Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.


The classical Siegel's formula in geometry of numbers states that for $f\in L^1(\mathbb R^d)$, let $\Lambda$ be a unimodular lattice, and let


$$\hat{f}(\Lambda):= \sum\_{v\in \Lambda\setminus 0} f(v).$$


Then $$\int\_X \hat{f} d\mu = \int\_{\mathbb R^d} f dv.$$


I wonder if there is such a generalization to this result:


For $g\in L^1((\mathbb R^d)^k)$, $1 \le k \le d$ and unimodular lattice $\Lambda$, let $\hat{g}^k(\Lambda):=\sum\_{(v\_1,\dots,v\_k)\in \Lambda^k \setminus 0} g(v\_1,\dots, v\_k)$. I speculate there might be such a generalization of Siegel's mean value formula:


$$\int\_X \hat{g}^k d\mu = \int\_{\mathbb R^d} \dots \int\_{\mathbb R^d} g(v\_1,\cdots,v\_k) dv\_1 \cdots dv\_k$$


Is this correct and proved somewhere in the literature? Or can anyone suggest a proof for it below if not too complicated from the classical Siegel's formula?



Proof of generalized Siegel's mean value formula in geometry of numbers
Such a generalization (roughly) exists, known as *Rodger's Integration Formula*.
See [Section 1.2 of Seungki Kim's Dissertation](https://stacks.stanford.edu/file/druid:ms598tw2644/thesis%20%281%29-augmented.pdf) for a reference.
Theorems 1.2 and 1.3 are of interest.



> 
> **Theorem 1.2**: (Rogers). Let $k < n$, and $\rho : (\mathbb{R}^n)^kk \to \mathbb{R}$ be a compactly supported,
> bounded, and Borel measurable function. Then
> $$
> \int\_{X}\sum\_{\substack{v\_1,\dots,v\_k\in L\setminus\{0\}\\\mathsf{rank}(\langle v\_1,\dots,v\_k\rangle) = k}}\rho(v\_1,\dots,v\_k)d\mu = \int\_{\mathbb{R}^n}\dots \int\_{\mathbb{R}^n}\rho(x\_1,\dots,x\_k)dx\_1\dots dx\_k.
> $$
> 
> 
> 


Up to the requirement that one sums over rank $k$ collections of vectors $(v\_1,\dots,v\_k)$, this is what you want.
One can remove this restriction (to get the generalization that you want), but the formula becomes more complex --- there is an additional (additive) term on the right-hand side.
See Theorem 1.3 of the dissertation for details.


