100 Unipotent classes of reductive groups

CHEVIE contains information about the unipotent conjugacy classes of a connected reductive group over an algebraically closed field k, and various invariants attached to them. The unipotent classes depend on the characteristic of k; their classification differs when the characteristic is not good (that is, when it divides one of the coefficients of the highest root). In good characteristic, the unipotent classes are in bijection with nilpotent orbits on the Lie algebra.

CHEVIE contains the following information attached to the class C of a unipotent element u:

• its centralizer CG(u), characterized by its reductive part, its group of components A(u):=CG(u)/CG(u)0, and the dimension of its radical.

• in good characteristic, its Dynkin-Richardson diagram.

• the Springer correspondence, attaching characters of the Weyl group or relative Weyl groups to each character of A(u).

The Dynkin-Richarson diagram is attached to a nilpotent element e of the Lie algebra g. By the Jacobson-Morozov theorem there exists an sl2 subalgebra of g containing e as the element (
01
00
)
. Let S be the torus (
h0
0h-1
)
of SL2 and let T be a maximal torus containing S so that S is the image of a one-parameter subgroup σ∈ Y(T). Consider the root decomposition g=∑α∈Σgα given by T; then α→⟨σ,α⟩ defines a linear form on Σ, determined by its value on simple roots. It is possible to choose a system of simple roots Π so that ⟨σ,α⟩ ≥ 0 for α∈Π, and then ⟨σ,α⟩∈{0,1,2} for any α∈Π. The Dynkin diagram of Π decorated by these values 0,1,2 is called the Dynkin-Richardson diagram of e, and in good characteristic is a complete invariant of its g-orbit.

Let B be the variety of all Borel subgroups and let Bu be the subvariety of Borel subgroups containing the unipotent element u. Then dim CG(u)=rankG+2dimBu and in good characteristic this dimension can be computed from the Dynkin-Richardson diagram: the dimension of the class of u is the number of roots α such that ⟨σ,α⟩∉{0,1}.

We describe now the Springer correspondence. Indecomposable locally constant G-equivariant sheaves on C, called local systems, are parameterized by irreducible characters of A(u). The ordinary Springer correspondence is a bijection between irreducible characters of the Weyl group and a large subset of the local systems which contains all trivial local systems (those parameterized by the trivial character of A(u) for each u). More generally, the generalized Springer correspondence associates to each local system a (unique up to G-conjugacy) cuspidal pair of a Levi subgroup L of G and a local system on an unipotent class of L, such that the set of local systems associated to a given cuspidal pair is parameterized by the characters of the relative Weyl group WG(L):=NG(L)/L. There are only few cuspidal pairs.

The Springer correspondence gives information on the character values of a finite reductive groups as follows: assume that k is the algebraic closure of a finite field 𝔽q and that F is the Frobenius attached to an 𝔽q-structure of G. Let C be an F-stable unipotent class and let u∈ CF; we call C the geometric class of u and the GF-classes inside CF are parameterized by the F-conjugacy classes of A(u), denoted H1(F,A(u)) (most of the time we can find u such that F acts trivially on A(u) and H1(F,A(u)) is then just the conjugacy classes). To an F-stable character φ of A(u) we associate the characteristic function of the corresponding local system (actually associated to an extension ~φ of φ to A(u).F); it is a class function Yu,φ on GF which can be normalized so that: Yu,φ(u1)=~φ(cF) if u1 is geometrically conjugate to u and its GF-class is parameterized by the F-conjugacy class cF of A(u), otherwise Yu,φ(u1)=0. If the pair u,φ corresponds via the Springer correspondence to the character χ of WG(L), then Yu,φ is also denoted Yχ. There is another important class of functions indexed by local systems: to a local system on class C is attached an intersection cohomology complex, which is a complex of sheaves supported on the closure C. To such a complex of sheaves is associated its characteristic function, a class function of GF obtained by taking the alternating trace of the Frobenius acting on the stalks of the cohomology sheaves. If Yψ is the characteristic function of a local system, the characteristic function of the corresponding intersection cohomology complex is denoted by Xψ. This function is supported on C, and Lusztig has shown that Xψ=∑φ Pψ,χ Yχ where Pψ,χ are integer polynomials in q and Yχ are attached to local systems on classes lying in C.

