Let $G$ be a connected, reductive group over a $p$-adic field $F$, which is unramified in the sense that $G$ is quasisplit to split over an unramified extension of $F$. Then $G$ is necessarily obtained from base change of a smooth reductive group scheme $H$ over $\mathcal O_F$ (meaning $H$ is an affine flat group scheme of finite type over $\mathcal O_F$, and $H \times_{\mathcal O_F} \mathcal O_F/\mathfrak p_F$ is a connected, reductive group over $\mathcal O_F/\mathfrak p_F$), and $G(\mathcal O_F) := H(\mathcal O_F)$ is a maximal hyperspecial compact subgroup of $K := G(\mathcal O_F)$.
The following is from Casselman's notes on the L-group (https://www.math.ubc.ca/~cass/research/pdf/miyake.pdf). The Hecke algebra $\mathscr H$ is the algebra of functions $f: G(F) \rightarrow \mathbb{C}$ of compact support which are bi $K$-invariant.
I have a few questions on this. I would be grateful if anyone could answer one or more of them.
Here are my questions so far.
1 . Convolution is defined by
$$f_1 \ast f_2(x) = \int\limits_{G(F)} f_1(y)f_2(xy^{-1})dy$$
I don't see how right convolution by elements in the Hecke algebra preserves the given property. I have been trying to use the Iwasawa decomposition $dg = db dk$ but so far I have been unable to get anywhere.
2 . Let $N_w^+ = wNw^{-1} \cap N$. This group is generated by those root subgroups which remain positive after $w^{-1}$ is applied. I don't see how the given integral over $N_w^+ \setminus N$ is well defined. I don't believe that $\phi_{\chi}(w^{-1}ng)$ as a function of $n$ is well defined on the coset space. If instead we use the function $n \mapsto \phi_{\chi}(w^{-1}ng)\chi(n)^{-1}$, then this is well defined: if we replace $n$ by $n_0n$ for some $n_0 \in N_w^+$, then
$$\phi_{\chi}(w^{-1}n_0ng)\chi(n_0n)^{-1} = \phi_{\chi}(w^{-1}n_0ww^{-1}ng) \chi(n_0n)^{-1}$$
Now write $\phi_{\chi}(w^{-1}n_0ww^{-1}ng) = \chi(w^{-1}n_0w)\delta_B(w^{-1}n_0w)^{1/2}\phi_{\chi}(w^{-1}ng)$. The modular character should be trivial on $N$. And some nice choice of representative $w$ will allow us to say that $\chi(w^{-1}n_0w) = \chi(n_0)$, and then this cancels with $\chi(n_0)^{-1}$.
So maybe that integration over $N_w \setminus N$ should be with the function $n \mapsto \phi_{\chi}(w^{-1}ng) \chi(n)^{-1}$.
3 . What exactly are the "simple inequalities" $\chi$ must satisfy so that the given integral converges?
4 . How can one show that the condition
$$\tau_w \phi_{\chi}(bk) = w\chi(b)\delta_B(b)^{1/2} \tau_w \phi_{\chi}(1)$$
holds?
I'm quite new at these things, and posting this mostly as an answer as it might become too big for a comment. I think your Haar measure on $G(F)$ is in particular a right Haar measure, and then (1) is the computation $$ (f*\varphi)(bgk)=\int_{G(F)}f(\gamma)\varphi(bgk\gamma^{-1})\,d\gamma= \int_{G(F)}f(\gamma k)\varphi(bgkk^{-1}\gamma^{-1})\,d\gamma = \int_{G(F)}f(\gamma)\varphi(bg\gamma^{-1})\,d\gamma, $$ where the change of variables is $\gamma\mapsto \gamma k$. This is a general fact: the functions $\varphi$ are by defintion $K$-fixed vectors in a principal series representation, and $K$-fixed vectors are always a module over the spherical Hecke algebra (in fact, convolution with any vector is projection onto this space, and this works for any open $K$ at all).
I think (2) is about intertwining operators between principal series representations. I have seen these defined by similar integrals but just over what you call $N_w^+$ in e.g. this paper.
I'm not sure about the rest. As you're probably aware, for intertwiners between principal series for $\mathrm{SL}(2,\mathbb{R})$, there is a similar integral depending on the character chosen to induce from. This is a complex number $s$, and the condition is something like $\Re(s)>0$. The paper I linked talks gives some formulas for $\mathrm{SL}_2(F)$, and I think the situation is analogous.