Does a left translation of an automorphic form satisfy left $K$-finiteness?


Let $F$ be a number field and $G$ is an algebraic group. Let $\phi$ is an automorphic form on $G$. Let $K$ be a maximal compact subgroup of $G(A)$.


Then $\phi$ has $K$-finiteness.


For arbitrary $x$, let $\phi\_x(g)=\phi(xg)$. Then I am wondering whether $\phi\_x$ has also $K$-finiteness.


Decompose $K=K\_{\infty}K^{\infty}$. I checked $K^{\infty}$-finiteness of $\phi\_x$ but cannot check $K\_{\infty}$-finiteness.


Does this hold? If so, how can we prove it?



Left translation of automorphic form satisfies $K$-finiteness?
Presumable (in an automorphic forms context with contemporary left-right conventions) you mean that $\varphi$ is *right* $K$-finite. This could apply to any (complex-valued) function $\varphi$ on a topological group $G$, with compact subgroup $K$. It means that the space of functions obtained by right translation $R\_k$ by $k\in K$ is finite-dimensional. These are functions $R\_k\varphi(g)=\varphi(gk)$.


Left translation $T\_x$ by $x$ stabilizes that collection of functions, because left and right translation commute: $T\_xR\_k\varphi=R\_kT\_x\varphi$.


