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I'm reading through this paper by Pouchin, and in it he makes a clam that some function space is isomorphic to the Hecke algebra. I'm trying to understand this and could really use some help.

Let $G = GL(n,\mathbb{F}_q)$ and let $B$ be the subgroup of upper triangular matrices. Let $X = G/B$ and let $\mathbb{C}_G(X \times X)$ be the space of G-invariant functions $f:X \times X \to \mathbb{C}$. Define a convolution product $(f \ast g)(F,F'') = \sum_{F' \in X} f(F,F') g(F',F'')$. I'm trying to show that under this convolution product, $\mathbb{C}_G(X \times X)$ is isomorphic to the Hecke algebra.

I've managed to show a couple things, including the $G$ orbits are parameterized by $S_n$, and (denoting the orbit corresponding to $\omega \in S_n$ as $\mathcal{O}_\omega$ and its characteristic function as $1_\omega$) that the orbit characteristic functions $\{1_\omega : \omega \in S_n \}$ form a basis for $\mathbb{C}_G(X\times X)$. Currently I'm working on showing $1_\omega \ast 1_\sigma = 1_{\omega \sigma}$ if $l(\omega\sigma) = l(\omega) + l(\tau)$ by showing that this is the case for $l(\omega s_i)$ for $s_i$ the transposition $(i \hspace{4mm}i+1)$, which is where I'm having a lot of difficulty.

I've found this paper by Daniel Bump where he proves this isomorphism (and the identity $1_\omega \ast 1_{s_i} = 1_{\omega s_i}$ with a different construction of the Hecke algebra. He constructs it as the space of $B$-bi-invariant functions from $G$ to $\mathbb{C}$, or equivalently the space of $\mathbb{C}$ valued functions over the double cosets $B \omega B$. I'm hoping that I can use the fact that this space and the orbits of $X \times X$ are both parameterized by $S_n$ to show that the Hecke algebra as the convolution algebra from the space of functions over $X \times X$ is isomorphic to this construction of the Hecke algebra, and that then Bump's paper will help me the rest of the way.

Could someone please help me show that the Hecke algebra constructed as the space of $G$-invariant functions from $X \times X \to \mathbb{C}$ is isomorphic to the Hecke algebra as constructed as the space of $B$-bi-invariant functions from $G \to \mathbb{C}$? Alternatively, could someone help me show $1_\omega \ast 1_{s_i} = 1_{\omega s_i}$ if $l(\omega s_i) = l(\omega) + 1$? It would be much appreciated.

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$G$-invariant functions $ X \times X$ to $\mathbb C$ are the same thing as functions $G \times G$ to $\mathbb C$ invariant under the right action of $B \times B$ and the left, diagonal action of $G$.

But functions invariant under the left diagonal action of $G$ have the form $f(g, h) = f' ( g^{-1} h)$ for a unique function $f'$ on $G$. This function $f'$ is left and right $B$-invariant if and only if $f$ is invariant under the right action of $B \times B$.

Up to some normalizing constants, this should give you the isomorphism you're looking for.

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