This is a very nice question! 

All of your examples are spherical quotients. Please take a look at 
[Sakellaridis - Spherical varieties and integral representations of $L$-functions](https://web.archive.org/web/20120714064540/http://andromeda.rutgers.edu/~sakellar/rs.pdf) and Yiannis' other papers 
for connections between spherical quotients and integral representations for L functions.