I'm reading Wallach's paper 'Asymptotic expansions of generalized matrix entries of representations of real reductive groups'(Lecture Notes in Math., 1024,287–369) and got confused by one statement there on asymptotics of Whittaker functions. Near the bottom of page 361, there is an inequality without proof $$|f_{\lambda,v}(x)|\le C_{n,v}x^{-n} for \ \ all \ \ x\ge 1 \ \ and \ \ all \ \ n=1,2,...$$
where $\lambda$ is the Whittaker functional on an irreducible generic smooth representation $\pi$, $v$ is a vector in $\pi$, $f_{\lambda,v}(e^t)=\lambda(\pi( exp \ \ tH)v)$, $H\in \mathfrak{a}$ and the smallest eigenvalue of $ad H$ on $\mathfrak{n}$ is 1 (see on top of page 358 of that paper), here $(P,A)$ is a standard parabolic pair with unipotent radical $N$ and $\mathfrak{a},\mathfrak{n}$ are Lie algebras of $A,N$ respectively.

I couldn't figure out this simple fact. Any help or reference are appreciated in advance.


Not sure about the level of generality in Wallach's paper, but if that Whittaker function is given by a Jacquet integral (Multiplicity One for Whittaker functions), you can integrate by parts many times and would give you the result.


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