Is the Steinberg representation always irreducible?

Question

Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}_n(\mathbb{F})$, denoted $T_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags $$0 \subsetneq V_0 \subsetneq \cdots \subsetneq V_k \subsetneq \mathbb{F}^n.$$ The space $T_n(\mathbb{F})$ is $(n-2)$-dimensional, and the Solomon-Tits theorem says that in fact $T_n(\mathbb{F})$ is homotopy equivalent to a wedge of $(n-2)$-dimensional spheres. The Steinberg representation of $\text{SL}_n(\mathbb{F})$, denoted $\text{St}_n(\mathbb{F})$, is $\widetilde{H}_{n-2}(T_n(\mathbb{F});\mathbb{C})$. This is one of the most important representations of $\text{SL}_n(\mathbb{F})$; for instance, if $\mathbb{F}$ is a finite field of characteristic $p$, then $\text{St}_n(\mathbb{F})$ is the unique nontrivial irreducible representation of $\text{SL}_n(\mathbb{F})$ whose dimension is a power of $p$.

The only proof I know that $\text{St}_n(\mathbb{F})$ is an irreducible representation of $\text{SL}_n(\mathbb{F})$ when $\mathbb{F}$ is a finite field uses character theory, and thus does not work for $\mathbb{F}$ infinite (in which case $\text{St}_n(\mathbb{F})$ is an infinite-dimensional representation of the infinite group $\text{SL}_n(\mathbb{F})$).

Question: For an infinite field $\mathbb{F}$, is $\text{St}_n(\mathbb{F})$ an irreducible representation of $\text{SL}_n(\mathbb{F})$? If not, is it at least indecomposable?

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EDIT 2: In the previous edit, I said that I accepted an answer that did not answer the question as stated. However, this has now changed since Andrew Snowden and I have written a paper giving a complete answer to this question.

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EDIT: I accepted an answer, but I am particularly interested in the field $\mathbb{Q}$, which is not covered by that answer. This case is interesting to me because it arises when studying the cohomology of $\text{SL}_n(\mathbb{Z})$; indeed, in this case the Tits building forms the boundary of the Borel–Serre bordification of the associated symmetric space and the Steinberg representation (as I defined it above) provides the "dualizing module" for $\text{SL}_n(\mathbb{Z})$. See Section 2 of my paper

• T. Church, B. Farb, A. Putman A stability conjecture for the unstable cohomology of $\text{SL}_n(\mathbb{Z})$, mapping class groups, and $\text{Aut}(F_n)$, in "Algebraic Topology: Applications and New Directions", pp. 55–70, Contemp. Math., 620, Amer. Math. Soc., Providence, RI. doi:10.1090/conm/620/12366, arXiv:1208.3216

for a discussion of this and references. It is also available on my webpage here.

Answer 1

Consider the case where $\mathbb F$ is a non-achimedean local field. Then following Borel and Serre (Cohomologie d'immeubles et de groupes $S$-arithmétiques, Topology, 1976), one may equip $T_n ({\mathbb F})$ with a topology coming from $\mathbb F$. With this topology, the cohomology in degree $n-2$ is indeed irreducible, this is the Steinberg representation of ${\rm SL}(n,{\mathbb F})$ in the sense of smooth (admissible) representations. However if you equip $T_n ({\mathbb F})$ with the topology of the geometric realization of your simplicial complex then you get a bigger representation which is not irreducible (I do not know wether it is indecomposable). For instance, for $n=2$, you get the space of complex functions on the projective line $P^1 ({\mathbb F})$ modulo constant functions. In contrast the Steinberg representation (in the smooth sense) is the space of locally constant functions on the projective line modulo constant functions.

Answer 2

Andrew Snowden and I managed to finally answer this question in our paper "The Steinberg representation is irreducible", available here. As you might guess from the title, we prove that the Steinberg representation over an infinite field is always irreducible. In fact, we prove something much more general that applies to arbitrary reductive groups over infinite fields, and also allows arbitrary coefficients for the Steinberg module.

It's worth also mentioning another recent paper by Galatius--Kupers--Randal-Williams called "$E_{\infty}$-cells and general linear groups of infinite fields", available here. One of their results says that the Steinberg representation for $\text{GL}_n$ (as discussed in this question) is indecomposable, i.e. is not the nontrivial direct sum of two subrepresentations. For infinite fields, the Steinberg representation is infinite-dimensional, so this is weaker than being irreducible. However, I think their proof is quite beautiful and worth reading even if it gives a weaker result.

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EDIT: My attention has been drawn to two earlier papers:

N. Xi, Some infinite dimensional representations of reductive groups with Frobenius maps, Sci. China Math. 57 (2014), no.~6, 1109--1120.

R. Yang, Irreducibility of infinite dimensional Steinberg modules of reductive groups with Frobenius maps, J. Algebra 533 (2019), 17--24.

These focus on connected reductive group $\mathbf{G}$ over the algebraic closure $k=\overline{\mathbb{F}}_q$ of a finite field $\mathbb{F}_q$. For instance, we could have $\mathbf{G}(k) = \text{GL}(n,k)$ as in the question. Their main theorem says that the Steinberg representation of $\mathbf{G}$ is irreducible with coefficients in any field. Xi's paper handles the case when the coefficients have characteristic $0$ or $\text{char}(k)$, and Yang's paper handles other characteristics.

Answer 3

Paul Broussous has indicated the way the Steinberg representation enters into the study of the group over a non-archimedean local field. I'll add some comments, in community-wiki format, to deal with the broader picture and fill in some references.

1) It's not clear to me why only the special linear group occurs in the formulation of the question, since this particular kind of representation lends itself well to a general treatment for semisimple (or reductive) algebraic groups. In any case, the historical development has mainly focused on just two situatins: first the finite groups of Lie type and later the analogous groups over non-archimedean local fields. It's important to consider what motivation there is for working over other fields, and especially what topology might be relevant besides the discrete topology. Infinite dimensional discrete representations of an infinite discrete group are usually quite hard to study.

2) In his early work, Steinberg himself worked mainly with finite groups (sometimes in the spirit of Chevalley's 1955 Tohoku paper). After experimenting with special cases, he found a succinct formulation of his construction in a short note here. His axiomatic formulation extends Chevalley's but his treatment always implicitly requires a finite group, in order to sum over various subgroups within the group algebra. But he can work over an arbitrary field, proving that his representation is irreducible precisely when the characteristic fails to divide the index of the finite "Borel subgroup" in the whole group. (There has been further work over the years on how the representation behaves when it isn't irreducible but the group is finite.)

3) Though Steinberg's construction is explicit (for finite groups), it doesn't easily yield all the character values over a field of characteristic 0. In fact these values are nice integers, first computed in absolute value by Srinivasan followed by later work clarifying the signs. Steinberg further showed in a 1963 paper that his representations occur naturally in the algebraic group context. In fact there is a family of them, one for each power of the characteristic $p$. Gradually the highest weight theory involving both algebraic and finite groups of Lie type was developed in a uniform way that integrated the Steinberg representation.

4) Later an alternative construction using the Tits building was developed: see especially the paper by Curtis-Lehrer-Tits here, but note that they used a slightly different version of the building. As they point out in their optional $\S8$, there are different possibilities here for the topology. But their main goal was to illuminate the determination of character values for the finite groups.

In each context people have defined "the Steinberg representation" in a precise way, but it's not clear to me what the optimal generality is for such a definition.