What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question how it occurred in my readings.

Let $G$ be the groups of points of a reductive group on a non-Archimedean local field, $N$ a unipotent subgroup and consider $\chi$ a character of $N$. Take the induced representation $\pi = \mathrm{Ind}_N^G(\chi)$.

1. There is a continuous decomposition of the form

$$\pi = \int_X \pi_x dx$$

where the $\pi_x$ are irreducible representations. I am searching for a more precise statement: what can be said about the space of parameters $X$, the underlying representations spaces of $\pi_x$ and the measure $dx$? What properties are known on $\pi_x$ in this specific setting (are they e.g. smooth? spherical if $\pi$ is spherical? etc.)

2. There is a spectral decomposition formula of the form, for any functions $\phi_1, \phi_2 \in \pi$, $$\langle \phi_1, \phi_2 \rangle = \int_X \langle A_x \phi_1, A_x \phi_2 \rangle dx$$ where the $A_x = V_\pi \to V_{\pi_x}$ are linear maps, more precisely intertwining operators so that $A_x \circ \pi = \pi_x \circ A_x$. Why is this true, and how far can we reconstruct $\phi$ from the knowledge of the $A_x \phi$?

I hope the question is precise enough and I will do my best to clarify it in comments otherwise. Any reference dealing with such theories is welcome, I do not know where to start. Thanks!

What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question how it occurred in my readings.

Let $G$ be the groups of points of a reductive group on a non-Archimedean local field, $N$ a unipotent subgroup and consider $\chi$ a character of $N$. Take the induced representation $\pi = \mathrm{Ind}_N^G(\chi)$.

1. There is a continuous decomposition of the form

$$\pi = \int_X \pi_x dx$$

where the $\pi_x$ are irreducible representations. I am searching for a more precise statement: what can be said about the space of parameters $X$, the underlying representations spaces of $\pi_x$ and the measure $dx$? What properties are known on $\pi_x$ in this specific setting (are they e.g. smooth? spherical if $\pi$ is spherical? etc.)

2. There is a spectral decomposition formula of the form, for any functions $\phi_1, \phi_2 \in \pi$, $$\langle \phi_1, \phi_2 \rangle = \int_X \langle A_x \phi_1, A_x \phi_2 \rangle dx$$ where the $A_x = V_\pi \to V_{\pi_x}$ are linear maps, more precisely intertwining operators so that $A_x \circ \pi = \pi_x \circ A_x$. Why is this true, and how far can we reconstruct $\phi$ from the knowledge of the $A_x \phi$?

Any reference dealing with such theories is welcome, I do not know where to start. Thanks!