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](https://doi.org/10.1090/conm/620/12366), arXiv:[1208.3216](https://arxiv.org/abs/1208.3216)


for a discussion of this and references. It is also available on my webpage [here](http://www.nd.edu/%7Eandyp/papers).



Is the Steinberg representation always irreducible?
Andrew Snowden and I managed to finally answer this question in our paper "The Steinberg representation is irreducible", available [here](https://arxiv.org/abs/2107.00794). 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](https://arxiv.org/abs/2005.05620). 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.


