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. These notions make sense and thus are implemented for any complex reflection group.
In terms of algebraic groups, an F-stable Levi subgroup of the reductive group G is d-split if and only if it is the centralizer of the Φd-part of its center. When d=1, we get the notion of a split Levi, which is the same as a Levi sugroup of an F-stable parabolic subgroup of G.
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 ]
gap> SplitLevis(ComplexReflectionGroup(5));
[ G5, Z3.(q-1), Z3<2>.(q-1), (q-1)^2 ]
99.6 CuspidalUnipotentCharacters
CuspidalUnipotentCharacters(WF[,d])
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).
A unipotent character γ of the corresponding finite reductive group
G is d-cuspidal if its Lusztig restriction to any proper d-split
Levi is zero. When d=0 we recover the usal notion of cuspidal character.
Equivalently the Φd-part of the generic degree of γ is equal
to the Φd-part of the generic order of the adjoint group of G.
This makes sense for any Spetsial complex reflection group and is
implemented for them. This also generalises to an arbitrary root of unity
E(d)^r (which should be given as the fraction r/d) by asking that the
generic degree of γ and the generic order of the adjoin group of
G have zeros of the same order at E(d)^r.
The function returns the list of indices of unipotent characters which are d-cuspidal. If d is omitted, it is taken to be 0.
gap> W:=CoxeterGroup("D",4);
CoxeterGroup("D",4)
gap> CuspidalUnipotentCharacters(W);
[ 14 ]
gap> CuspidalUnipotentCharacters(W,3);
[ 1, 2, 5, 7, 8, 9, 12, 14 ]
gap> CuspidalUnipotentCharacters(ComplexReflectionGroup(4),3);
[ 3, 6, 7, 10 ]
CuspidalPairs(W[,d[,ad]])
returns the pairs (LF,λ) where LF is a d-split Levi (with
d-center of dimension ad if ad is given) and λ is a
d-cuspidal character of LF. If d is omitted it is assumed to be 1,
which means ordinary cuspidal pairs. The character λ is returned as
its index amongst unipotent characters. d can represent an arbitrary root
of unity (if integral, it represents E(d) and if a fraction r/d
represents E(d)^r).
gap> CuspidalPairs(CoxeterGroup("F",4));
[ [ F4, 31 ], [ F4, 32 ], [ F4, 33 ], [ F4, 34 ], [ F4, 35 ],
[ F4, 36 ], [ F4, 37 ], [ B2<3,2>.(q-1)^2, 6 ], [ (q-1)^4, 1 ] ]
gap> CuspidalPairs(ComplexReflectionGroup(4),3);
[ [ G4, 3 ], [ G4, 6 ], [ G4, 7 ], [ G4, 10 ], [ (q-1)(q-E3), 1 ] ]
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