Local component of global irreducible representation of GL_2(A_F)

Question

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!

Answer 1

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.