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Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL_2(\mathbb{F})$ that is $\pi=Ind_B^{GL_2}(\chi_1\otimes\chi_2)$ for two characters $\chi_1$ and $\chi_2$ of the maximal torus. Then I came to know from the book "Automorphic Forms on Adele group" by Gelbert that due to Casselman , it follows that conductor of $\pi=$ (conductor of $\chi_1$)$\times$(xonductor of $\chi_2$).

I want to know if there is similar result for $GL_n(\mathbb{F})$ for $n>2$. That is if $\pi=Ind_B^{GL_n}(\chi_1\otimes\chi_2\otimes\dots\otimes\chi_n)$ is a principal series representation of $GL_n(\mathbb{F})$, then is it true that conductor of $\pi= \prod$(conductor of $\chi_i$)?

Please refer some good paper for this.

Thank you in advance.

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    $\begingroup$ Yes, this is true and well known to experts. It should follow from a combination of two papers of Jacquet, Piatetski-Shapiro, and Shalika. I’ll try to find the exact references later. $\endgroup$
    – Peter Humphries
    Commented Dec 10, 2019 at 11:14
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    $\begingroup$ Let $\tilde{\gamma}=\gamma/\epsilon$, where $\gamma$ and $\epsilon$ are the gamma and the epsilon factors, respectively. Then the conductor $C(\pi)$ of an irreducible admissible representation of $\mathrm{GL}_n(F)$ can be defined by $$\tilde{\gamma}(1/2-s,\pi)=\tilde{\gamma}(1/2,\pi)C(\pi)^s+O_\pi(s),\quad s\to 0.$$ Both $\gamma$ and $\epsilon$ factors, hence $\tilde{\gamma}$ are multiplicative over isobaric sum, i.e. if $\pi$ is isobaric sum of $\chi_i$'s then $\tilde{\gamma}(s,\pi)=\prod_{i}\tilde{\gamma}(s,\chi)$. The conclusion follows by expanding the above display by multiplicativity. $\endgroup$
    – Subhajit Jana
    Commented Dec 10, 2019 at 14:50
  • $\begingroup$ Not exactly a reference, Subhajit! ;) $\endgroup$
    – Peter Humphries
    Commented Dec 10, 2019 at 14:56
  • $\begingroup$ @SubhajitJana please provide a reference, that will be more helpful. $\endgroup$
    – user15243
    Commented Dec 10, 2019 at 15:05
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    $\begingroup$ I'm writing up an answer now $\endgroup$
    – Peter Humphries
    Commented Dec 10, 2019 at 15:06

1 Answer 1

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Here is the story as I know it. The more general setup is for induced representations of Langlands type. These are certain generic representations of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field. The construction of such representations is the following.

Suppose that $n = n_1 + \cdots + n_r$ with $n_1,\ldots,n_r \geq 1$. Let $\sigma_j$ be a discrete series representation of $\mathrm{GL}_{n_j}(F)$ (necessarily unitary and tempered), from which we can construct a (no longer necessarily unitary or tempered) generic irreducible admissible smooth representation $\pi_j = \sigma_j \otimes \left|\det\right|^{t_j}$ of $\mathrm{GL}_{n_j}(F)$. We form the representation $\pi_1 \otimes \cdots \otimes \pi_r$ of $\mathrm{GL}_{n_1}(F) \times \cdots \times \mathrm{GL}_{n_r}(F)$, which we view as a Levi subgroup in $\mathrm{GL}_n(F)$. Then by normalised parabolic induction, we can induce this to a representation $\pi = \pi_1 \boxplus \cdots \boxplus \pi_r$ of $\mathrm{GL}_n(F)$.

