# Haar measure decomposition using orbital integrals

Let $$G$$ be a unimodular locally compact group, $$N,A \le G$$ be unimodular closed subgroups. Suppose that $$A$$ normalizes $$N$$. Let $$N_0 \le N$$ be a compact open subgroup. Suppose that a function $$f : N \rightarrow \mathbb{C}$$ is supported on $$\bigcup_{a \in A} a N_0 a^{-1}$$. Is there a formula to compute $$\int_N f(x) dx$$in this case in terms of integration over $$A$$ and $$N_0$$? For example, if $$G$$ is a finite group, we have $$\sum_{x \in N} f\left( x \right) = \sum_{a \in A} \sum_{x \in N_0} \frac{1}{\left| C_x \left( A \right) \right|} f\left( a x a^{-1} \right),$$ where $$C_x\left( A \right) = \left\{ a \in A \mid ax=xa \right\}$$is the centralizer of $$x$$ in $$A$$.

I am interested in particular in the case where $$G$$ is a matrix group (over a non-Archimedean field), $$N$$ is a unipotent radical of a parabolic subgroup of $$G$$, and $$A$$ is a subgroup of the Levi part of the same parabolic subgroup.

• I suspect that (assuming you are also willing to take $f$ locally constant and compactly supported, as usual in such applications) you'll have much better luck in your specific case than in general, because the orbit stratification is so much nicer. I don't know off the top of my head even in that special case, though. – LSpice Oct 15 at 14:41
• Actually, wait: if $f$ is compactly supported, then, since a compact subset of $\operatorname{Int}(A)(N_0)$ is contained in $\operatorname{Int}(a)(N_0)$ for a specific $a \in A$, it's easy. So maybe you don't want to assume compactly supported? – LSpice Oct 15 at 14:44
• Yeah, $f$ in my case should not be compactly supported, but it is locally constant. – darkl Oct 15 at 14:57
• Aha. Is there some other hypothesis that guarantees that $\int_N f(x)\mathrm dx$ converges? – LSpice Oct 15 at 15:34
• (Actually I guess I just mean more generally, could you say some more about how this problem arises? Orbital integrals on $p$-adic groups is pretty close to my research, and I might be able to say more if I knew the context.) – LSpice Oct 15 at 16:58