# Proof of generalized Siegel's mean value formula in geometry of numbers

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?

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.