Lusztig and Shoji have given an algorithm to compute the matrix Pψ,χ, which is implemented in CHEVIE. The relationship with characters of G(𝔽q), taking to simplify the ordinary Springer correspondence, is that the restriction to the unipotent elements of the almost character Rχ is equal to qbχ Xχ, where bχ is dim Bu for an element u of the class C such that the support of χ is C. The restriction of the Deligne-Lusztig characters Rw to the unipotents are called the Green functions and can also be computed by CHEVIE. The values of all unipotent characters on unipotent elements can also be computed in principle by applying Lusztig's Fourier transform matrix (see the section on the Fourier matrix) but there is a difficulty in that the Xχ must be first multiplied by some roots of unity which are not known in all cases (and when known may depend on the congruence class of q modulo some small primes).

We illustrate these computations on some examples:

    gap> W:=CoxeterGroup("A",3,"sc");
    CoxeterGroup("A",3,"sc")
    gap> uc:=UnipotentClasses(W);
    UnipotentClasses( A3 )
    gap> Display(uc);
    1111<211<22<31<4
       u |D-R dBu B-C         C(u) A3() A1(2A1)/-1 .(A3)/I .(A3)/-I
    ________________________________________________________________
    4    |222   0 222       q^3.Z4  1:4       -1:2      I:      -I:
    31   |202   1 22.    q^4.(q-1)   31
    22   |020   2 2.2    q^4.A1.Z2 2:22      11:11
    211  |101   3 2.. q^5.A1.(q-1)  211
    1111 |000   6 ...           A3 1111

In CoxeterGroup("A",3,"sc") the "sc" specifies that we are working with the simply connected group, that is sln; another syntax for the same group is RootDatum("sl",4). The first column in the table gives the name of the unipotent class, which here is a partition describing the Jordan form. The partial order on unipotent classes given by Zariski closure is given before the table. The column D-R, displayed only in good characteristic, gives the Dynkin-Richardson diagram for each class; the column dBu gives the dimension of the variety Bu. The column B-C gives the Bala-Carter classification of u, that is in the case of sl4 it displays u as a regular unipotent in a Levi subgroup by giving the Dynkin-Richardson diagram of a regular unipotent (all 2's) at entries corresponding to the Levi and . at entries which do not correspond to the Levi. The column C(u) describes the group CG(u): a power qd describes that the unipotent radical of CG(u) has dimension d (thus qd rational points); then follows a description of the reductive part of the neutral component of CG(u), given by the name of its root datum. Then if CG(u) is not connected, the description of A(u) is given

using another vocabulary:
a cyclic group of order 4 is given as Z4, and a symmetric group on 3 points would be given as S3.

For instance, the first class 4 has CG(u)0 unipotent of dimension 3 and A(u) equal to Z4, the cyclic group of order 4. The class 22 has CG(u) with unipotent radical of dimension 4, reductive part of type A1 and A(u) is Z2, that is the cyclic group of order 2. The other classes have CG(u) connected. For the class 31 the reductive part of CG(u) is a torus of rank 1.

Then there is one column for each Springer series, giving for each class the pairs a:b where a is the name of the character of A(u) describing the local system involved and b is the name of the character of the (relative) Weyl group corresponding by the Springer correspondence. At the top of the column is written the name of the relative Weyl group, and in brackets the name of the Levi affording a cuspidal local system; next, separated by a / is a description of the central character associated to the Springer series (omitted if this central character is trivial): all local systems in a given Springer series have same restriction to the center of G. To find what the picture becomes for another algebraic group in the same isogeny class, for instance the adjoint group, one simply discards the Springer series whose central character becomes trivial on the center of G; and each group A(u) has to be quotiented by the common kernel of the remaining characters. Here is the table for the adjoint group:

    gap> Display(UnipotentClasses(CoxeterGroup("A",3)));
    1111<211<22<31<4
       u |D-R dBu B-C         C(u) A3()
    ____________________________________
    4    |222   0 222          q^3    4
    31   |202   1 22.    q^4.(q-1)   31
    22   |020   2 2.2       q^4.A1   22
    211  |101   3 2.. q^5.A1.(q-1)  211
    1111 |000   6 ...           A3 1111

Here is another example:

    gap> W:=CoxeterGroup("G",2);;
    gap> Display(UnipotentClasses(W));
    1<A1<~A1<G2(a1)<G2
         u |D-R dBu B-C   C(u)                    G2() .(G2)
    _________________________________________________________
    G2     | 22   0  22    q^2                phi{1,0}
    G2(a1) | 20   1  20 q^4.S3 21:phi{1,3}' 3:phi{2,1}  111:
    ~A1    | 01   2  .2 q^3.A1                phi{2,2}
    A1     | 10   3  2. q^5.A1              phi{1,3}''
    1      | 00   6  ..     G2                phi{1,6}

which illustrates that on class G2(a1) there are two local systems in the principal series of the Springer correspondence, and a further cuspidal local system. Also, from the B-C column, we see that that class is not in a proper Levi, in which case the Bala-Carter diagram coincides with the Dynkin-Richardson diagram.

