General linear groups whose irreducible representations are finite-dimensional Let $n\ge 2$ be an integer, $R$ a noetherian commutative ring and $K$ a commutative field. Is it true that all irreducible representations over $K$ of $GL_n(R)$ are finite-dimensional only if $R$ is finite?
And if we require that this finiteness condition is true for all (possibly algebraically closed) fields $K$ of a given characteristic?
 A: Proposition: let $R$ be a commutative noetherian ring, $K$ a (commutative) field and $n\ge 2$ an integer. If any irreducible representation of $GL_n(A)$ over $K$ is finite-dimensional, then $R$ is finite.
Steps:
(1) By a classical theorem of Formanek (J. Alg. 1973), if $G$ is a non-abelian free group, the group algebra $K[G]$ is primitive. This shows that every group containing a non-abelian free group has an infinite-dimensional irreducible representation over $K$.
(2) It is well-known that $GL_2(\mathbb{Z})$ contains a non-abelian free group. So, from (1), if the ring $R$ has characteristic $0$, then $GL_n(R)$ ($n\ge 2$) has always an infinite-dimensional irreducible representation over $K$.
(3) An old work by Nagao (J. Inst. Polytech. Osaka City Univ. 1959) shows that $GL_2(k[T])$ contains a non-abelian free group for every field $k$. So, from (1), if $R$ has prime characteristic $p$ and contains an element which is transcendental over $\mathbb{Z}/p$, then $GL_n(R)$ ($n\ge 2$) has always an infinite-dimensional irreducible representation over $K$.
(4) If $R$ is a locally finite field which is not finite, then $GL_n(R)$ ($n\ge 2$) has always an infinite-dimensional irreducible representation over $K$. One can prove this by using the recent work by Andrew Putman and Andrew Snowden on Steinberg representation for infinite fields that A. Putman mentioned as a comment, or classical results by Hartley (Quart. J. Math. 1977 when $K$ has characteristic $0$, Rocky Mountain J. Math. 1983 else).
(5) By noetherian induction, to prove the proposition, it is enough to prove that $GL_n(R)$ ($n\ge 2$) has an infinite-dimensional irreducible representation over $K$ when $R$ is infinite and $R/I$ is finite for every non-zero ideal $I$ of $R$.
(5a) If $R$ is not a domain, let $x, y\in R\setminus\{0\}$ such that $xy=0$. Then $R/(x)$ and $R/(y)$ are finite by induction assumption, and the exact sequence of $R$-modules $R/(y)\to R\to R/(x)\to 0$ implies that $R$ must be finite.
(5b) Assume that $R$ is a domain. By step (2), one may assume that $R$ has prime characteristic $p$. By step (3), one also may assume that every element of $R$ is algebraic over $\mathbb{Z}/p$. But in this case, $R$ must be a locally finite field, so one concludes by (4).
