## Finite Hecke algebras

Last update: 23 December 2012

## Weylish presentation

Let $q$ be an indeterminate and let $𝕂=ℤ\left[q,{q}^{-1}\right]$. The Hecke algebra ${H}_{0}$ is presented by generators ${T}_{w}$, $w\in {W}_{0}$, and relations

$Tw1 Tw2= Tw1w2, ifℓ(w1w2) =ℓ(w1)+ ℓ(w2), TsiTw= (q-q-1)Tw+ Tsiw, ifℓ(siw)< ℓ(w) (0≤i≤n).$

The conversion between the two presentations is given by the relations

$Tw= Tsi1… Tsiℓ, if w∈W0 and w=si1… siℓ is a reduced word, (1.22)$

## Coxeterish presentation

Let ${t}^{\frac{1}{2}}$ be an indeterminate and let $𝕂=ℤ\left[{t}_{i}^{\frac{1}{2}},{t}_{i}^{-\frac{1}{2}}\right]$. HOW SHOULD WE DEAL WITH THE ISSUE OF MULTIPLE PARAMETERS--PERHAPS AN EXERCISE??
The finite Hecke algebra ${H}_{0}$ is the algebra over $𝕂$ given by generators ${T}_{{s}_{1}},{T}_{{s}_{2}},\dots ,{T}_{{s}_{n}}$ and relations $Tsi2 =( ti 12 - ti-12 ) Tsi+1, and Tsi Tsj Tsi⋯ ⏟ mij factors = Tsj Tsi Tsj⋯ ⏟ mij factors for i≠j,$ where $\frac{\pi }{{m}_{ij}}={𝔥}^{{\alpha }_{i}}\measuredangle {𝔥}^{{\alpha }_{j}}$ is the angle between ${𝔥}^{{\alpha }_{i}}$ and ${𝔥}^{{\alpha }_{j}}$.

The algebra ${H}_{0}$ has $𝕂$-basis $\left\{{T}_{w}\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}w\in {W}_{0}\right\}$.

## Convolution algebra presentation

Let ${𝔽}_{q}$ be a finite field with $q$ elements, $G =G( 𝔽q) a finite Chevalley group over 𝔽q ∪| Ba Borel subgroup ∪| T a maximal torus.$ The Weyl group of $G$ is ${W}_{0}=N/T,whereN=\left\{g\in G\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}gT{g}^{-1}=T\right\}is the normalizer ofTinG.$

(a) Let $w\in {W}_{0}$. Then $BwB ⋅ BsjB = { BwsjB, if wsj> w, BwB ∪ BwsjB, if wsj< w,$ (b) Bruhat decomposition: $G= ⨆w∈W0 BwB.$ (c) The characteristic functions $\left\{{T}_{w}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}w\in W\right\}$ of the double cosets $BwB$ are a basis of the Hecke algebra $H=C\left(B\G/B\right)$ and $Tw Tsj = { Twsj, ifwsj >w, q Twsj + (q-1)Tw, ifwsj

For the moment, we refer to affflags1.14.07.pdf for the proof.

## Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. These important facts about Iwahori-Hecke algebras are found in Bourbaki ????. The original papers are [Iw] Iwahori ????, and [IM] Iwahori-Matsumoto ????. One can also see Steinberg Lecture notes ?????.

References?