Introduction

For the arithmetic study of varieties over finite fields powerful cohomological methods are available which in particular shed much light on the nature of the corresponding zeta functions. For algebraic schemes over spec ℤ and in particular for the Riemann zeta function no cohomology theory has yet been developed that could serve similar purposes. For a long time it had even been a mystery how the formalism of such a theory could look like. This was clarified in [D1]. However until now the conjectured cohomology has not been constructed.

There is a simple class of dynamical systems on foliated manifolds whose reduced leafwise cohomology has several of the expected structural properties of the desired cohomology for algebraic schemes. In this analogy, the case where the foliation has a dense leaf corresponds to the case where the algebraic scheme is flat over spec ℤ e.g. to spec ℤ itself. In this situation the foliation cohomology which in general is infinite dimensional is not of a topological but instead of a very analytic nature. This can also be seen from its description in terms of global differential forms which are harmonic along the leaves. An optimistic guess would be that for arithmetic schemes χ there exist foliated dynamical systems X whose reduced leafwise cohomology gives the desired cohomology of χ. If χ is an elliptic curve over a finite field this is indeed the case with X a generalized solenoid, not a manifold, [D3].

We illustrate this philosophy by comparing the “explicit formulas” in analytic number theory to a transversal index theorem.

ABSTRACT We study branching laws for a classical group G and a symmetric subgroup H. Our approach is by introducing the branching algebra, the algebra of covariants for H in the regular functions on the natural torus bundle over the flag manifold for G. We give concrete descriptions of certain subalgebras of the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs (G, H) are associated in pairs, (G, H) and (H′, G′).

Our results may be regarded as a further development of classical invariant theory as described by Weyl [64], and extended previously in [14]. They show that the framework of classical invariant theory is flexible enough to encompass a wide variety of calculations that have been carried out by other methods over a period of several decades. This framework is capable of further development, and in some ways can provide a more precise picture than has been developed in previous work.

Introduction

The Classical Groups.

Hermann Weyl's book, The Classical Groups [64], has influenced many researchers in invariant theory and related fields in the decades since it was written. Written as an updating of “classical” invariant theory, it has itself acquired the patina of a classic. The books [11] and [55] and the references in them give an idea of the extent of the influence. The current authors freely confess to being among those on whom Weyl has had major impact.

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.