This is an interesting question. For background, Cartier (in the Corvallis volume) proves that when K is special, there's a Satake-like isomorphism. That gives (1) and (2). But I've never tried to work out the structure of the Hecke algebra for a non-special maximal parahoric, e.g., in a ramified unitary group. Someone should do that, if it hasn't been done before! Note that H(G,K) will be a subalgebra of the Iwahori Hecke algebra when K is any parahoric, so explicit generators and relations can be used.

You're talking about the Iwahori-Hecke algebra, I presume? Maybe look at the answers to the question here: mathoverflow.net/questions/40305/…

Thanks for adding the graphs to your answer. I think that, as your graphs show, the mediant-endpoint-rational-center bins have the effect of reducing the effect of the left/right discrepancy around rational numbers. The rational-endpoint-Farey-bins highlight this discrepancy. Both seem to show interesting information, so I'm glad we've got both now! p.s. I use Python+TikZ. What about you?

@AaronMeyerowitz: No -- the bins are definitely not equal in width, as their endpoints comprise all rational numbers with denominators <= 60 (when reduced). So yes, I normalized the height. It seemed like a good way to pick up on any phenomena special to bin-edges at rational numbers.

Oh - now I understand the "quarter-squares" bit. Makes sense! I've updated the question to use "Farey-binning" and bring out these spikes a bit more.

This binning issue is subtle -- very cool! I've updated the question a bit to show the effect of "Farey binning". I think this should highlight the somewhat unusual behavior of this distribution near rational numbers, without showing preference for certain denominators.

Thanks for the edits and explanations. Maybe I can figure out the spikes from that formula for the number of $n$ with $\{ \sqrt{n} \} \in (\alpha, \beta)$.

Excellent! I think this answers Question #1. I'm going to leave the question open for a little while to see if anyone answers (parts of) Question #2. In the meantime, I wonder about your answer -- can such estimates of exponential sums explain the visible difference between the flatness of the $\sqrt{n}$ and the noise of the $\sqrt{p}$ distributions?

I understand the source of the spike at zero (the squares, as you say). But I don't understand the "75 quarter-squares" bit. Do you have an explanation for the somewhat chaotic spiking around 0.5?

@VesselinDimitrov Thanks! I might think about this more, though my skills in this context are limited to experiments rather than serious analytic number theory.

Also, does the trivial argument for equidistribution of $\sqrt{n}$ mod $1$ give the flatness? One gets a $O(N^{-1/2})$ bound on "discrepancies", but I don't know off the top of my head how that compares to uniform sampling.

Thank you for the insights @VesselinDimitrov. I thought that under RH, one had some control over the number of primes in an interval of length $Y^{0.5 + \epsilon}$ around $Y$. I guess this isn't good enough for the equidistribution of $\{ \sqrt{p} \}$? Is it the $\epsilon$ or something deeper?