# On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂

@article{Dawydiak2018OnLA, title={On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂}, author={Stefan Dawydiak}, journal={arXiv: Representation Theory}, year={2018} }

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…

## One Citation

### Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula

- Mathematics
- 2021

Let W̃ be an extended affine Weyl group, H be the corresponding affine Hecke algebra over the ring C[q,q], and J be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis {tw}. Viewing…

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