In this post I want to describe a remarkable representation associated to finite groups of Lie type. For this, let be a connected reductive group over a finite field with elements, and let be the unipotent radical of some Borel -subgroup of . Steinberg constructed an irreducible representation of of dimension . For convenience, we will assume that is split, although the experts can rest assured that everything goes through if one replaces absolute root systems by relative root systems. If is a (split) maximal -torus of and is the system of simple roots corresponding to the pair , then we define
where is the parabolic subgroup of containing corresponding to . From this definition, it is unclear that this is a character of (rather than a generalized character), let alone irreducible, and it is not clear what its dimension should be.
Example. Let , so that consists of a single element and . By definition, is the space of -valued functions on , while may be understood as the space of constant functions on the same space. Thus one can regard as the (-dimensional) space of complex-valued functions on of average value . The fact that is irreducible is still not obvious, but it follows in this case from arguments with Frobenius reciprocity and Mackey theory. (If anyone can think of a simple argument, I would be happy to hear it!)
Example. The Steinberg representation is not intrinsic to the abstract group , as it can happen that a finite group can be given the structure of a group of Lie type in more than one way. For example, , and this is the unique simple group of order 168. (Here denotes the quotient of by its center; it does not denote , which has order and is not simple.) Thus there can be more than one Steinberg representation associated to a given group. In this case, there is one Steinberg representation of dimension 7 and another of dimension 8. (The fact that the Steinberg representation of descends to can be deduced from the definition using the Bruhat decomposition and an inclusion-exclusion argument.)
The proof that has properties as described works essentially as follows:
- Show that if is the alternating character of the Weyl group of , and if is the subgroup of generated by the simple reflections along roots in , then . This is proved by introducing a simplicial complex attached canonically to , showing that (the geometric realization of) is a sphere, recognizing as the representation of on the top homology of , and then using the Hopf trace formula.
- Use point 1 and the Bruhat decomposition (+ adjacent theory) to deduce that or its negative is irreducible.
- Compute and conclude in particular from point 2 that is irreducible. This is proved by introducing a simplicial complex attached canonically to , showing that (the geometric realization of) is homotopic to the wedge of -many spheres and the using the Hopf trace formula to recognize as the representation of on the top homology of .
In this post I will only prove points 1 and 2, leaving point 3 for a later post in which I will discuss buildings more fully. (One can show point 3 using elementary methods avoiding buildings, but the combinatorics are somewhat intricate and it seems to me that there is more intuition to be gained from the topological argument. For details, see the proof of Theorem 2(b) in Curtis, The Steinberg Character of a Finite Group with a -pair.) A motivating point to keep in mind is that the Weyl group is often thought of as an analogue of where is replaced by “the field with one element”. For us this only has the heuristic value that many true statements about admit analogous statements for , and the latter are often viewed as “degenerate” forms of the former. For example, there are universal polynomials in which compute the order of , and specializing leads to formulas for the order of . See the introduction of this ArXiV document, for example. In the situation above, we view the Coxeter complex of the Weyl group as being a degenerate form of the spherical building of and the alternating character of as being a degenerate form of the Steinberg character of . As we will see in a later post, the spherical building of is built up from subcomplexes called apartments, each of which is canonically isomorphic to the Coxeter complex of . (Note also that has dimension , and specializing gives the dimension of the alternating character .) In this way, the basic facts about the Steinberg representation are deduced from the degenerate and nondegenerate situations.
Before beginning to prove points 1 and 2 from above, I want to remark that the Steinberg representation shows up in many unexpected places in algebraic geometry and the theory of algebraic groups. I intend to write a post explaining this at a later date, but for the moment let it suffice to say that it can be shown that the reduction of modulo comes from a representation of the algebraic group , and it holds a very special place among these representations. In particular, the fact that it can be defined concretely as a representation of a finite group means that its dimension can be determined, while in general the dimensions of the simple representations of are very mysterious. As an example application, the Steinberg representation is the crucial ingredient in the proof that quotients of normal affine varieties by reductive groups (exist as schemes and) remain affine in characteristic ; the same kinds of arguments (plus a lot of technique in reductive group schemes over general bases) lead to a very interesting characterization of reductivity over general bases, see Alper, Adequate moduli spaces and geometrically reductive group schemes, Cor. 9.7.7. The Steinberg representation is also an ingredient in the proof of Kempf’s vanishing theorem, a statement about the vanishing of the higher cohomology of certain line bundles on flag varieties.
