Consider the group $GL(n,F_q)$ for finite field $F_q$, consider its irreducible representations over complex numbers.

**Questions** Is my understanding correct that the dimensions of all such irreps are polynomials in $q$ with integer coefficients having zeros only at roots of unity and zero ?

I understand that the answer should be contained in Green's 1955 paper THE CHARACTERS OF THE FINITE GENERAL LINEAR GROUPS, but as for me paper is difficult for extracting information. The case $q=2$ should be treated with care as demenstrated in comment from user148212.

**Questions 2** If anwer is yes - is there any conceptual/nice reason for it ?

(R. Stanley comment below perfectly explains the part about roots).

**Questions 3** If someone can give some nice formula for such dimensions that would be quite helpful.

**Questions 4** For other finite groups of Lie type is there any similar phenomena ?

**Questions 5** From the perspective of $F_1$ it would be nice to know is there always a manifold such that number of $F_q$ points is given by these polynomials ? (Flags are of that type).

Let me give some examples known to me supporting the positive answer to the question:

For $GL(2,F_q)$ dimensions are : $1$ (det-like irreps) , $q+1$ (principal series), $q-1$ (cuspidal), $q$ (Steinberg = irregular principal series). See e.g. MO273764, MO271389.

In general "regular princinpal series" - irreps induced from non-trivial characters of the Borel subgroup will have dimension $[n]_q!$. Just because $GL/Borel = Flag$ manifold has such number of points.

Cuspidal irreps: the degree of a cuspidal character of $GL(n, q)$ is $(q − 1)(q^2 − 1)· · ·(q^{n-1}-1)$ (see page 135 Corollary 5.4.5. of very nice thesis "Character Tables of the General Linear Group and Some of its Subroups" containing huge amount of concrete information).

For the so-called unipotent irreps there is q-analogue of "hook formula". The degrees of the unipotent characters are “polynomials in q”: $ q^{d(λ)} \frac{(q^n − 1)(q^{n−1} − 1)· · ·(q − 1)}{ \prod_{h(λ)}(q^h − 1) }$ with a certain d(λ) ∈ N, and where h(λ) runs through the hook lengths of λ. See nice survey by G. Hiss FINITE GROUPS OF LIE TYPE AND THEIR REPRESENTATIONS (top page 26, section 3.2.6).

From above source - section 3.2.7: The degrees of the unipotent characters of $GL(5,q)$ for table $(5)$ dim = $ 1 $, for table $(4, 1)$ dim = $q(q + 1)(q^2 + 1)$ for table $(3, 2)$ dim = $q^2(q^4 + q^3 + q^2 + q + 1)$ for table $(3, 1^2)$ dim = $q^3(q^2 + 1)(q^2 + q + 1)$ for table $ (2^2, 1)$ dim = $q^4(q^4 + q^3 + q^2 + q + 1)$ for table $(2, 1^3)$ dim = $q^6(q + 1)(q^2 + 1)$ for table $(1^5)$ dim = $ q^{10}$

characters for GL(3), GL(4) has been computed by R. Steinberg The representations of GL(3,q), GL(4,q), PGL(3,q), PGL(4,q) Canad. J. Math. 3(1951), 225-235. Which "This paper is part of a Ph.D. thesis written at the University of Toronto under the direction of Professor Richard Brauer". The degrees of the irreducible characters of GL3(q): $(q − 1)^2(q + 1)$, $(q − 1)(q^2 + q + 1)$, $ (q + 1)(q^2 + q + 1)$, $q^2 + q + 1$, $q(q^2 + q + 1)$, $q(q + 1)$, $q^3$, $1$. See e.g. G.Hiss quoted above section 3.3.6 page 28.