Finite Hecke algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 23 December 2012

Weylish presentation

Let q be an indeterminate and let 𝕂 =[q, q-1]. The Hecke algebra H0 is presented by generators Tw, wW0, and relations

Tw1 Tw2= Tw1w2, if(w1w2) =(w1)+ (w2), TsiTw= (q-q-1)Tw+ Tsiw, if(siw)< (w) (0in).

The conversion between the two presentations is given by the relations

Tw= Tsi1 Tsi, if wW0 and w=si1 si is a reduced word, (1.22)

Coxeterish presentation

Let t12 be an indeterminate and let 𝕂 =[ ti 12 , ti-12 ]. HOW SHOULD WE DEAL WITH THE ISSUE OF MULTIPLE PARAMETERS--PERHAPS AN EXERCISE??
The finite Hecke algebra H0 is the algebra over 𝕂 given by generators Ts1, Ts2, , Tsn and relations Tsi2 =( ti 12 - ti-12 ) Tsi+1, and Tsi Tsj Tsi mij factors = Tsj Tsi Tsj mij factors for ij, where π mij = 𝔥αi𝔥αj is the angle between 𝔥αi and 𝔥αj.

The algebra H0 has 𝕂-basis { Tw | wW0}.

Convolution algebra presentation

Let 𝔽q 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 W0=N/T, where N={gG | gTg-1 =T} is the normalizer of T in G.

(a) Let wW0. Then BwB BsjB = { BwsjB, if wsj> w, BwB BwsjB, if wsj< w, (b) Bruhat decomposition: G= wW0 BwB. (c) The characteristic functions {Tw | wW} of the double cosets BwB are a basis of the Hecke algebra H= C(B\G/B) and Tw Tsj = { Twsj, ifwsj >w, q Twsj + (q-1)Tw, ifwsj <w.

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 ?????.



page history