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?

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.

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.