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

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.

• Is "S" actually "T"? – paul garrett Jul 10 '17 at 19:11
• Yes, sorry about that – darkl Jul 10 '17 at 19:16

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.
• Do you really need any assumptions on $V$? A compactly supported, locally constant function is a combination of constant multiples of characteristic functions, i.e., lies in $\mathrm C_{\mathrm c}^\infty \otimes_{\mathbb C} V$, and for such functions integration is easy. – LSpice Jul 11 '17 at 16:32