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$\DeclareMathOperator\SL{SL}$The volume of $\SL_n(\mathbb{R})/\SL_n(\mathbb{Z})$ can be computed under the natural measure that it inherits from $GL_n(\mathbb{R})$. Two formulae seem to be known. $$\operatorname{vol}(\SL_n(\mathbb{R})/\SL_n(\mathbb{Z}))= \zeta(2)\zeta(3) \dotsb \zeta(n).$$ This is available in notes by Paul Garrett [2]. In [1], the volume is given as $$\operatorname{vol}(\SL_n(\mathbb{R})/\SL_n(\mathbb{Z}))= \frac{\sqrt{2}}{\prod_{i=1}^{n-1}2^{i-1}i!}\zeta(2)\zeta(3) \dotsb \zeta(n).$$

Main question.

1.) Are there any such formulæ known for the quotient of $\SL_n(K\otimes_{\mathbb{Q}} \mathbb{R})/\SL_n(\mathcal{O}_K)$? I expect that it would be an expression involving Dedekind zeta functions. Here $K$ is a number field and $\mathcal{O}_K$ is the ring of integers.

Another question.

2.) Why are the two expressions different? Is one of them wrong or are they under different scaling of the Haar measures? Which of them is the volume with respect to the natural measure on $\SL_n(\mathbb{R})$?

[1] Paula - Comparison of volumes of Siegel sets and fundamental domains of $\operatorname{SL}_n(\mathbb Z)$

[2] Garrett - Volume of $\operatorname{SL}_n(\mathbb Z)\backslash\operatorname{SL}_n(\mathbb R)$ and $\operatorname{Sp}_n(\mathbb Z)\backslash\operatorname{Sp}_n(\mathbb R)$

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    $\begingroup$ Concerning 1.) have a look at numdam.org/item/PMIHES_1989__69__91_0.pdf $\endgroup$ Jun 8, 2021 at 20:11
  • $\begingroup$ What is the natural measure on $\operatorname{SL}_n(\mathbb R)$? \\ Name of @StefanWitzel's reference: Prasad - Volumes of $S$-arithmetic quotients of semi-simple groups. $\endgroup$
    – LSpice
    Jun 8, 2021 at 20:43
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    $\begingroup$ One obvious choice of a measure of $SL_n(\mathbb{R})$ is to work with the Frobenious inner product on $\mathfrak{sl}_n(\mathbb{R})$ and use it to generate a metric tensor. This is probably the same as the Hausdorff measure it has as a subspace of $GL_n(\mathbb{R})$ which carries the measure $dg |\det(g)|^{-n}$. If I'm not wrong, Siegel worked with the measure $\mu_{SL_n(\mathbb{R})}(U)=\mu_{GL_n(\mathbb{R})}(\cup_{t\in (0,1]}tU)$. This too might be the same thing. $\endgroup$ Jun 8, 2021 at 21:58

1 Answer 1

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$\DeclareMathOperator\SL{SL}$The same argument (due to Siegel, in a classical form, of course), adelized, gives the analogous computation for any number field, and, yes, the corresponding Dedekind zeta appears. (To know that the adelic analogue computes the same thing, rather than the adelic quotient for $\SL_n$ being larger than the classical one, probably we invoke Strong Approximation, just to be safe.)

Certainly there is the issue of describing a reasonable normalization of the Haar measures. Of course, in a silly way, we could normalize them to get any result we wanted. For that matter, from a contemporary viewpoint, it is more stylish to define "Tamagawa measures" on semisimple groups $G$, designed so that the measure of $G_k\backslash G_{\mathbb A}$ is (often) $1$, or some other simple integer. With hindsight, Siegel's computation gives hints about how to do this. (See Weil's book "Adeles and algebraic groups".)

Operationally, from a neo-classical viewpoint, the important thing is to make natural choices of measures on $P_v$ and $K_v$, where $P_v$ is the $v$-adic minimal parabolic and $K_v$ is the $v$-adic maximal compact, which assemble to give a measure on $G_v$. It is natural to give $K_v$ measure $1$ at non-archimedean places. The measure on $P_v$ is made from a local multiplicative measure on $k_v^\times$, which has the natural normalization that $\mathfrak{o}_v^\times$ has measure $1$. At places $v$ not ramified over $\mathbb Q$, the additive measure naturally gives $\mathfrak{o}_v$ measure $1$. At absolutely ramified places, there is also a natural normalization as in Iwasawa–Tate theory, and this introduces a power of the discriminant, unsurprisingly.

