arXiv subjects

Tobias Simon

Publications and source records attributed to Tobias Simon.

4 recordsLinked to original sources

arXiv2024

Polynomial growth of holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown

The Kr\"otz-Stanton Extension Theorem states that the orbit map of a K-finite vector in a Hilbert representation of a linear Lie group extends to a holomorphic map to a principal fibre bundle over the complex crown domain associated to the Riemannian symmetric space $G/K$. We extend this theorem to arbitrary connected semisimple Lie groups and prove polynomial growth estimates at the boundary. Using this, we show that the boundary values of these holomorphic extensions exist in the space of distribution vectors.

math.RT↗

arXiv2023

Factorial type I KMS states of Lie groups

Motivated by the study of KMS conditions for C*- or W*-dynamical systems defined by covariant unitary representations of topological groups, we consider Gibbs states of a finite-dimensional Lie group $G$ and prove that these are precisely the factorial type I KMS states. For an element $X\in \textbf{L}(G)$ and an irreducible unitary representation $ρ$ of $G$ satisfying $\text{tr}(e^{i\partialρ(X)})=1$, the corresponding Gibbs state is defined as $φ(g)=\text{tr}(ρ(g)e^{i\partialρ(X)})$. We prove that under the mild assumption that $ρ$ has discrete kernel, the condition $\text{tr}(e^{i\partialρ(X)})<~\infty$ implies that the generator $X$ is an inner point of the set $\text{comp}(\mathfrak{g})$ of elliptic elements in $\mathfrak{g}$. This allows us to obtain a complete characterization of Lie algebras $\mathfrak{g}$, representations $ρ$ with discrete kernel and generators $X$ such that $\text{tr}(e^{i\partialρ(X)})<\infty$.

math.RT↗

arXiv2020

Singular curves and Baker-Akhiezer functions

We present the concept of Baker-Akhiezer functions on singular complex curves. For this purpose, we translate the algebraic presentation of such curves in [Se, Chapter~IV] into the analytic setting. Generalised divisors and their interplay with partial desingularisations are the fundament of the construction of Baker-Akhiezer functions.

math.AG↗

arXiv2020

Burchnall-Chaundy Theory

The Burchnall-Chaundy theory concerns the classification of all pairs of commuting ordinary differential operators. We phrase this theory in the language of spectral data for integrable systems. In particular, we define spectral data for rank 1 commutative algebras $A$ of ordinary differential operators. We solve the inverse problem for such data, i.e. we prove that the algebra $A$ is (essentially) uniquely determined by its spectral data. The isomorphy type of $A$ is uniquely determined by the underlying spectral curve.

math.SP↗