Jacquet, Piateski-Shapiro, and Shalika defined new vectors for generic representations of $GL(n,F)$, where $F$ is a non-archimedean local field. I know that this notion has been extended to $GSp(4,F)$. Is there an extension to other $p$-adic groups?



New vectors for $p$-adic groups
The theory of new vectors for ${\rm GSp}(4)$ has been written by Schmidt and Roberts :


Local Newforms for GSp(4).
Springer Lecture Note in Mathematics, vol. 1918 (2007) 


See also Schmidt's webpage :


<http://www2.math.ou.edu/~rschmidt/>


The definition is trickier than in the case of ${\rm GL}(N)$


By the way : there is a mistake in Jacquet/Piateski-Shapiro/Shalika. It was pointed out and fixed by Matringe :


arXiv:1201.5506 Essential Whittaker functions for GL(n). Nadir Matringe. 


See also Jacquet's webpage.


New vectors are also known for generic representations of reductive groups of small ranks (in fact of rank $1$) : ${\rm SL}(2)$, unitary groups.


There is no general theory (except for spherical representations).


