Representations of the groups GH,H/K,n

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 June 2010

Representations of the groups GH,H/K,n

The irreducible representations of the group GH,1,n ca be derived from Clifford theory or via the tower 1⊆G12⊆G1⊆G32⊆G2⊆G52⊆G3⊆…, where Gn=GH,1,n=Hn⋊Sn  and  Gn+12=GH,1,n×GH,1,1, so that Gn+12 is a Levi "subgroup" of Gn+1. The tower of G^ has

1. vertices on level n: mutlipartitions λ=λαα∈H^ with n boxes total,
2. vertices on level n+12: pairs λ□α, where λ∈G^n,α∈H^.
3. edges from level n to level n+12:λ→λ□α for each α∈H^.
4. edges from level n+12 to level n:μ□α→ν if ν is obtained from μ by adding a box to μα.

For each 0≤m≤r-1 the elements tim,1≤i≤n, form a conjugacy class in Gr1n and the elements timtj-mij,1≤i<j≤n,0≤m≤r-1, form another conjugacy class in Gr1n. Thus the elements zsm=∑i=1ntim,0≤m≤n-1,  and  zl=1r∑m=0r-1∑1≤i<j≤ntimtj-mij, are elements of ZℂGr1n. So zsm and zl must act by a constant on any irreducible representation Sλ of Gr1n. Define x1=0, xk=∑1≤i<j≤k0≤l≤r-1tiltj-lij-∑1≤i<j≤k-10≤l≤r-1tiltj-lij=1r∑1≤i<k0≤l≤r-1tiltk-lik,for  2≤k≤n and yk=∑i=1kti-∑i=1k-1ti=tk,for  1≤k≤n.

The elements x1,…,xn and y1,…,yn all commute with each other and the action of these elements on the irreducible representation Sλ of Gr1n is given by ykvT=sTkvT  and  xkvT=cTkvT, for all standard tableaux T.

		Proof.
	The proof is via induction on k using the relations xk=skxk-1sk+∑l=0r-1yk-1lskyk-1-l  and  yk=skyk-1sk. The base cases x1vT=0=cT1vT  and  y1vT=sT1vT are immediate from the defintions. Then ykvT=skyk-1sk=sksTk-1skTT+1+skTTsTkvskT=sksTkskvT+sTk-1-sTkskTTvT=sTkvT+0=sTkvT, and xkvT=skxk-1sk+1r∑l=0r-1skyk-1-lyklvT=skcTk-1skTTvT+cTk1+skTTvskT+1r∑l=0r-1sTk-lsTklvT=skcTkskvT+cTk-1-cTkskTTvT+1r∑l=0r-1sTk-1sTklvT=skcTkskvT+-1+1vT,ifsTk=sTk-1,skcTkskvT+0+0vT,ifsTk≠sTk-1,=cTkvT.□

Define an action of H on H^ by hα=χh⊗α, and extend this to an action of H on the simple GH,1,n modules by mP⊗vT↦mhP⊗vhT.

1. The simple GH,H/K,n modules are indexed by pairs λ_μ where λ_∈K\G^n,μ∈K^n, where Kλ is the stabiliser of λ in K.
2. The simple GH,H/K,n module, for any fixed representative λ of the coset λ_, GK,nλ_μ=pμGnλ,where  pμ=∑k∈Kλχμk-1k, is the minimal idempotent of Kλ corresponding to the module Kλμ.
3. As a GH,H/K,nKλ bimodule Gλ=⊗μ∈K^λGnλμ⊗Kλμ.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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