At archimedean places $v$, thinking in terms of Iwasawa–Tate theory gives a natural measure on $P_v$. The chief ambiguity here is about $K_v$. One choice is to relate the measure to the natural measure of spheres. Another is to give $K_v$ measure $1$. And other choices that adapt to induction over the size $n$. These lead to different constants, attributible to these choices at archimedean places.

EDIT: in response to a question about why computing "the" volume of $\SL_n(\mathbb{Q})\backslash\SL_n({\mathbb A})$ should be essentially the same thing as that of $\SL_n(\mathbb Z)\backslash\SL_n(\mathbb R)$, and so on… The most typical (?) thing that is said is that (by Strong Approximation) $\SL_n(\mathbb A)=\SL_n(\mathbb Q)\SL_n(\mathbb{R})\prod_{p<\infty}\SL_n(\mathbb Z_p)$, from which \begin{gather*} \SL_n(\mathbb Q)\backslash\SL_n(\mathbb A)/\prod_{p<\infty}\SL_n(\mathbb Z_p) \approx (\SL_n(\mathbb Q)\cap \SL_n(\mathbb R)\prod_{p<\infty}\SL_n(\mathbb Z_p))\backslash\SL_n(\mathbb R) \\ \;=\; \SL_n(\mathbb Z)\backslash\SL_n(\mathbb R). \end{gather*} In particular, a right $\SL_n(\mathbb R)$-invariant measure on one gives the same on the other. The measures on the various $X_N$ can be normalized to make everything consonant… A possibly disappointing aspect of this viewpoint is that it gives no origin story, despite being an orthodox viewpoint.

Another viewpoint, which in the long term may be more explanatory, is in terms of projective limits of classical quotients $X_N=\Gamma_N\backslash\SL_n(\mathbb R)$, where $\Gamma_N$ is the principal congruence subgroup of level $N$. When $M$ divides $N$, there is the natural surjection $X_N\to X_M$. Ordering positive integers by divisibility, the projective limit (a type of non-abelian solenoid) is naturally an object of interest. Already we could look at the proj lim of circles $\mathbb R/N\mathbb Z$, or even the proj lim of circles $\mathbb R/2^n\mathbb Z$. These solenoids were studied from a topological viewpoint in the early 1940s by Eilenberg and MacLane. The latter case gives $(\mathbb R\times \mathbb Q_2)/\mathbb Z[{1\over 2}]^\Delta$, where $\mathbb Q_2$ is the $2$-adic numbers. An interesting operational point is that $\mathbb Q_2$ acts on the family of circles $\mathbb R/2^n\mathbb Z$, although it does not act on the individuals.

Similarly the proj lim of $\mathbb R/N\mathbb Z$ is $\mathbb A/\mathbb Q$. That is, in fact every $\mathbb Q_p$ acts on that family of circles, though not on individuals.

Similarly, $\SL_n(\mathbb Q_p)$ acts on the projective family of classical arithmetic quotients $X_N$, though not on individuals.

(The emergence of adelic things in some areas of physics also seems to be not just by fiat of definition, but because one takes a projective limit of "charge lattices", whatever those are. :) I don't know anything beyond those words…)

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    $\begingroup$ @LSpice, Thanks for your edits! Reviewing them, I did also notice that I'd written $G_\mathbb{R}$ when obviously I wanted the adele group... Cheers. $\endgroup$ Jun 8, 2021 at 21:04
  • $\begingroup$ I wondered, but didn't want to make a substantive change like that. $\endgroup$
    – LSpice
    Jun 8, 2021 at 21:11
  • $\begingroup$ :) And, oop, a K_v outside of math mode. :) One more edit, I guess. :) $\endgroup$ Jun 8, 2021 at 21:14
  • $\begingroup$ "To know that the adelic analogue computes the same thing..." Could you please add more details on this. My knowledge on adelic things is limited but I would really like to understand how taking "natural" choices for measures based on Iwasawa decomposition at each valuation $v$ of the adelic group gives the same measure that $SL_n(\mathbb{R})$ os $SL_n(K_\mathbb{R})$ naturally has (see my comment on the question above). For $\mathfrak{sl}_n(K_\mathbb{R})$, one could take a natural quadratic form on $K_\mathbb{R}$ and extend to $\mathfrak{sl}_n(K_\mathbb{R})$. $\endgroup$ Jun 8, 2021 at 22:29
  • $\begingroup$ @Breakfastisready, I'll make an edit perhaps tomorrow about this. Your question has significance for many people, I'm sure... $\endgroup$ Jun 8, 2021 at 23:17

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