Let $G$ be a unimodular totally disconnected topological group (for instance, the $F$-points of a reductive group over a nonarchimedean local field). Fix a Haar measure $dx$ on $G$, normalized such that $mu(G)=1$. Then one can form the Hecke algebra $H:=H(G)$ to be defined as the convolution algebra of locally constant functions on $G$, where convolution is given by $$f_1star f_2(g) =int_G f_1(x)f_2(x^{-1}g) dx.$$

Note that this algebra is associative, noncommutative, and nonunital. There are smaller rings inside, denoted $H(G)_Ksubset H(G)$, where $K$ is compact open, consisting of $K$-biinvariant functions. Each $H(G)_K$ has units given by $e_K:=text{vol}(K)^{-1}1_K$ where $1_K$ is the characteristic function for $K$. An $H$-module is a $mathbb{C}$-algebra homomorphism $pi: Hto text{End}(V)$. Further, we say that such an $H$-module is **non-degenerate** if for all $vin V$ there exists compact open $K$ such that $e_K(v)=v$ where $e_K(v)$ refers to the action of $e_K$ on $v$.

It is well-known that there is a categorical equivalence between non-degenerate $H$-modules and smooth representations of $G$. For instance, see Bushnell-Henniart. I am trying to realize this correspondence in a more explicitly way. Bushnell-Henniart remarks that given a non-degenerate $H$-module $V$, we can construct a smooth representation of $G$ by taking choosing compact open $K$ such that $e_Kv=v$ and then setting $pi(g)(v):=text{vol}(K)^{-1}1_{gK}(v)$ where the right hand side indicates the action of $H$ on $V$. My confusion is that I do not see how this is a well-defined action. First:

(i) Given $gin G$, Why does $pi(g)(v)$ not depend on the choice of $K$?

(ii) How does one see that $pi(g_1g_2)(v)=pi(g_1)pi(g_2)(v)$?

For (i), certainly this is clear if $g=e$, and the general statement should be similar. But I do not see how to make this work. For (ii), it is clear that one must choose $K$ to be sufficiently small, but the details elude me.

Any and all remarks would be appreciated.