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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.

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  • $\begingroup$ 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. $\endgroup$ – LSpice Oct 15 at 14:41
  • $\begingroup$ 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? $\endgroup$ – LSpice Oct 15 at 14:44
  • $\begingroup$ Yeah, $f$ in my case should not be compactly supported, but it is locally constant. $\endgroup$ – darkl Oct 15 at 14:57
  • $\begingroup$ Aha. Is there some other hypothesis that guarantees that $\int_N f(x)\mathrm dx$ converges? $\endgroup$ – LSpice Oct 15 at 15:34
  • $\begingroup$ (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.) $\endgroup$ – LSpice Oct 15 at 16:58

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