As mentioned above, we will need to introduce the Coxeter complex of a Coxeter system , a certain simplicial complex attached to . Before doing so, let us recall several facts about Coxeter systems. There will be very few proofs; we refer (vaguely) to [B, Chap. V, Sec. 1] for all results below. First, recall that by definition, is a Coxeter system when is a group, is a subset of , and has a presentation of the form
where , , and whenever . Note that distinct elements and commute if and only if . We will deal only with the case that is finite, though there are important cases (appearing in Bruhat-Tits theory, for example) in which infinite are important. Coxeter groups are to be thought of as groups generated by reflections, and indeed as long as is finite there is a canonical faithful (finite-dimensional) real representation of with the properties that is a discrete subgroup of and each element of is a reflection on . (Using this, it is easy to see that is finite if and only if this representation can be equipped with an inner product so that is a subgroup of the associated orthogonal group.)
Example. Consider the dihedral group for . Clearly the pair is a Coxeter system, and it can be realized concretely as the subgroup of generated by the orthogonal reflections along the -axis and along the line radians counterclockwise from the -axis. If then there is also a dihedral group , but it is slightly more complicated to describe the representation for . (In this case and do not act by orthogonal reflections.)
Let be the Weyl group of , i.e., the quotient , where is the normalizer of in . This is a constant -group, which we will often identify with its underlying group of -points. A fundamental fact (and the reason this section is here) is that if is the Weyl group of the reductive group and is the set of orthogonal reflections (with respect to a -invariant inner product on ) along roots in , then the pair is a Coxeter system. This follows from the general formalism of Tits systems, see [B, Chap. V, Sec. 2].
Example. Taking (or ), we can deduce that is a Coxeter system, where is the symmetric group on letters and is the set of adjacent transpositions for . In fact, is generated by the with the following relations:
- for all ,
- for all , and
- whenever .
If then we may write for some . If is minimal among all such decompositions, then we will call a reduced decomposition of and we will define the length of to be equal to . For any decomposition of , there is some subsequence such that is a reduced decomposition of . Thus in particular every decomposition for which there exists no proper such subsequence is of length . While there may be several different minimal decompositions of a given element, the set does not depend on this choice.
Example. In we have , and these are both reduced decompositions. (This generalizes entirely to when .)
For every subset , there is a subgroup of generated by all of the elements of . This is itself a Coxeter group, and we have , i.e., any element of which can be written as a product of elements of is itself an element of . In particular, the subgroup determines the subset . The discussion above on minimal decompositions shows also that .
Example. For any Coxeter system and any we have . If are distinct then is a dihedral group (of finite or infinite order according to whether or not).
Example. If as in a previous example and for some , then .
We are now ready to define the Coxeter complex . It is defined as follows: the vertices of are precisely the cosets , where as above is the subgroup of generated by . The facets of are then defined to be those sets of vertices such that . We will denote such a facet by . If we have chosen an ordering of , then we will let . If is a subset of , then we will say that the facet has type . Note that every facet is of this form for some and some subset of .
Example. If as above, then is a square: order by . We have and , and the facets are precisely the vertices and the chambers and . It is easily checked that this is a square. (Draw a picture!) In general, the Coxeter complex of is a -gon.
Notice that acts on by simplicial automorphisms, via . We will often not distinguish between and its geometric realization. One of the main results of [B, Chap. V] is that is a triangulation of the -dimensional Euclidean sphere. (See the footnote at the end of this post.) In particular, if (so ) then its (integral) homology in degree is and in all other degrees; if then its homology is in degrees and , and it is in all other degrees. The main theorem in this section relies on a computation of the character of the representation of on .
