x INTRODUCTIO N
These notes adopt th e view that th e Iwahori-Hecke algebra s rather tha n the
g-Schur algebra s are the objects o f central importance . Thi s i s partly a matte r
of persona l tast e an d partl y expedience ; othe r authors , suc h a s Donki n [48 ] and
Green [74] , travel i n the reverse direction . On e consequence o f our perspective i s
that ou r definition o f the Iwahori-Hecke algebr a ma y strike som e reader s a s being
contrived; t o remedy thi s w e now provide additiona l motivation .
First recal l that, a s an abstract group , the symmetric grou p has a presentatio n
with generator s s±,.. ., s
n
-i an d relation s
si = 1, fo r i = 1, 2,... , n - 1,
SiSj = SjSi, fo r 1 i j 1 n 2,
SiSi+iSi = Si+iSiSi
+
i, fo r i = 1, 2,... , n - 2 .
Identifying Si with th e transposition (z, i + 1) shows tha t &
n
i s a quotien t o f the
group W wit h th e presentation above ; a little mor e wor k verifie s tha t W = & n.
Now fix a ring R and let q be an element of R. Th e Iwahori-Hecke algebra J$? =
J%R,q(&n) is the associative algebr a wit h generator s Xi,... , Tn_i an d relation s
(T
2
-g)(T
2
+ l ) = 0 , fo r i = 1,2,... , n - l ,
TtT3 = TjTi, fo r 1 i j - 1 n - 2 ,
T.T^T, =T
i +
iT
i
r
i
+i , fo r 2 = 1,2,... , n - 2 .
In particular , J^ 7 and i^6n ar e isomorphic whe n q = 1. Thus , J f i s a deformation
of the group rin g i26
n
o f the symmetric group ; that is , the Iwahori-Hecke algebra s
of 6
n
ar e a famil y o f algebra s whic h 'loo k like ' th e grou p rin g o f the symmetri c
group excep t tha t th e multiplication i s 'deformed' b y q.
Prom th e presentatio n o f ffl i t seem s likel y tha t M* will collaps e fo r som e
choices o f the parameter q\ in fact, w e will sho w i n chapter 1 that, independentl y
of q, the Iwahori-Hecke algebr a i s always a free i^-modul e o f rank n\ = \&
n
\. Th e
best explanatio n o f why the ran k o f Jif i s independent o f q is that th e Iwahori -
Hecke algebra s appea r naturall y i n the representation theor y o f the general linea r
groups; we now describe ho w this come s about .
Assume tha t q is a prim e powe r an d le t G = GL
n
(q) b e th e genera l linea r
group ove r th e field o f q elements . Le t B = B(q) b e a Bore l subgrou p o f G ;
thus, B is conjugate to the subgroup of upper triangular matrice s in G. Le t Ind
B
(l)
be th e induce d it!G-representatio n o n th e righ t coset s o f B i n G an d le t H
q
=
End^G (lnd
B
(l)) b e the endomorphis m algebr a o f thi s module . Amazingly , th e
algebras H
q
an d J4fR,q(Sn) ar e canonically isomorphic ; so, Jif i s also a deformatio n
of the endomorphism algebr a H
q
\
Here i s a rough proo f o f the isomorphism J4?R, q(£n) H q. Fo r any subset X
of G le t [X]
J2xexx' a n e
l
e m e n
t o f RG. The n Ind B(l) i s fre e a s a n RG-
module wit h basi s [Bg], where g run s ove r a se t o f righ t cose t representative s
of B i n G. Therefore , H
q
ha s basis [BgB], wher e g run s ove r th e (B,B)-doub\e
coset representatives . No w 6n i s the Weyl grou p o f G , s o G = \J
wee
BwB b y
the Bruha t decomposition ; her e we identify
n
wit h th e subgroup o f permutatio n
matrices i n G. Therefore , H
q
i s R-free wit h basi s { [BwB] \ w G 6n }.
As above , le t si,...,s
n
_ i b e generator s o f 6
n
an d writ e £(w) = k i f k i s
minimal suc h tha t w = Si
1
... Si
k
fo r som e 1 i
3
n. The n fo r al l w G (5n and
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