Principal series component of Gelfand-Graev representation

Let $G$ be a connected reductive group defined over non- archimedean local field $F$. $B$ minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. $\psi$ non-degenerate character of $U(F)$ $\lambda$ $T(F)$. $(K,\rho )$ Bushnell-Kutzko type associated to the Bernstein block $G(F)$ determined by pair $(T,\lambda )$. We study $\rho$-isotypical component $(c\text {-ind}_{U(F)}^{G(F)}\psi )^{\rho }$ induced space $c\text {-ind}_{U(F)}^{G(F)}\psi$ functions compactly supported mod $U(F)$. show that is cyclic module for Hecke algebra $\mathcal {H}(G,\rho When split, we describe it more explicitly in terms make assumptions on residue characteristic $F$ later also center depending Our results generalize main result Chan Savin [Math. Z. 288 (2018), pp. 125â??133] who treated case $\lambda =1$ split.