The Borel-Weil theorem for complex projective space

(with Michael Eastwood)

Abstract

Our aim, in this article, is to give a complete classification of the irreducible finite-dimensional holomorphic representations of the complex general linear group. We shall construct them as spaces of holomorphic sections of certain vector bundles over complex projective space. This construction is essentially due to Borel and Weil (with exposition due to Serre).

Our account is complete in the sense that we shall assume that the reader is familiar with basic analysis and, especially, the theory of functions of several complex variables (for example, as in Gunning or Wells). However, we shall assume the reader knows practically nothing of representation theory. This combination is apparently not so uncommon. (For example, one of us (ME) fell into this category for many years.)

The Borel-Weil Theorem is much more general than the version we shall explain in this article. Moreover, the literature already contains many fine proofs (see, for example, Knapp). Everything in this article is well-known. However, we hope that, by sticking to the most familiar of groups (the general linear group) and the most familiar of spaces (complex projective space), we can avoid some of the hurdles which present themselves in a more general treatment.

The existence of such hurdles is already lamented in Knapp's book where, also, the strategy is to explain via examples. A similar strategy is found in Fulton and Harris' book. We highly recommend these sources for further reading in representation theory.

The text is punctuated with exercises. We hope that they will be found helpful but certainly they are not essential to the exposition and can be omitted on first reading. For this reason, they are enclosed in boxes. In an ideal world, paper would be engineered with these boxes replaced by icons. Pressing on an icon would reveal the exercises. Some of the exercises use well-known notation or terminology which is not explained in the text. Towards the end of the article we discuss some variations, in particular Bott's cohomological variation and how to adapt the construction to other groups. These final sections have no details and are meant only as an introduction to further reading.

One of us (ME) remembers his very first undergraduate tutorial at Hertford College back in 1970. ``Our tutor was Brian Steer and my fellow tutee was Paul Woodruff. Brian told us about groups acting sets. By our second tutorial, two weeks later, we were expected to have understood Sylow's Theorems from this point of view. Of course, knowing no better, we thought this was a perfectly reasonable request and so we just did it. Given this sort of start, we went on to learn an enormous amount from Brian during the next three years!'' The Borel-Weil Theorem concerns one of the finest examples of groups acting on sets. We dedicate this article to Brian Steer on the occasion of his sixtieth birthday.

We thank Michael Murray for several useful conversations.

Appears in Invitations to Geometry and Topology, Oxford Graduate Texts in Mathematics (2002).

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