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I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\alpha_2,\alpha_3$ refer to the associated Langlands parameters. It is known that \begin{split}&\alpha_1=k-1,\alpha_2=0,\alpha_3=-(k-1) \text{, when }f \text{ is a holomorphic form of weight } k; \\ &\alpha_1=2t_j,\alpha_2=0,\alpha_3=-2t_j \text{, when }f \text{ is a Maass form of Laplace eigenvalue } 1/4+t^2_j.\end{split}

My question is how about the symmetric cube lifts $\operatorname{sym}^3f$ ? If one now assumes that $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are the associated Langlands parameters of $\operatorname{sym}^3f$, what are the exact shapes of these parameters when $f$ being a holomorphic form (resp. a Maass form)?

If some expert knows something on this question, please give some comments or guide a reference.

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    $\begingroup$ I am not sure what "If some expert leans something on this question" means. The context suggested you meant "If some expert knows something tour this question", so I edited accordingly. I hope that was all right. $\endgroup$
    – LSpice
    Commented Jul 17, 2022 at 23:40
  • $\begingroup$ @LSpice Many thanks! $\endgroup$
    – hofnumber
    Commented Jul 18, 2022 at 5:58

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To understand this question better one should remember what Langlands parameters actually are. A Langlands parameter isn't just a list of numbers: these numbers are the components of a map from some auxiliary group (the real Weil group, or its subgroup $\mathbb{C}^\times$) into the diagonal torus of $\operatorname{GL}_n$. With this definition, it's obvious what the symmetric cube, etc, is: it's just given by composing a Langlands parameter into $\operatorname{GL}_2$ with the symmetric cube map from $\operatorname{GL}_2$ to $\operatorname{GL}_4$, and it's an easy exercise to work out what this does to a given $2 \times 2$ diagonal matrix.

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  • $\begingroup$ Dear Prof. Loeffler, thanks for explanation, definitely I was concerned about the Langlands parameters just from the point of view of the functional equation of the $L$-function of $L(s, \text{sym}^3f)$. Particularly, I need the exact forms of the Gamma factors of the $L$-function from the symmetric cube lifts of the $GL_2$-cusp forms. So, could you please give some more specific information on the associated parameters $\alpha_1,\alpha_2,\alpha_3,\alpha_4$? This is really what I am concerned about. Much obliged! $\endgroup$
    – hofnumber
    Commented Jul 18, 2022 at 8:03
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    $\begingroup$ You seem unwilling to put in any work yourself here. $\endgroup$
    – David Loeffler
    Commented Jul 18, 2022 at 8:17
  • $\begingroup$ Dear Prof. Loeffler, I have, yes, searched many papers, however it seems that there is no any account on the associated parameters $\alpha_1,\alpha_2,\alpha_3,\alpha_4. $ And, to be frankly, I am not familiar with the group representations. It's maybe not an easy exercise to work out the Langlands parameter to any given $2\times 2 $ diagonal matrix, for which I really need some help from the top experts like you here. $\endgroup$
    – hofnumber
    Commented Jul 18, 2022 at 9:53
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    $\begingroup$ Dear @hofnumber - you say that you have seached many papers, and you have asked several questions on this site that seem quite specialized or technical, but it is unclear from your questions and comments what previous background you have. It is suprising that you say you are not familar with group representations, if you are asking questions about Langlands parameters or Maass forms. Perhaps it would be more productive if you asked more basic questions about the concepts involved? $\endgroup$
    – Yemon Choi
    Commented Jul 18, 2022 at 18:08
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    $\begingroup$ The computation can be carried out precisely as Prof. Loeffler described. This is essentially Linear Algebra. There is actually a very nice paper by Michel and Cogdell 'On the complex moments of the symmetric square L-function at s=1' where precisely this computation is carried out in great detail. (See section 3 of the paper: doi.org/10.1155/S1073792804132455 ). $\endgroup$
    – Windom Earle
    Commented Jul 18, 2022 at 18:25

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