Theorem: If is the alternating character of , the homomorphism determined by the condition for all , then
Proof. Let . If then the result is obvious, so we will assume . Choose an ordering . For each , , let denote the free abelian group generated by the -facets of . For each subset of , let denote the set of -facets of of type . Let be the character of the permutation representation of on ; for each as above, let denote the character of the permutation representation of on ; and let denote the character of the representation of on . Trivially, . By the Hopf trace formula, we have
Evidently is the trivial representation since is connected. Moreover, , as we can see by considering the fundamental -cycle . Then we have
Now if , are complementary subsets of then is the subgroup of fixing each , . So . As we have
and a simple rearrangement gives
Comparing characters of and
In this section we will prove point 2 using fairly formal methods in finite group theory along with some comparisons between the subgroup structures of and . First, if we identify the sets of simple roots and of orthogonal reflections along simple roots, then we have for all subsets . (Although it is not sensible to multiply elements of with elements of , these double cosets are still sensible objects because normalizes and is contained in .) In particular, the Bruhat decomposition states that . It follows from the same formalism that if are two subsets then the number of -double cosets is equal to the number of -double cosets. To give a flavor of the methods involved (and because this will be needed), I give a proof of this statement below.
Lemma: The map , , is a bijection.
Proof. First, the map is well-defined: for example, is then we have
by one of the axioms of a Tits system. The Bruhat decomposition shows that this map is surjective, so we need only show that for every , . For this note that
We claim . Since one inclusion is obvious, it suffices to show the inclusion . Let , and write for some (as we can do by definition of ). By [B, Chap. V, Sec. 2, Lem. 1], we have
The right hand side is clearly contained in , so we are done. It suffices now to show , but indeed this follows from precisely the same argument as above. So the first displayed equation is true and the Lemma has been proved.
We are now ready to compare and . For each subset of (), we let and .
Theorem: The mapping is an isometry (with respect to the usual inner product) from the complex vector space generated by the characters of to the complex vector space generated by the characters of . If is an irreducible character of for integers , then or its negative is an irreducible character of . In particular, or its negative is an irreducible character of .
Proof. The second statement follows from the first: namely, write as an (integral) linear combination of irreducible characters of and note that the isometry statement implies that exactly one of the coefficients in this linear combination is nonzero, and this nonzero coefficient is either equal to 1 or -1. The final statement follows from the theorem in the previous section.
It is now enough to show that for all subsets of . In fact, we will show that is equal to the number of -double cosets in . The same method will show an analogous result for . As the number of -double cosets is equal to the number of -double cosets (as noted in the Lemma above), the result follows.
First, note that if is a finite group acting transitively on two sets and , and if , , then the number of orbits of acting on is equal to the number of -double cosets in , where and are the stabilizers of and in , respectively. If and are the characters of the permutation representations of on and , then we note that is the number of fixed points of acting on . By Burnside’s lemma, it follows that the number of orbits of acting on is equal to
Apply these observations to , , , , , and similarly for , to conclude.
Footnote. It is not entirely trivial to extract from [B, Chap. V] the fact that is a triangulation of an -dimensional sphere. The main point to show is that (the geometric realization of) can alternatively described in the following way: equip with a -invariant inner product, so that is generated by orthogonal reflections. There is a system of hyperplanes in consisting of those hyperplanes fixed by some reflection in , and this system satisfies the axioms outlined in the beginning of [B, Chap. V, Sec. 3]. Equip the unit sphere (with respect to the given inner product) with the triangulation coming from this system of hyperplanes. Using the results of [B, Chap. V, nos. 3.2, 3.3], one can show that in fact is isomorphic to this triangulation of .
References. [B] = Bourbaki, Groupes et algèbres de Lie, Chaps. IV, V, and VI