Let Wφ be a reflection coset on a vector space V and Lwφ a reflection subcoset where L is a parabolic subgroup (the fixator of a subspace of V). There are several interesting cases where the relative group NW(Lwφ)/L, or a subgroup of it normalizing some further data attached to L, is itself a reflection group.
A first example is the case where φ=1 and w=1, W is the Weyl group of a finite reductive group GF and the Levi subgroup LF corresponding to L has a cuspidal unipotent character. Then NW(L)/L is a Coxeter group acting on the space X(ZL)⊗ℝ. A combinatorial characterization of such parabolic subgroups of Coxeter groups is that they are normalized by the longest element of larger parabolic subgroups (see Lus76, 5.7.1).
A second example is when L is trivial and wφ is a ζ-regular element, that is the ζ-eigenspace Vζ of wφ contains a vector outside all the reflecting hyperplanes of W. Then NW(Lwφ)/L=CW(wφ) is a reflection group in its action on Vζ.
A similar but more general example is when Vζ is the ζ-eigenspace of some element of the reflection coset Wφ, and is of maximal dimension among such possible ζ-eigenspaces. Then the set of elements of Wφ which act by ζ on Vζ is a certain subcoset Lwφ, and NW(Lwφ)/L is a reflection group in its action on Vζ (see LS99, 2.5).
Finally, a still more general example, but which only occurs for Weyl groups or Spetsial reflection groups, is when L is a ζ-split Levi subgroup (which means that the corresponding subcoset Lwφ is formed of all the elements which act by ζ on some subspace Vζ of V), and λ is a d-cuspidal unipotent character of L (which means that the multiplicity of ζ as a root of the degree of λ is the same as the multiplicity of ζ as a root of the generic order of the semi-simple part of G); then NW(Lwφ,λ)/L is a complex reflection group in its action on Vζ.
Further, in the above cases the relative group describes the decomposition of a Lusztig induction.
When GF is a finite reductive group, and λ a cuspidal unipotent character of the Levi subgroup LF, then the GF-endomorphism algebra of the Harish-Chandra induced representation RLG(λ) is a Hecke algebra attached to the group NW(L)/L, thus the dimension of the characters of this group describe the multiplicities in the Harish-Chandra induced.
Similarly, when L is a ζ-split Levi subgroup, and λ is a d-cuspidal unipotent character of L then (conjecturally) the GF-endomorphism algebra of the Lusztig induced RLG(λ) is a cyclotomic Hecke algebra for to the group NW(Lwφ,λ)/L. The constituents of RLG(λ) are called a ζ-Harish-Chandra series. In the case of rational groups or cosets, corresponding to finite reductive groups, the conjugacy class of Lwφ depends only on the order d of ζ, so one also talks of d-Harish-Chandra series. These series correspond to l-blocks where l is a prime divisor of Φd(q) which does not divide any other cyclotomic factor of the order of GF.
The CHEVIE functions described in this chapter allow to explore these situations.
RelativeDegrees(WF [,d])
Let WF be a reflection group or a reflection coset. Here d specifies a
root of unity ζ: either d is an integer and specifies ζ
=E(d)
or is a fraction smaller a/b with 0<a<b and specifies ζ
=E(b)^a
. If omitted, d is taken to be 0, specifying ζ=1. Then
if Vζ is the ζ-eigenspace of some element of WF, and is of
maximal dimension among such possible ζ-eigenspaces, and W is the
group of WF then NW(Vζ)/CW(Vζ) is a reflection group in its
action on Vζ. The function RelativeDegrees
returns the reflection
degrees of this complex reflection group, which are a subset of those of
W.
