Finite Hecke algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 December 2012

Weylish presentation

Let $q$ be an indeterminate and let $𝕂=\mathbb{Z}[q,q^{-1}]$. The Hecke algebra $H_0$ is presented by generators $T_w$, $w\in W_0$, and relations

$$\begin{array}{cc}{T}_{w_1}{T}_{w_2}=T_{w_1{w}_2},& \text{if}\ell \left(w_1{w}_2\right)=\ell \left(w_1\right)+\ell \left(w_2\right),\\ T_{s_i}{T}_w=(q-q^{-1})T_w+T_{s_{i}w},& \text{if}\ell \left(s_{i}w\right)<\ell \left(w\right)(0\le i\le n)\text{.}\end{array}$$

The conversion between the two presentations is given by the relations

$$\begin{array}{cc}{T}_w=T_{s_{i_1}}\dots T_{s_{i_{\ell }}},\text{if}w\in W_0\text{and}w=s_{i_1}\dots s_{i_{\ell }}\text{is a reduced word,}& \text{(1.22)}\end{array}$$

Coxeterish presentation

Let $t^{\frac{1}{2}}$ be an indeterminate and let $𝕂=\mathbb{Z}[t_i^{\frac{1}{2}},t_i^{-\frac{1}{2}}]$. 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 $$T_{s_i}^2=(t_i^{\frac{1}{2}}-t_i^{-\frac{1}{2}})T_{s_i}+1,and{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_i}{T}_{s_j}{T}_{s_i}\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_j}{T}_{s_i}{T}_{s_j}\cdots }}}\text{for}i\ne j,$$ 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 $\left\{T_w\right|w\in W_0\}$.

Convolution algebra presentation

Let ${𝔽}_q$ be a finite field with $q$ elements, The Weyl group of $G$ is $W_0=N/T,$ where$ N=\{g\in G|gT{g}^{-1}=T\}$is\; the\; normalizer\; of$ T$in$ G$.$$

(a) Let $w\in W_0$. Then $$BwB\cdot B{s}_{j}B=\{\begin{array}{cc}Bw{s}_{j}B,\hfill & \text{if}w{s}_j>w,\hfill \\ BwB\cup Bw{s}_{j}B,\hfill & \text{if}w{s}_j<w,\hfill \end{array}$$ (b) Bruhat decomposition: $$G=\underset{w\in W_0}{⨆}BwB.$$ (c) The characteristic functions $\left\{T_w\right|w\in W\}$ of the double cosets $BwB$ are a basis of the Hecke algebra $H=C\left(B\backslash G/B\right)$ and $$T_w{T}_{s_j}=\{\begin{array}{ll}{T}_{w{s}_j},& \text{if}w{s}_j>w,\\ q{T}_{w{s}_j}+(q-1)T_w,& \text{if}w{s}_j<w.\end{array}$$

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

References?

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