Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular characters
Let $\Delta_B$ be the modulus-character of Borel subgroup , and $B(F)$. We define $$ H (bk) = \Delta_B(b), \qquad b \in B(F), \; k \in G(\mathfrak{o}).$$
Let $I$ be the Iwahori, how can we compute for $w \in G(F)$ the value $$ \int\limits_{I} \int\limits_{I} H(i_1 w i_2) d i_1 d i_2$$ i_2 = \int\limits_{I} H(i_1 w) d i_1 $$ or the value $$ \int\limits_{G(o)} \int\limits_{G(o)} H(k_1 w k_2) d k_1 d k_2?$$k_2 = \int\limits_{G(o)} H(k_1 w) d k_1 ?$$
In the case $GL(2)$, there is the Iwahori decomposition, which becomes probably useful to compute this integral. Is this perhaps related to the constant $D$, which turns up in front of the constant term/Abel transform ($D=$determinant of $ad( Lie(G)) -1$ acting on $Lie(G)/Lie(T)$)?
Similar things can be asked for reductive Lie groups with the maximal compact subgroup instead $G(o)$. What is the answer there?
