99 Eigenspaces and $d$-Harish-Chandra series

Let 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 is a ζ-regular element, that is the ζ-eigenspace Vζ of 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 , and is of maximal dimension among such possible ζ-eigenspaces. Then the set of elements of 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.

Subsections

  1. RelativeDegrees
  2. RegularEigenvalues
  3. PositionRegularClass
  4. EigenspaceProjector
  5. SplitLevis

99.1 RelativeDegrees

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 (d11),...,(dnn) are the generalized degrees of WF they are the di such that ζdii.

    gap> W:=CoxeterGroup("E",8);
    CoxeterGroup("E",8)
    gap> RelativeDegrees(W,4);
    [ 8, 12, 20, 24 ]

99.2 RegularEigenvalues

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 ]

99.3 PositionRegularClass

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

99.4 EigenspaceProjector

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

99.5 SplitLevis

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 ]

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gap3-jm
18 Jun 2018