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For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet.

Though I thought that it is finite set, in some paper, it is written that there are infinitely many automorphic cuspidal representations in the associated packet.

How is it possible?

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  • $\begingroup$ What is the precise context in which it is asserted to be infinite? $\endgroup$
    – Kimball
    Nov 27, 2021 at 13:39
  • $\begingroup$ @Kimball, Page 40 line 2~3 of the following paper. math.purdue.edu/~liu2053/S6.pdf $\endgroup$
    – Andrew
    Nov 27, 2021 at 14:50
  • $\begingroup$ I'm pretty sure this is not for a general linear group $GL_n$, but for some other reductive group -- apparently $Sp_{2(m-n)}$. (For $GL_n$ I believe all $A$-packets are finite, because they're singletons!) In general a global packet is a product of local packets, and all the local guys are finite, but since the set of places is infinite, the global packet can be infinite -- you can construct explicit examples using restriction to $SL_2$ of automorphic reps of $GL_2$ (the packet has more than one element at any unramified place where the Hecke eigenvalue is 0). $\endgroup$ Nov 29, 2021 at 7:50
  • $\begingroup$ @Loeffler, Thank you for the comment. May I ask some stupid quesiton? If $\pi$ is a unitary irreducible cuspidal automorphic representation of $GL(2)$, I am wondering whether there are infinitely many places $v$ such that $\pi_v$ is discrete series representation. Because the local A-packet for $Mp(2)$ associated to $\pi_v$ is singleton iff $\pi$ is not discrete series, if such places $v$, which makes $\pi_v$ discrete series, are infinitely many, then there are infinitely many cuspidal representations in the global Vogan A-packet for $Mp(2)$ associated to $\pi$. $\endgroup$
    – Andrew
    Nov 29, 2021 at 12:54
  • $\begingroup$ @Loeffler, I think $S_{\pi}$, the set of places $v$ such that $\pi_v$ is square integrable, is finite because unramified representation can never be square-integrable. Since the number of elements of the global Vogan A-packet of $\pi$ for $Mp(2)$ is $2^{|S_{\pi}|}$, the number of global cuspidal representations in the packet seems also finite. $\endgroup$
    – Andrew
    Nov 29, 2021 at 21:42

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