Fusion process and representations of affine Hecke algebras

82 Pages·2009·0.64 MB·English

by  Paolo Papi

Fusion process and representations of affine Hecke
algebras
Paolo Papi
SapienzaUniversit`adiRoma
joint work with V. Guizzi and M. Nazarov
PaoloPapi FusionprocessandaffineHeckealgebras
Upshot
We want to provide a detailed account of a combinatorial
construction (after Cherednik) which exhibits the irreducible (finite
dimensional) modules of the affine Hecke algebra of GL as cyclic
n
modules over the Hecke algebra of GL .
n
The main theorem we are going to discuss has been (roughly)
stated by Cherednik without proof.
PaoloPapi FusionprocessandaffineHeckealgebras
Affine Hecke algebras: Bernstein presentation
The affine Hecke algebra H(cid:101)n (of GLn) is the C(q)-algebra with the
following presentation
PaoloPapi FusionprocessandaffineHeckealgebras
Affine Hecke algebras: Bernstein presentation
Generators for H(cid:101)n
T ,...,T
1 n−1
X±1,...,X±1
1 n
Relations H(cid:101)n
T T T = T T T , 1 ≤ k ≤ n−2,
k k+1 k k+1 k k+1
T T = T T , 1 < |k −l|,
k l l k
(T −q)(T +1) = 0, 1 ≤ k ≤ n−1,
k k
X X = X X , 1 ≤ k,l ≤ n,
k l l k
X T = T X , l (cid:54)= k,k +1,
l k k l
T X T = qX , 1 ≤ k ≤ n−1.
k k k k+1
PaoloPapi FusionprocessandaffineHeckealgebras
Affine Hecke algebras: Bernstein presentation
The “finite” Hecke algebra Hn is the subalgebra of H(cid:101)n
generated by the T ; it has the following presentation, which shows
i
it as a q deformation of C[S ].
n
PaoloPapi FusionprocessandaffineHeckealgebras
Affine Hecke algebras: Bernstein presentation
Generators for H
n
T ,...,T
1 n−1
Relations H
n
T T T = T T T , 1 ≤ k ≤ n−2,
k k+1 k k+1 k k+1
T T = T T , 1 < |k −l|,
k l l k
(T −q)(T +1) = 0.
k k
PaoloPapi FusionprocessandaffineHeckealgebras
The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n.
H(cid:101)n = Hn ⊗An as vector spaces
Theorem
The center of H(cid:101)n consists precisely of symmetric polynomials in
X±1,...,X±1.
1 n
Corollary
Any irreducible H(cid:101)n-module is finite dimensional
Remarks
Recall the defining relations or H(cid:101)n:
T T T =T T T , T T =TT ,
k k+1 k k+1 k k+1 k l l k
(T −q)(T +1)=0, X X =XX ,
k k k l l k
XT =T X, T X T =qX
l k k l k k k k+1
PaoloPapi FusionprocessandaffineHeckealgebras
Theorem
The center of H(cid:101)n consists precisely of symmetric polynomials in
X±1,...,X±1.
1 n
Corollary
Any irreducible H(cid:101)n-module is finite dimensional
Remarks
T T T =T T T , T T =TT ,
k k+1 k k+1 k k+1 k l l k
(T −q)(T +1)=0, X X =XX ,
k k k l l k
XT =T X, T X T =qX
l k k l k k k k+1
The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n.
H(cid:101)n = Hn ⊗An as vector spaces
PaoloPapi FusionprocessandaffineHeckealgebras
Corollary
Any irreducible H(cid:101)n-module is finite dimensional
Remarks
T T T =T T T , T T =TT ,
k k+1 k k+1 k k+1 k l l k
(T −q)(T +1)=0, X X =XX ,
k k k l l k
XT =T X, T X T =qX
l k k l k k k k+1
The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n.
H(cid:101)n = Hn ⊗An as vector spaces
Theorem
The center of H(cid:101)n consists precisely of symmetric polynomials in
X±1,...,X±1.
1 n
PaoloPapi FusionprocessandaffineHeckealgebras
Remarks
T T T =T T T , T T =TT ,
k k+1 k k+1 k k+1 k l l k
(T −q)(T +1)=0, X X =XX ,
k k k l l k
XT =T X, T X T =qX
l k k l k k k k+1
The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n.
H(cid:101)n = Hn ⊗An as vector spaces
Theorem
The center of H(cid:101)n consists precisely of symmetric polynomials in
X±1,...,X±1.
1 n
Corollary
Any irreducible H(cid:101)n-module is finite dimensional
PaoloPapi FusionprocessandaffineHeckealgebras