This is known as an induced representation of Whittaker type (in particular, it has a Whittaker model); we say that $\pi$ is the isobaric sum of $\pi_1,\ldots,\pi_r$. If $\Re(t_1) \geq \cdots \geq \Re(t_r)$, then this is said to be an induced representation of Langlands type (which is isomorphic to its Whittaker model). These may not be irreducible, but the irreducible ones are generic irreducible admissible smooth representations, and conversely every generic irreducible admissible smooth representation is isomorphic to an induced representation of Langlands type.

The case you are interested in is $n_1 = \cdots = n_r = 1$, so that each $\pi_j$ is simply a character.

The theorem you are interested in is the following.

The conductor exponent $c(\pi)$ of an induced representation of Langlands type $\pi = \pi_1 \boxplus \cdots \boxplus \pi_r$ is equal to $c(\pi_1) + \cdots + c(\pi_r)$.

Here the conductor exponent is as in my answer here:

On the consistency of the definition of the conductor for automorphic forms

In "Conducteur des représentations du groupe linéaire" by Jacquet, Piatetski-Shapiro, and Shalika, this conductor exponent is defined (see in particular Section 5). In "Rankin-Selberg Convolutions" by the same authors, further properties are shown. In particular, a combination of Theorem 3.1, Proposition 8.4, and Proposition 9.4 shows that the epsilon factor $\varepsilon(s,\pi,\psi)$ satisfies $$\varepsilon(s,\pi,\psi) = \prod_{j = 1}^{r} \varepsilon(s,\pi_j,\psi),$$ where $\psi$ is an additive character of $F$.

(More precisely, they show the same multiplicativity properties hold for $L(s,\pi)$ and $\gamma(s,\pi,\psi)$, from which the result for $\varepsilon(s,\pi,\psi)$ follows.)

Finally, Théorème 5 of the first paper states that $$\varepsilon(s,\pi,\psi) = \varepsilon\left(\frac{1}{2},\pi,\psi\right) q^{-c(\pi)\left(s - \frac{1}{2}\right)}$$ (at least for unramified $\psi$), at which point the result follows.

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    $\begingroup$ Would you like to include the archimedean case? Otherwise, I might spell my comment out with a bit details. $\endgroup$
    – Subhajit Jana
    Commented Dec 11, 2019 at 8:47
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    $\begingroup$ @SubhajitJana depends on whether you want me to include the archimedean case for the algebraic conductor exponent or the analytic conductor $\endgroup$
    – Peter Humphries
    Commented Dec 11, 2019 at 9:51
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    $\begingroup$ I was originally thinking about the analytic conductor, but if you like you may include either of them. $\endgroup$
    – Subhajit Jana
    Commented Dec 11, 2019 at 10:05
  • $\begingroup$ @PeterHumphries Can I clarify for the expression $$\epsilon(s, \pi, \psi) = \epsilon( \frac{1}{2}, \pi, \psi) q^{-c(\pi) (s - \frac{1}{2})}$$ it is found in Jacquet's "Conducteur des représentations du groupe linéaire" paper? As the paper only indicates that epsilon factor can be written as a monomial, which follows from the local FE. Furthermore, do you happen to have the reference to show that $|\epsilon(s, \pi, \psi)| =1$ for any irreducible admissible representation $\pi$, with $\psi$ being unramified? This is from Cogdell et. al's "Lectures on Automorphic L-functions". Thank you! $\endgroup$
    – L-JS
    Commented Jan 25 at 17:46
  • $\begingroup$ @L-JS It is stated in Theorem 3.3 (4) of "Zeta functions of simple algebras" by Godement and Jacquet that $\epsilon(s,\pi,\psi)$ is equal to $q^{-ms}$ times a nonzero constant. From here, see Section 5 of "Conducteur des représentations du groupe linéaire" to see that $m = c(\pi)$. The fact that $|\epsilon(1/2,\pi,\psi)| = 1$ when $\psi$ is unramified and $\pi$ is a generic irreducible admissible unitary representation follows by applying the local functional equation twice for the Godement-Jacquet zeta integral. $\endgroup$
    – Peter Humphries
    Commented Jan 25 at 22:59

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