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.

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