As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:


For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient representation of $\pi$, there is a canonical pairing $B:\pi \times \pi ^\vee \to \mathbb{C}$ such that $B(v,w):=w(v)$.


Then the matrix coefficient of $\pi$ is a function of $G$ defined like this;


$\{B(gv,w):G \to \mathbb{C}\}\_{v\in \pi, w\in \pi ^\vee}$.


But some papers seems to define matrix coefficient not using this canonical pairing but some inner product of $\pi$. 


For example, in the global Gan-Gross-Prasad conjecture tells about tempered representations, that is tempered at all places.


In Ichino-Ikeda's paper 'On the periods of automorphic form and Gross-Prasad conjecture',or Neal Harris's paper 'The refined Gross-Prasad conjecture on unitary group' they define the notion of matrix coefficient of $\pi\_v$, the local represenation of global automorphic representation $\pi$, as the $<g\cdot v,w>\_{v,w\in \pi}$ where $<,>$ is some inner product of $\pi\_v$. And then they define the notion 'temperedness' when all the matrix coefficient are $L^{2+\epsilon}(G)$. But obviously, this definition of temperedness seems depend on the choice of inner product $<,>$ and it seems different with the usual tempered notion, because the usual tempered notion concerns the canonical inner product of $\pi\_v$ and $\pi^{\vee}\_v$.


Is they are the same?


I am also wondering that if a given representation $\pi$ is unitary, can we use hermitian form as the natural local inner product of $V$ and define matrix coefficient using it?


Please deliver me from this confusing.


Thank you for reading my question. 



On the definition of matrix coefficient
Given a representation π you can define the complex conjugate representation by taking the same vector space on which the new mult by a complex number is the old multiplication by the conjugate of the complex number. An invariant Hermitian form on π gives an identification of the contragredient of π with the complex conjugate of π. Therefore, the two matrix coefficients that you have defined agree. 


(This was initially given as a comment; since Yemen Choi suggested that it be given as an answer, I do so! 


