This volume grew out of the conference in honour of Hermann Weyl that took place in Bielefeld in September 2006.

Weyl was born in 1885 in Elmshorn, a small town near Hamburg. He studied mathematics in Göttingen and Munich, and obtained his doctorate in Göttingen under the supervision of Hilbert. After taking a teaching post for a few years, he left Göttingen for Zürich to accept a Chair of Mathematics at the ETH Zürich, where he was a colleague of Einstein just at the time when Einstein was working out the details of the theory of general relativity. Weyl left Zürich in 1930 to become Hilbert's successor at Göttingen, moving to the new Institute for Advanced Study in Princeton, New Jersey after the Nazis took power in 1933. He remained there until his retirement in 1951. Together with his wife, he spent the rest of his life in Princeton and Zürich, where he died in 1955.

The Collaborative Resarch Centre (SFB 701) Spectral Structures and Topological Methods in Mathematics has manifold connections with the areas of mathematics that were founded or influenced by Weyl's work. These areas include geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics.

In 1913, Weyl published Die Idee der Riemannschen Fläche (‘The Concept of a Riemann Surface’), giving a unified treatment of Riemann surfaces.

Introduction

In 1926 Hermann Weyl published a paper that contains his character formula for irreducible finite dimensional complex representations of complex and real semi-simple Lie groups and their Lie algebras. It can also be interpreted as a character formula for connected compact groups and for semi-simple algebraic groups in characteristic 0. (Here I am using modern terminology; when Weyl wrote his paper, terms like “Lie groups” were not yet in use.)

When we look at Weyl's character formula as a statement for Lie algebras, then it is a theorem on purely algebraic objects. However, Weyl used analytic methods to prove it. Not surprisingly, people looked for algebraic proofs. These attempts were finally successful and led also to useful reformulations of Weyl's formula. This development will be described in the first section of this survey.

The other topic to be discussed will be the search for analogues to Weyl's formula in more general cases. To start with, a finite dimensional complex semi-simple Lie algebra has an abundance of irreducible representations that are infinite dimensional. It was natural to look for character formulae for at least some families of representations sharing features of the finite dimensional ones — for example those generated by a highest weight vector.

Furthermore, it was also natural to go beyond finite dimensional complex semi-simple Lie algebras. There are several algebraic objects that share many structural features with these Lie algebras and that have similar representation theories.

Preface

We are here for a conference in honor of Hermann Weyl and so I may be allowed, before touching the main topic of my talk, to speak about my personal reminiscences of him.

It was in the year 1952. I was 24 and had my first academic jobat Müchen when I received an invitation from van der Waerden to give a colloquium talk at Zürich University. In the audience of my talk I noted an elder gentleman, apparently quite interested in the topic. Afterwards – it turned out to be Hermann Weyl – he approached me and proposed to meet him next day at a specific point in town. There he told me that he wished to know more about my doctoral thesis, which I had completed two years ago already but which had not yet appeared in print. Weyl invited me to join him on a tour on the hills around Zürich. On this tour, which turned out to last for several hours, I had to explain to him the content of my thesis which contained a proof of the Riemann hypothesis for function fields over finite base fields. He was never satisfied with sketchy explanations, his questions were always to the point and he demanded every detail. He seemed to be well informed about recent developments.

This task was not easy for me, without paper and pencil, nor blackboard and chalk. So I had a hard time. Moreover the pace set by Weyl was not slow and it was not quite easy to keep up with him, in walking as well as in talking.

Introduction

The theory of affine buildings reveals fascinating links between group theory, Euclidean geometry and number theory. In particular, reflections, the Weyl chambers of root systems and valuations of fields all play a central role in their classification.

The study of affine buildings was begun by Bruhat and Tits in [4] and the classification of affine buildings of rank at least four was completed by Tits in [11]. When combined with the classification of Moufang polygons carried out in [12], the Bruhat-Tits classification covers also affine buildings of rank three under the assumption in this case that the building at infinity is Moufang.

Our goal here is to give a very brief overview of this work. All the details can be found in the forthcoming book [14] (as well, of course, as in [4] and [11]).

In this article we regard buildings exclusively as certain edge-colored graphs. For different points of view, see [1]. Other excellent sources of results about affine buildings are [3], [5] and [8].

