In studying automourphic representation, I want to know whether my understanding is on the right way.


Let $\pi$ be a irreducible cuspidal representation of $GL\_2(A\_F)$. 


Then $\pi\_v$, the local component of $\pi$, is the unique subquotient of the induced representation of $\rho(\chi|\cdot|^s, \chi|\cdot|^{-s})$ where $\chi$ is unitary character and $0\le s \le \frac{1}{2}$. Is it right?


Then the work of Henry Kim and Freydoon Shahidi in $[$ Functorial products for $GL\_2 \times GL\_3$ and the symmetric cube for $GL\_2$ $]$ is to ensuring $s<\frac{5}{34}$?


Since I am just beginner in this area, any comment will be appreciated!



Local component of global irreducible representation of GL_2(A_F)
Not quite. The local component $\pi\_v$ is one of four types. 


1. If $\pi\_v$ belongs to the *tempered principal series*, then it is of the form $\rho(\chi\_1,\chi\_2)$, where $\chi\_1$ and $\chi\_2$ are arbitrary unitary characters (not necessarily equal).
2. If $\pi\_v$ belongs to the *complementary series*, then it is of the form $\rho(\chi|\cdot|^s, \chi|\cdot|^{-s})$, where $\chi$ is a unitary character, and $0\leq s<\frac{1}{2}$. The quoted work of Kim-Shahidi shows that $s\leq\frac{5}{34}$, while Blomer and Brumley (Ann. of Math. 174 (2011), 581-605) proved that $s\leq\frac{7}{64}$.
3. If $\pi\_v$ is *special* (also called a *Steinberg representation*), then it is the unique irreducible infinite-dimensional subrepresentation of $\rho(\chi|\cdot|^{1/2}, \chi|\cdot|^{-1/2})$, where $\chi$ is a unitary character.
4. Finally, $\pi\_v$ can be a *supercuspidal representation*.


You can learn about these things in Chapter 4 of Bump: Automorphic forms and representations, especially in Sections 4.5 to 4.8.


I should add that unramified representations belong to Cases 1 and 2, but the converse is not true.


**Remark.** The above was meant for a non-archimedean place $v$. For an archimedean place $v$ things are a bit different (e.g. for $v$ real, there is the principal series, the complementary series, and the discrete series, cf. Theorem 2.7.1 in Bump's book). Besides Bump's book, I can also recommend Goldfeld-Hundley's two volumes as a good introduction to the subject.


