Representations of orthogonal groups over the field of two elements

Question

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.

Answer 1

The question is brief, but there are a great many unknown features of the modular representation theory of these families of groups (for the prime $p=2$). The approach via algebraic groups pioneered by Steinberg is conceptually attractive but far from able to deal effectively with small primes at this point. There might also be approaches via finite group theory and combinatorics. It's hard to predict what will ultimately give the most explicit results.

Since the chosen prime 2 divides the order of each finite group involved, complete reducibility of representations breaks down badly. So there is an open-ended problem of determining the indecomposable representations which are not irreducible; in your case there are infinitely many of these up to isomorphism. Thus far the best systematic results have been obtained just for irreducible representations (= simple modules for the group algebra). So I'll make a few comments about these.

1) As Nick Gill indicates, there is some supporting literature. Note that the useful 1967-68 lectures on Chevalley groups given at Yale by the late Robert Steinberg are still downloadable from his webpage at UCLA. Also, the reference of mine which is most useful is neither of those listed; instead try my survey with many references in the more recent LMS Lecture Note Series No. 326 (2006) Modular Representatins of Finite Groups of Lie Type. For example, 5.3-5.4 deal in your case with the abstract group isomorphism between symplectic groups $Sp_{2n}$ and odd spin groups (the 2-fold covers of the groups $SO_{2n+1}$) In characteristic 2 there are further identifications of related finite groups.

2) One of Steinberg's main contributions was his proof that irreducible (rational) representations of the ambient algebraic group restrict naturally to irreducible representations of the finite subgroups of Lie type. Along with this is his tensor product theorem, which reduces the study of the latter representations in your case to those whose "highest weight" in Lie theory has $n$ coordinates equal to 0 or 1.

3) Unfortunately, most of the successful further theory involving algebraic groups (especially building on Lusztig's ideas) only applies so far when the given prime is "sufficiently large". In particular, for $p=2$ there is little guidance from the general theory. Even so, there is some hope that affine Weyl groups of Langlands dual type (here relative to root lattices multiplied by suitable powers of 2) will turn out to predict the key data required for the modular representations of these finite groups.

4) Keep in mind that, from the Lie theory viewpoint, the groups you are interested in are of two distinct types: $B_n$ (which in characteristic 2 have group isomorphisms with type $C_n$ finite groups) and $D_n$. Moreover, the Lie theoretic approach tends to start with simply connected groups (here Spin groups) and then yield results in a fairly mechanical way for the related groups you consider. How to express the results is not immediately clear, since for the algebraic group a "formal character" involves just a knowledge of weight multiplicities, while for the finite groups one has to re-interpret the data as a Brauer character.

5) The modular version of the Atlas of Finite Groups has a lot of detail about relatively small groups of all types, assembled by explicit computations.

Answer 2

You can find a nice, concise introduction in $\S5.4$ of Kleidman and Liebeck's The subgroup structure of the finite classical groups. They focus on the modular theory over $\overline{\mathbb{F}_2}$, but also include a section on how to "descend" to a smaller field (see pp.'s 192, 193.)

For further information, you may wish to consult

• Carter, Finite groups of Lie type: conjugacy classes and complex characters;
• Humphreys, Linear algebraic groups;
• Humphreys, Ordinary and modular representations of Chevalley groups;
• Steinberg, Lectures on Chevalley groups;
• Steinberg, Endomorphisms of linear algebraic groups.

Many of these texts are very general, so you will need to dig around to find the relevant bits. I have e-copies of some of the texts above including, in particular, Kleidman and Liebeck. Email me if you want a copy.