with a field of Toeplitz algebras. It builds Toeplitz bundles over the osculating group data, leverages Kirillov theory and Weyl quantization to identify the symbol algebra Σ(G) with 𝒯0 and relates C*(T_HM) to C*_0(T_HM) via Morita-type equivalences, thereby connecting hypoelliptic analysis to Toeplitz operator theory. The theory specializes cleanly to H-type groups, where symplectic bundles trivialize and the symbol calculus becomes a locally trivial tensor product with Clifford-module structure, yielding explicit isomorphisms and exact sequences. Collectively, the results offer a coherent, geometrically grounded method to study principal symbols and filtered calculus on polycontact and H-type manifolds, with concrete descriptions of symbol algebras, representations, and Morita equivalences.
Abstract
We show that the C*-algebra of a regular 2-step nilpotent lie group can be recovered using continuous fields of Toeplitz algebras and a crossed product. We generalize this result to polycontact manifolds in the sense of van Erp which are endowed with fields of such groups. We also investigate those manifolds with a more rigid structure, namely those modeled on H-type groups. In all those cases, there is a certain pseudodifferential calculus named filtered calculus, we show that the algebra of principal symbols can also be recovered from the field of Toeplitz algebras.