We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.
Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods
Institute of Mathematical Sciences, NYU Shanghai（纽约大学上海校区数学科学研究所）；Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité（巴黎左岸儒勒·庞加莱数学研究所，巴黎西岱大学）；Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay（凡尔赛数学实验室，凡尔赛大学，法国国家科学研究中心，巴黎萨克雷大学）；School of Engineering, Mathematics and Physics, University of East Anglia（东英吉利大学工程、数学与物理学院）；Department of Mathematics, Faculty of Science, Kyoto University（京都大学理学部数学科）
In this paper we study left and right n- regular functions that originally were introduced in [FL4]. When n=1, these functions are the usual quaternionic left and right regular functions . We show that n- regular functions satisfy most of the properties of the usual regular functions , including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy-Fueter type reproducing formulas. Arguably, these Cauchy-Fueter type reproducing formulas for n- regular functions are quaternionic analogues of Cauchy's integral formula for the n-th order pole e…
Locally analytic vectors in the completed cohomology of quaternionic Shimura curves
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