Buildings

Let Σ be an edge-colored graph and let I denote the set of colors appearing on the edges of Σ. We call |I| the rank of Σ. For each subset J of I let ΣJ be the graph obtained from Σ by deleting all the edges whose color is not in J (but without deleting any vertices). A J-residue of Σ for some subset J of I is a connected component of the graph ΣJ. Thus two distinct J-residues (for a fixed subset J of I) are always disjoint.

Introduction

One of the most influencial contributions of Hermann Weyl to mathematical physics has been his paper Gruppentheorie und Quantenmechanik [We27] from 1927 and its extended version, the book [We28] which was published a year later and carries the same title. The main topic of this part of Hermann Weyl's work is the mathematics of quantum mechanics. After the fundamental papers by Heisenberg and Schrödinger on the foundations of quantum mechanics had appeared in the twenties of the last century this was the central question studied in mathematical physics at that time and which to a certain degree still is present in all attempts to construct mathematically rigorous theories unifying quantum mechanics and general realtivity.

In his article Gruppentheorie und Quantenmechanik, Hermann Weyl essentially introduced two novel aspects to the mathematics of quantum mechanics, namely the following:

1. (i) The representation theory of (compact) Lie groups on Hilbert spaces was applied to mathematically determine atomic spectra.

2. (ii) A conceptually clear quantization method was proposed which associates quantum mechanical operators to classical observables which mathematically are represented by appropriate functions of the space and momentum variables. Nowadays, this quantization scheme is named after his inventor Weyl quantization.

In this paper I will elaborate only on the second aspect, since the representation theory of compact Lie groups has already been covered in detail in other contributions to these proceedings.

Abstract

Up to the year 1910 there had been many significant mathematical contributions to the theory of linear ordinary differential, and of linear integral equations. Many of these advances were based on the original studies initiated by Sturm and Liouville commencing in 1829. In the closing years of the 19th century the work lead by the Göttingen school of mathematics gave a much needed overview of these significant and varied contributions to mathematical analysis.

The contributions of Hermann Weyl, in and around the year 1910, to the theory of Sturm-Liouville theory heralded the modern analytical and spectral study of boundary value problems. In particular the paper written for Mathematischen Annalen in 1910 stands today as a landmark not only in Sturm-Liouville theory, but in the development of mathematical analysis in the 20th century.

This paper discusses the work of Weyl, and indeed of the Göttingen school of mathematics, in introducing the now familiar terms of Sturm-Liouville theory; limit-point and limit-circle endpoint classifications; the point, continuous and essential spectra; singular eigenfunction expansions; and the interplay of these results with the development of quantum theory in physics.

Introduction

The investigation of second-order linear ordinary differential equations has a long and fascinating history, extending back to the middle of the 18th century. It shaped the concept of a function, led to Cantor's set theory and influenced the theories of measure and integration. It was essential to solving the initial-boundary-value problems for partial differential equations, for example the heat and wave equations, by separation of the variables.

Introduction

This text grew out of an attempt to understand a remark by Harish-Chandra in the introduction of [12]. In that paper and its sequel he determined the Plancherel decomposition for Riemannian symmetric spaces of the non-compact type. The associated Plancherel measure turned out to be related to the asymptotic behavior of the so-called zonal spherical functions, which are solutions to a system of invariant differential eigenequations. Harish-Chandra observed: ‘this is reminiscent of a result of Weyl on ordinary differential equations’, with reference to Hermann Weyl's 1910 paper, [29], on singular Sturm–Liouville operators and the associated expansions in eigenfunctions.

For Riemannian symmetric spaces of rank one the mentioned system of equations reduces to a single equation of the singular Sturm–Liouville type. Weyl's result indeed relates asymptotic behavior of eigenfunctions to the continuous spectral measure but his result is formulated in a setting that does not directly apply.

In [23], Kodaira combined Weyl's theory with the abstract Hilbert space theory that had been developed in the 1930's. This resulted in an efficient derivation of a formula for the spectral measure, previously obtained by Titchmarsh. In the same paper Kodaira discussed a class of examples that turns out to be general enough to cover all Riemannian symmetric spaces of rank 1.

It is the purpose of this text to explain the above, and to describe later developments in harmonic analysis on groups and symmetric spaces where Weyl's principle has played an important role.