# Iwahori-Hecke algebra of $GL_2$

I had make an old post about this, but since seeing it now shows that I had no idea how to explain my question (because I didn't understand it) I'll try to do it normally here.

So I am studying this paper https://arxiv.org/pdf/math/0309168.pdf on Iwahori-Hecke algebras, and I am trying to apply this in the group $G=GL_2(\mathbf{Q}_p)$. Setting $K=GL_2(\mathbf{Z}_p)$ and $I=K\cap B$ where $B$ is the Borel subgroup of upper triangular matrices, I would like to understand two algebras: the algebra $H=C_c(I \setminus G / I)$ and the algebra $H_K=C_c(K\backslash G / K).$ The confusion in the previous post came from the fact that they both mainly go under the name "the Hecke algebra". Now, for $GL_2$, from high-tech things such as the Satake isomorphism we can see that $H_K\cong Z(H)\cong \mathbf{C}[s_1^{\pm},s_2^{\pm}].$ But I am also curious what $H$ exactly is in this case.

My thought is: Using lemma 1.7.1, it is enough to find just $R,H_0$. Now $H_0$ is $C_c(I \setminus K / Ι)$. These double cosets seem like they should be just $I, wI$ where $w$ is the permutation matrix (the nontrivial element of the Weyl group). So $H_0=\mathbf{C}^2$, since complex functions in these two elements are just pairs of complex numbers. $R=\mathbf{C}[Χ_*]$, where $X_*$ are essentially the elements $\begin{pmatrix}p^m & 0\\ 0 & p^n \end{pmatrix},$ so we can identify $R$ with $\mathbf{C}[s_1^{\pm},s_2^{\pm}]$, where the coefficient of the $z_1^mz_2^n$ term corresponds to the value of the function in $\begin{pmatrix}p^m & 0\\ 0 & p^n \end{pmatrix}$.

Now the lemma should give $H=R\otimes H_0\cong \mathbf{C}^2[s_1^{\pm},s_2^{\pm}]\cong (\mathbf{C}[s_1^{\pm},s_2^{\pm}])^2,$ but this is absurd since this algebra is cmmutative and we know that the center will be $H_K$. My impression would be that $H$ should something similar to what we found, but maybe something like $H=\mathbf{C}[s_1^{\pm},s_2^{\pm}]\rtimes S_2$ where $S_2$ acts by permuting the variables, which would imply that the center is what we want, and is kind of similar to $(\mathbf{C}[s_1^{\pm},s_2^{\pm}])^2$. The problem is that although I feel that my above argument was messy, I can't pinpoint the exact mistake, which may be obvious.

Could you tell me my mistakes in the above, and also help me calculate $H$?

The Hecke algebra $$C_c(I\backslash G/I)$$ is calculated in Bushnell-Henniart The Local Langlands Conjecture for GL(2) Section 17.2.