# 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}
}
• 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…
1 Citations
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

## References

SHOWING 1-10 OF 11 REFERENCES

Introduction Coxeter groups Partial order on $W$ The algebra ${\mathcal H}$ The bar operator The elements $c_w$ Left or right multiplication by $c_s$ Dihedral groups Cells Cosets of parabolic
• Mathematics
• 1979
here l(w) is the length of w. In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions
• Mathematics
• 2018
Let G be a split reductive p-adic group, I ⊂ G be an Iwahori subgroup, H(G) be the Hecke algebra and C(G) ⊃ H(G) be the Harish-Chandra Schwartz algebra. The purpose of this note is to define (in
• Mathematics
• 1998
Abstract. Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this
It is shown that, for a class of graphs X including all vertex-transitive graphs, if perfect state transfer occurs at time τ , then H(τ) is a scalar multiple of a permutation matrix of order two with no fixed points.

• 2018

### Periodic W -graphs

• Represent. Theory
• 1997

• 2018

### Advanced studies in pure math

• Cells in affine Weyl groups
• 1985