On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂

  title={On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂},
  author={Stefan Dawydiak},
  journal={arXiv: Representation Theory},
  • Stefan Dawydiak
  • Published 4 October 2018
  • Mathematics
  • arXiv: Representation Theory
Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis $\{t_w\}$ of $J$, and further invert the… 
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