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Let $G$ be a reductive group over a number field $\mathbb{Q}$ and $K$ be a maximal compact subgroup of $G$. Let $\Gamma$ be an arithmetic subgroup of $G(\mathbb{Q})$.

Let $\mathfrak{g}$ be the complexfied Lie alogebra of $G$ and $Z(\mathfrak{g})$ the center of the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$.

Then one can define the automorphic form on $\Gamma \backslash G(\mathbb{R})$ as function on $\Gamma \backslash G(\mathbb{R})$ satisfying 'smooth', 'right $K$-finite', 'moderate growth', '$Z(\mathfrak{g})$-finite' conditions.

Let $X$ be an element in $\mathfrak{g}$ and $\phi$ is an automorphic form $\Gamma \backslash G(\mathbb{R})$.

Then some book says that $X\phi$ satisfies also $K$-finite conditions. But there is no proof on this and I can't check it well.

Would you let me know why $X\phi$ is also $K$-finite?

It seems that it would use some compatibility property of $\mathfrak{g}$-actions and $K$-actions, (i.e., $kX\phi=((Ad(k)X))k\phi$ for $k\in K$, $X\in \mathfrak{g}$.)

Any comments are appreciated!

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  • $\begingroup$ I know close to nothing about the subject but are you sure you want to denote your number field by $\mathbb{Q}$? $\endgroup$ Dec 13, 2021 at 14:17
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    $\begingroup$ @SylvainJULIEN, Yes! $\endgroup$
    – Andrew
    Dec 13, 2021 at 14:34
  • $\begingroup$ It's a bit odd, as $\mathbb{Q}$ usually denotes the field of rational numbers, and not an arbitrary finite extension thereof. $\endgroup$ Dec 13, 2021 at 14:36
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    $\begingroup$ @SylvainJULIEN, You re right! For simplicity, I took the rational number field $\mathbb{Q}$ as a number field here. But my question can be samely asked to arbitrary number field. $\endgroup$
    – Andrew
    Dec 13, 2021 at 15:47

1 Answer 1

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Consider the map $\mathfrak{g}\otimes C^\infty(\Gamma\backslash G)\to C^\infty(\Gamma\backslash G)$, $X\otimes f\mapsto Xf$. The group $K$ acts on both sides and the map is equivariant. If $f$ lies in a finite-dimensional $K$-module $M$, then $Xf$ lies in the finite-dimensional $K$-module that is the image of $\mathfrak{g}\otimes M$.

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  • $\begingroup$ Oh, I see! Since $\mathfrak{g}$ is finite dimension, the image should be also fin. dimension. Thank u very much! $\endgroup$
    – Andrew
    Dec 13, 2021 at 19:48

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