What makes Gaussian distributions special? Local field version?

This question is inspired by the recent one about Gaussian measures over the reals:

What makes Gaussian distributions special?

I would be interested in a similar list of characterizations for the probability measure, on the field of $$p$$-adic numbers $$\mathbb{Q}_p$$, whose density with respect to the standard additive Haar measure is given by the indicator function of the ring of $$p$$-adic integers $$\mathbb{Z}_p$$.

• Probably worth linking to this paper: arxiv.org/abs/math/9803046 Commented Feb 3, 2021 at 4:10
• Thanks! If you have time it would be good if you could extract one of the characterizations in the paper and post that as an answer. Commented Feb 3, 2021 at 20:27
• Have you checked the refs in mathoverflow.net/questions/339214/… ? Commented Feb 28, 2021 at 3:58
• I would say that it ought to be $\Bbb{C}$-valued owing to comments in my post below. Commented Mar 3, 2021 at 4:19
• This question reminds me of Sally and Taibleson - Special functions on local fields. Commented Mar 9, 2021 at 4:58

1 Answer

Perhaps one answer is the appearance of the Gaussian in the oscillator (a.k.a. Weil) representation --- which makes sense even in non-zero characteristic. You're probably aware of the construction of this infinite dimensional representation for the double cover of $$\mathrm{SL}_2(\Bbb{R})$$. The story for $$\mathrm{SL}_2 \big( \Bbb{F}_q \big)$$ where $$q$$ a power of a prime strongly parallels the continuous story except, of course, it's finite:

Choose any non-square $$\delta \in \Bbb{F}_q$$ and form the quadratic extension $$\Bbb{F}_q(\delta) \cong \Bbb{F}_q^2$$. As usual, we identify elements of $$\Bbb{F}_q(\delta)$$ as linear combinations $$z = a + \sqrt{\delta} b$$ with $$a, b \in \Bbb{F}_q$$ subject the usual formulae for addition and multiplication; conjugation and norm are expressed as $$\bar{z} = a - \sqrt{\delta} b$$ and $$\mathrm{N}(z) = a^2 - \delta b^2$$ respectively. We'll need to choose (any) non-trivial additive character $$\psi: \Bbb{F}_q \longrightarrow \Bbb{C}^*$$ together with any multiplicative character $$\chi: \Bbb{F}_q(\delta)^* \longrightarrow \Bbb{C}^*$$ which generates the character group of $$\Bbb{F}_q(\delta)^*$$. Morally $$\psi$$ plays the role of the exponential function $$\exp: \Bbb{R} \longrightarrow \Bbb{R}^*$$ in this finite context. Now set

$$$$W_\chi := \ \left\{ \begin{array}{l} \displaystyle \text{all functions} \ f: \Bbb{F}_q(\delta) \longrightarrow \Bbb{C} \ \ \text{such that} \\ \displaystyle f(wz) = \overline{\chi(w)} \, f(z) \ \text{whenever \mathrm{N}(w)=1} \end{array} \right\}$$$$

which is $$q-1$$ dimensional. The oscillator representation $$\varrho_\chi: \mathrm{SL}_2(\Bbb{F}_q) \longrightarrow \mathrm{GL}(W_\chi)$$ is determined by the action of the follow elements, which generate $$\mathrm{SL}_2(\Bbb{F}_q)$$:

$$$$\begin{array}{ll} \displaystyle \varrho_\chi { \scriptstyle \begin{pmatrix} x & 0 \\ 0 & x^{-1} \end{pmatrix}} f(z) &\displaystyle = \ f(xz) \\ \displaystyle \displaystyle \varrho_\chi { \scriptstyle \begin{pmatrix} 1 & \ \ y \\ 0 & \ \ 1 \end{pmatrix}} f(z) &\displaystyle = \ \underbrace{\psi \big( y \, \mathrm{N}(z) \big)}_{\text{the Gaussian G_y(z)}} \cdot f(z) \\ \displaystyle \displaystyle \varrho_\chi { \scriptstyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} f(z) &\displaystyle = \ -q \widehat{f}(-\bar{z}) \end{array}$$$$

where $$\widehat{f}$$ is the Fourier transform of $$f$$ with respect to the additive structure of $$\Bbb{F}_q(\sqrt{\delta})$$. Like the continuous Gaussian, $$G_y(z)$$ is an eigenfunction of this Fourier transform.

