Distributions on the mirabolic subgroup which are left invariant to the unipotent radical

Question

I'm trying to find a reference or a proof to the following statement used by Matringe, N., 2012. Cuspidal representations of GL (n, F) distinguished by a maximal Levi subgroup, with F a non-archimedean local field. Comptes Rendus Mathematique, 350(17-18), pp.797-800. (used in the proof of Proposition 2.1/Proposition 3.1) and Kable, A.C., 2004. Asai L-functions and Jacquet's conjecture. American journal of mathematics, 126(4), pp.789-820. (used in the proof of Proposition 1).

Let $F$ be a $p$-adic field.

We denote by $P_n$ the subgroup of $ \mathrm{GL}_{n}\left(F\right) $ of matrices having $ \begin{pmatrix}0 & \dots & 0 & 1\end{pmatrix} $ as their last row.

We denote $G_{n-1}=\left\{ \begin{pmatrix}g\\ & 1 \end{pmatrix}\mid g\in\mathrm{GL}_{n-1}\left(F\right)\right\} $ and $ U_{n}=\left\{ \begin{pmatrix}I_{m-1} & v\\ & 1 \end{pmatrix}\mid v\in F^{m-1}\right\} $. It is known that $ P_{n}=G_{n-1}\ltimes U_{n} $.

Let $ V $ be a vector space over $ \mathbb{C}$ and let $ \tilde{T}:C_{c}^{\infty}\left(P_{n},V\right)\rightarrow\mathbb{C} $ be a distribution such that $ \tilde{T}$ is invariant to left translations of $ U_n $, i.e. $ \left\langle \tilde{T},\lambda\left(u\right)f\right\rangle =\left\langle \tilde{T},f\right\rangle $ for every $ u \in U_n $ and $ f \in C_{c}^{\infty}\left(P_{n},V\right) $. Then there exists a distribution $ S $ on $ C_{c}^{\infty}\left(G_{n-1},V\right) $ with compact support, such that $$ \left\langle \tilde{T},f\right\rangle =\int_{G_{n-1}}\left[\int_{U_{n}}f\left(ug\right)d\mu_{U_{n}}\left(u\right)\right]dS\left(g\right) $$

Here $ \mu_{U_{n}} $ is an Haar measure of $ U_{n} $.

I think I could prove this is $ U_n $ was an open-compact subgroup of $ P_n $, but it is neither of these. I guess the statement is not unique for $ P_n $, and that a similar statement holds for any semi-direct product.

Answer 1

For a totally disconnected group $G$ and closed subgroup $H$, the averaging map $\alpha:C^\infty_c(G)\to C^\infty_c(H\backslash G)$ is surjective, certainly for scalar-valued functions, and I think for $V$-valued functions where $V$ is quasi-complete, locally convex, so that Gelfand-Pettis (or some equivalent kind of) vector-valued integrals work well. Here $C^\infty_c$ means compactly supported and locally constant, as usual for t.d. situations, and "distribution" means a continuous linear functional on $C^\infty_c$ (the latter with the canonical LF-space topology as strict colimit of finite-dimensional spaces).

The $H$-invariance of a distribution $u$ on $C^\infty_c(G)$ implies that $u$ factors through the averaging map to $C^\infty_c(H\backslash G)$, that is, gives a continuous linear functional on the latter.

There is no promise that that distribution on $H\backslash G$ is compactly supported, though. Do you really need or expect that? I don't think it would be true without additional hypotheses.