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First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.

In the world of quantum mathematics, the letter $q$ is a standard symbol that parametrizes deformation/quantization of a theory. Examples include (affine, double affine,…) Hecke algebra, (elliptic) quantum group.

The letter $q$ appears in a (seemingly) completely different way in arithmetic. Namely, it appears as the number of elements in the finite field $\mathbb{F}_q$, or fancier, eigenvalues of Frobenius automorphisms. In some cases, the formal limit $q\to 1,\mathbb{F}_q\to\mathbb{F}_1$ can be thought as some semiclassical limit.

There is an example when quantum $q$ equals to arithmetic $q$, that is, the famous Kazhdan–Lusztig conjecture. The conjecture (now theorem) indicates a way to calculate characters of irreducible highest weight modules over a reductive Lie algebra $\mathfrak{g}$ in terms of characters of Verma modules. The conjecture was proved by Beilinson–Bernstein and Kashiwara–Brylinski by their localization theorems. In the localization theorem, simple modules correspond to some IC sheaves, Verma modules correspond to some standard sheaves, and the parameter $q$ corresponds to the Tate twist $(1)$, $v=q^{\frac{1}{2}}$ corresponds to $(\frac{1}{2})$.

My naïve understanding is that the localization theorem is the key in this problem; the deformation parameter $q$ is the shadow of some “lifted mixed geometry”.

I wonder if there are more examples where quantum $q$ equals to arithmetic $q$, either in a similar (localization theorem, mixed geometry) or different flavor.

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I think this is actually an extremely common situation in algebraic combinatorics (where the $q$ in a $q$-analog has meaning as both a "deformation parameter" and a prime power corresponding to a finite field). Here is just one very simple example.

For commuting variables $x$ and $y$ the binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$. We can "deform" this theory by making the variables $q$-commute in the sense that $xy = qyx$. Then the $q$-binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}_q x^{n-k}y^{k}$, where $\binom{n}{k}_q$ is the usual $q$-binomial coefficient. (So $\binom{n}{k}_q= \frac{[n]_q!}{[n-k]_q![k]_q!}$ with $[n]_q = \frac{1-q^n}{1-q}$ and $[n]_q!=[n]_q[n-1]_q\cdots[1]_q$.)

On the other hand, $\binom{n}{k}_q$ for $q$ a prime power can also be interpreted as the number of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. Sending $q\to 1$ we recover the interpretation of $\binom{n}{k}$ as the number of $k$ element subsets of an $n$ element set.

Like I said, this binomial coefficient example is very simple, but it is typical of many examples in algebraic combinatorics. See for example the desiderata (i)-(vi) for a $q$-analog listed in the Reiner-Stanton-White AMS Notices paper "What is... cyclic sieving?" (https://www.ams.org/notices/201402/rnoti-p169.pdf).

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    $\begingroup$ For an explanation of the connection between $q$ as a variable and as a prime power in $q$-binomial coefficients, see D. E. Knuth's paper, Subspaces, subsets, and partitions, doi.org/10.1016/0097-3165(71)90022-7. $\endgroup$
    – Ira Gessel
    Commented Mar 22 at 15:55
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    $\begingroup$ I thought of this too, but I think it may not answer the OP's question. Here's another way to phrase what I think is the intended question. In how many of these examples from algebraic combinatorics does the $q\to 1$ limit tie in with what number theorists think of as $\mathbb{F}_1$? That is, the $q$ arising in algebraic combinatorics is often visibly related to quantization, but it's often not so obviously related to $\mathbb{F}_1$. $\endgroup$
    – Timothy Chow
    Commented Mar 23 at 1:17
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    $\begingroup$ @TimothyChow: to me “bare linear algebra” over $\mathbb{F}_1$ is the theory of finite sets, their subsets, etc. as one develops in elementary combinatorics. I think it’s true that number theorists want to do more than bare linear algebra over $\mathbb{F}_1$ (e.g. they want to keep track of the Frobenius automorphism, they want to do algebraic geometry, …) but that doesn’t mean it’s not “the same” $\mathbb{F}_1$. $\endgroup$
    – Sam Hopkins
    Commented Mar 23 at 2:22
  • $\begingroup$ Related: Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients? $\endgroup$
    – Timothy Chow
    Commented Aug 31 at 16:25
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Some geometric constructions of $U_q(GL(n))$ via the finite fields - might fit the answer. The "q" plays both roles - finite field $F_q$ and deformation parameter $q$ for quantum group. I was always puzzled by such a coincidence.

Such constructions seems to be quite numerous. Few early examples:

A geometric setting for the quantum deformation of GL(n) A. A. Beilinson, G. Lusztig, R. MacPherson Duke Math. J. 61(2): 655-677 (October 1990). https://doi.org/10.1215/S0012-7094-90-06124-1

The Hall Algebra Approach to Quantum Groups CM Ringel https://www.math.uni-bielefeld.de/~ringel/opus/elam.pdf

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    $\begingroup$ In particular, (Iwahori–)Hecke algebras appear naturally in both contexts: in number theory etc (related to $\mathbb{F}_q$; I don't know anything about this), and as the centraliser of $U_q \,\mathfrak{gl}_r$ on $V^{\otimes n}$ with $V=L_{\omega_1}$ in quantum Schur–Weyl duality. $\endgroup$
    – Jules Lamers
    Commented Mar 22 at 15:34
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If $G$ is a unimodular locally compact group and $K$ is a compact subgroup, then the Hecke algebra of the pair $(G,K)$ is the algebra of compactly supported functions on $K\backslash G/K$ with a convolution product. When $G$ is a reductive group over a nonarchimedean local field and $K$ is an Iwahori subgroup, this algebra coincide with a specialization of an affine Iwahori-Hecke algebra (which is a deformation of the group algebra of an affine Weyl group) where the deformation parameter is set to the size of the residue field.

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