As far as I know, the construction that I've outlined (and which I learned from Amritanshu Prasad's online notes) can be carried out for $$\mathrm{SL}_2\big( \Bbb{Z}/q\Bbb{Z} \big)$$ where $$q= p^N$$ is still a power of a fixed prime $$p$$. Furthermore a coherent choice of additive and multiplicative characters $$\psi_N$$ and $$\chi_N$$ can be made for each $$N \geq 1$$ so that these oscillator representations agree with the inverse system

$$$$1 \longleftarrow \Bbb{Z}/p\Bbb{Z} \stackrel{\ \text{mod p}}{\longleftarrow} \Bbb{Z}/p^2\Bbb{Z} \stackrel{\ \text{mod p^2}}{\longleftarrow} \Bbb{Z}/p^3\Bbb{Z} \longleftarrow \, \cdots$$$$

thus allowing the oscillator representations $$\varrho_{\chi_N}$$ to be exported to the $$p$$-adic integers $$\Bbb{Z}_p$$, which will inherent some kind of Gaussian-like function $$\varprojlim G_{\bf y}$$ where $${\bf y}= (y_1,y_2,y_3, \dots)$$ and $$y_N = y_{N+1} \, \text{mod p^N}$$ for each $$N \geq 1$$.

• I would follow this up with the observation that Young's convolution inequality $\| f * g\|_r \leq c_{p,q} \| f \|_p \, \| g \|_q$ with ${1 \over p} + {1 \over q} = 1 + {1 \over r}$ is maximized when $f$ and $g$ are Gaussians. One might ask whether or not the same characterization holds for the $\Bbb{F}_q$-Gaussians $G_y$. Commented Mar 3, 2021 at 4:06
• In a similar vein, the Babenko-Beckner inequality $\| \widehat{f} \|_q \leq \tilde{c}_{p,q} \| f \|_p$ with ${1 \over p} + {1 \over q} = 1$ and $1 < p \leq q$ is maximized by Gaussians. Does this characterize $\Bbb{F}_q$-Gaussians ? Is there some analogue at least ? Commented Mar 3, 2021 at 4:33
• Thank you for a very nice answer! I remember reading a long time ago about Segal-Shale-Weil and the metaplectic representation, Mehler's formula etc. Do you know if one gets the Fourier transform as a particular value of the corresponding unitary representation, like at $t=\frac{\pi}{4}$ if I remember correctly for the Oscillator? If so are the eigenfunctions determined, like Hermite functions in the Archimedean case? Commented Mar 3, 2021 at 14:58
• I also hope someone will write an answer along the line of your comments, namely some charaterization of the indicator function of $\mathbb{Z}_p$ as some kind of maximizer/minimizer. Commented Mar 3, 2021 at 14:59
• If $\delta = -1$ is a non-square in $\Bbb{F}_q$ then the Fourier transform is indeed a special value of the action the subgroup $\mathrm{SO}_2(\Bbb{F}_q)$ consisting of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ \delta b & a \end{pmatrix}$ with $a^2- \delta b^2 =1$. As far as Hermite functions are concerned, wouldn't you need a good candidate for a laplace operator $\Delta: \Bbb{C} \big[\Bbb{F}_q (\sqrt{ \delta }) \big] \longrightarrow \Bbb{C} \big[\Bbb{F}_q (\sqrt{ \delta }) \big]$ as well as an appropriate $\Bbb{C}$-valued potential function to play the role of $|z|^2$ ? Commented Mar 3, 2021 at 16:33