Convergent star products on cotangent bundles of Lie groups

Abstract

For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on $$T^*G$$ T ∗ G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on $$T^*G$$ T ∗ G , for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $$\hbar $$ ħ . This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on $${\mathfrak {g}}^*$$ g ∗ . The passage to the Weyl-ordered star product, i.e. the Gutt star product on $$T^*G$$ T ∗ G , is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $$\hbar $$ ħ .

Spectral theory of the invariant Laplacian on the disk and the sphere – a complex analysis approach

“…By and large, we follow Rudin's [37] treatment of invariant subspaces of eigenfunctions of the hyperbolic Laplacian, 10 M. Heins, A. Moucha, and O. Roth but again completely working in the holomorphic setting; we briefly comment on the similarities and differences between our and Rudin's approach in Remark 8.5. We close the paper with Section 9, which connects the Poisson-Fourier modes to the invariant differential operators of Peschl-Minda type studied in [19].…”

“…We note in passing that the subdomains Ω + and Ω − arise naturally in the study of the Fréchet space structure of H(Ω) (see [18]) and also for studying invariant differential operators of Peschl-Minda type acting on H(Ω) (see [19]). The following result shows that they are also useful for describing the Möbius invariant subspaces of the eigenspaces of the invariant Laplacian Δ zw .…”

“…In the next section, we give an account of the main results of this work and their ramifications for the spectral theory of the hyperbolic and spherical Laplacian as well as an outline of the structure of the remaining sections. The accompanying papers [18,19,27,32] are related to other aspects of the function theory of the set Ω, the complement of the complexified unit circle, and its applications. Our interest in this set and its inhabitants, the holomorphic functions on Ω, first arose in connection with previous work [7,10,12,28,41,43] on canonical Wick-type star products in strict deformation quantization of the unit disk and the Riemann sphere, and from our desire to understand the somehow mysterious role played by Ω and, in particular, by its function-theoretic properties in this regard.…”

“…Our interest in this set and its inhabitants, the holomorphic functions on Ω, first arose in connection with previous work [7,10,12,28,41,43] on canonical Wick-type star products in strict deformation quantization of the unit disk and the Riemann sphere, and from our desire to understand the somehow mysterious role played by Ω and, in particular, by its function-theoretic properties in this regard. A partial explanation was given in [19], where it was indicated that invariant differential operators of Peschl-Minda type, on the one hand, effectively facilitate and unify the study of the star products on the disk and the sphere, and on the other hand, are perhaps best understood as operators acting on the spaces of holomorphic functions on Ω and its three distinguished subdomains. We started wondering whether and how the most basic differential operator acting on Ω, the invariant Laplacian Δ zw , and its spectral theory possibly fit into this emerging picture.…”

A Formal Equivalence of Deformation Quantization and Geometric Quantization (of Higher Groupoids) and Non-Perturbative Sigma Models

“…Many authors (aside from the ones mentioned throughout this paper) have worked on related topics, eg. convergent products, see [1], [9], [20], [29] [39], [43]. Also, [40] contains a list of open problems 4 regarding exactly the topics in this paper.…”