Consider cell decomposition of flag variety of $\mathbb{H}^{n}$ into orbits under the action of $B(\mathbb{H})$ - group of upper triangular matrices with coefficients in $\mathbb{H}$. I think cells should have the same structure and dimension over $\mathbb{H}$ as it was in the case of $\mathbb{C}$. But what happens to the corresponding Hecke algebra? Is there something known about it?
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