The point is that these degrees are obtained quickly by invariant-theoretic computations: if (d1,ε1),...,(dn,εn) are the generalized degrees of WF they are the di such that ζdi=εi.
gap> W:=CoxeterGroup("E",8); CoxeterGroup("E",8) gap> RelativeDegrees(W,4); [ 8, 12, 20, 24 ]
RegularEigenvalues(W)
Let W be a reflection group or a reflection coset. A root of unity
ζ is a regular eigenvalue for W if some element of <W> has a
ζ-eigenvector which lies outside of the reflecting hyperplanes. The
function RelativeDegree
returns a list describing the regular eigenvalues
for W. If all the primitive n-th roots of unity are regular
eigenvalues, then n is put on the result list. Otherwise the fractions
a/n are added to the list for each a such that E(n)a is a primitive
n-root of unity and a regular eigenvalue for W.
gap> W:=CoxeterGroup("E",8);; gap> RegularEigenvalues(W); [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30 ] gap> W:=ComplexReflectionGroup(6);; gap> L:=Twistings(W,[2])[4]; Z3[I]<2>.(q-I) gap> RegularEigenvalues(L); [ 1/4, 7/12, 11/12 ]
PositionRegularClass(WF [,d])
Let WF be a reflection group or a reflection coset. Here d specifies a
root of unity ζ: either d is an integer and specifies ζ
=E(d)
or is a fraction smaller a/b with 0<a<b and specifies ζ
=E(b)^a
. If omitted, d is taken to be 0, specifying ζ=1. The
root ζ should be a regular eigenvalue for WF (see
RegularEigenvalues). The function returns the index of the conjugacy
class of WF which has a ζ-regular eigenvector.
gap> W:=CoxeterGroup("E",8);; gap> PositionRegularClass(W,30); 65 gap> W:=ComplexReflectionGroup(6);; gap> L:=Twistings(W,[2])[4]; Z3[I]<2>.(q-I) gap> PositionRegularClass(L,7/12); 2
EigenspaceProjector(WF, w ,d)
Let WF be a reflection group or a reflection coset. Here d specifies a
root of unity ζ: either d is an integer and specifies ζ
=E(d)
or is a fraction smaller a/b with 0<a<b and specifies ζ
=E(b)^a
. The function returns the unique w-invariant projector on the
ζ-eigenspace of w.
gap> W:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> w:=EltWord(W,[1..3]); ( 1,12, 3, 2)( 4,11,10, 5)( 6, 9, 8, 7) gap> EigenspaceProjector(W,w,1/4); [ [ 1/4+1/4*E(4), 1/2*E(4), -1/4+1/4*E(4) ], [ 1/4-1/4*E(4), 1/2, 1/4+1/4*E(4) ], [ -1/4-1/4*E(4), -1/2*E(4), 1/4-1/4*E(4) ] ] gap> RankMat(last); 1
SplitLevis(WF [, d [,ad]])
Let WF be a reflection group or a reflection coset. If W is a
reflection group it is treated as the trivial coset Spets(W)
.
Here d specifies a root of unity ζ: either d is an integer and
specifies ζ=E(d)
or is a fraction a/b with 0<a<b and
specifies ζ=E(b)^a
. If omitted, d is taken to be 0, specifying
ζ=1.
A Levi is a subcoset of the form W1F1 where W1 is a parabolic subgroup of W, that is the centralizer of some subspace of V.
The function returns a list of representatives of conjugacy classes of d-split Levis of W. A d-split Levi is a subcoset of WF formed of all the elements which act by ζ on a given subspace Vζ. If the additional argument ad is given, it returns only those subcosets such that the common ζ-eigenspace of their elements is of dimension ad.
gap> W:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> SplitLevis(W,4); [ A3, (q+1)(q^2+1) ] gap> 3D4:=CoxeterCoset(CoxeterGroup("D",4),(1,2,4)); 3D4 gap> SplitLevis(3D4,3); [ 3D4, A2<1,3>.(q^2+q+1), (q^2+q+1)^2 ] gap> W:=CoxeterGroup("E",8); CoxeterGroup("E",8) gap> SplitLevis(W,4,2); [ D4<3,2,4,5>.(q^2+1)^2, (A1xA1)<5,7>x(A1xA1)<2,3>.(q^2+1)^2, 2(A2xA2)<3,1,5,6>.(q^2+1)^2 ]
gap3-jm