show/hide this revision's text 2 normalization corrected

I am not sure, if you are still interested in this, but here is the general computation:

Let $\phi \in C_c^\infty (GL_n(F))$, and let $\pi$ be a super-cuspidal representation of a Levi subgroup $M$ of a parabolic $P$ with unipotent radical $N$, and let $\pi_0 = Ind_{P}^{GL_n(F)} \pi$ be the normalized induced representation (assume unitary, irreducible for safety). Let $K$ be compact open subgroup with $GL_n(F) = P K$.

  1. Define $ \phi^K (x) = \int\limits_K \phi(k^{-1}xk) d k,$
  2. Define $ A\phi^K(m) = \Delta_P(m)^{-1/2} Delta_P(m)^{1/2} \int\limits_{N} \phi^K(mn)d n$ for $m \in M$
  3. Then we have that $A\phi^K \in C_c^\infty(M)$ and the formula $$ tr \pi_0(\phi) = tr \pi( A \phi^K).$$

The same formula is also useful, if the representation is not irreducible (unitarizability is not really an issue, and admissibility follows from the Iwasawa decomposition), but one has to normalize and decompose according to the $K$-isotypes.

So in your situation, you get a Fourier transform of $A \phi^K$. This in addition with Moshe Adrian answer computes all the irreducible, unitary principal series representation, at least in prinicple.

show/hide this revision's text 1

I am not sure, if you are still interested in this, but here is the general computation:

Let $\phi \in C_c^\infty (GL_n(F))$, and let $\pi$ be a super-cuspidal representation of a Levi subgroup $M$ of a parabolic $P$ with unipotent radical $N$, and let $\pi_0 = Ind_{P}^{GL_n(F)} \pi$ be the induced representation (assume unitary, irreducible for safety). Let $K$ be compact open subgroup with $GL_n(F) = P K$.

  1. Define $ \phi^K (x) = \int\limits_K \phi(k^{-1}xk) d k,$
  2. Define $ A\phi^K(m) = \Delta_P(m)^{-1/2} \int\limits_{N} \phi^K(mn)d n$ for $m \in M$
  3. Then we have that $A\phi^K \in C_c^\infty(M)$ and the formula $$ tr \pi_0(\phi) = tr \pi( A \phi^K).$$

The same formula is also useful, if the representation is not irreducible (unitarizability is not really an issue, and admissibility follows from the Iwasawa decomposition), but one has to normalize and decompose according to the $K$-isotypes.

So in your situation, you get a Fourier transform of $A \phi^K$. This in addition with Moshe Adrian answer computes all the irreducible, unitary principal series representation, at least in prinicple.