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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?

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  • $\begingroup$ I edited your post, I hope you don't mind. $\endgroup$ – GH from MO Feb 21 '12 at 22:57
  • $\begingroup$ No problem GH, but there is a french proverb saying that "Lorsque l'on tombe, ce n'est pas le pied qui a tort. " $\endgroup$ – Rajkarov Feb 22 '12 at 3:47
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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).

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