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][1], 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). 
 

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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][2], [MO271389][3].

* 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][4] "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][5]".
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][6] (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][7]. 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.




  [1]: http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0072878-2/S0002-9947-1955-0072878-2.pdf
  [2]: https://mathoverflow.net/q/273764/10446
  [3]: https://mathoverflow.net/a/271389/10446
  [4]: http://researchspace.ukzn.ac.za/xmlui/bitstream/handle/10413/978/Character%20tables%20thesis.pdf?sequence=1
  [5]: https://en.wikipedia.org/wiki/Hook_length_formula
  [6]: http://www.math.rwth-aachen.de/~Gerhard.Hiss/Preprints/StAndrewsBath09.pdf
  [7]: http://dx.doi.org/10.4153/CJM-1951-027-x