The characteristics 2 and 3 are not good for G2. To get the unipotent classes and the Springer correspondence in bad characteristic, one gives a second argument to the function UnipotentClasses:

    gap> Display(UnipotentClasses(W,3));
    1<A1,(~A1)3<~A1<G2(a1)<G2
         u |dBu   C(u)       G2() .(G2) .(G2) .(G2)
    ________________________________________________
    G2     |  0 q^2.Z3 1:phi{1,0}         E3: E3^2:
    G2(a1) |  1 q^4.Z2 2:phi{2,1}   11:
    ~A1    |  2    q^6   phi{2,2}
    A1     |  3 q^5.A1 phi{1,3}''
    (~A1)3 |  3 q^5.A1  phi{1,3}'
    1      |  6     G2   phi{1,6}

The function ICCTable gives the transition matrix between the functions Xχ and Yψ.

    gap> Display(ICCTable(UnipotentClasses(W)));
    Coefficients of X_phi on Y_psi for G2

                |G2 G2(a1)(21) G2(a1) ~A1 A1   1
    _____________________________________________
    Xphi{1,0}   | 1          0      1   1  1   1
    Xphi{1,3}'  | 0          1      0   1  0 q^2
    Xphi{2,1}   | 0          0      1   1  1  P8
    Xphi{2,2}   | 0          0      0   1  1  P4
    Xphi{1,3}'' | 0          0      0   0  1   1
    Xphi{1,6}   | 0          0      0   0  0   1

Here the row labels and the column labels show the two ways of indexing local systems: the row labels give the character of the relative Weyl group and the column labels give the class and the name of the local system as a character of A(u): for instance, G2(a1) is the trivial local system of the class G2(a1), while G2(a1)(21) is the local system on that class corresponding to the 2-dimensional character of A(u)=A2.

Subsections

  1. UnipotentClasses
  2. ICCTable

100.1 UnipotentClasses

UnipotentClasses(W[,p])

W should be a CoxeterGroup record for a Weyl group or RootDatum describing a reductive algebraic group G. The function returns a record containing information about the unipotent classes of G in characteristic p (if omitted, p is assumed to be any good characteristic for G). This contains the following fields:

group:

a pointer to W

p:

the characteristic of the field for which the unipotent classes were computed. It is 0 for any good characteristic.

orderClasses:

a list describing the Hasse diagram of the partial order induced on unipotent classes by the closure relation. That is .orderclasses[i] is the list of j such that Cj Ci and there is no class Ck such that Cj Ck Ci.

classes:

a list of records holding information for each unipotent class (see below).

springerSeries:

a list of records, each of which describes a Springer series of G.

The records describing individual unipotent classes have the following fields:

name:

the name of the unipotent class.

parameter:
a parameter describing the class (for example, a partition describing the Jordan form, for classical groups).

Au:

the group A(u).

dynkin:

present in good characteristic; contains the Dynkin-Richardson diagram, given as a list of 0,1,2 describing the coefficient on the corresponding simple root.

red:

the reductive part of CG(u).

dimBu:

the dimension of the variety Bu.

The records for classes contain additional fields for certain groups: for instance, the names given to classes by Mizuno in E6, E7, E8 or by Shoji in F4.

The records describing individual Springer series have the following fields:

levi:

the indices of the reflections corresponding to the Levi subgroup L where lives the cuspidal local system ι from which the Springer series is induced.

relgroup:

The relative Weyl group NG(L,ι)/L. The first series is the principal series for which .levi=[] and .relgroup=W.

locsys:

a list of length NrConjugacyClasses(.relgroup), holding in i-th position a pair describing which local system corresponds to the i-th character of NG(L,ι). The first element of the pair is the index of the concerned unipotent class u, and the second is the index of the corresponding character of A(u).

Z:

the central character associated to the Springer series, specified by its value on the generators of the centre.

    gap> W:=CoxeterGroup("A",3,"sc");;
    gap> uc:=UnipotentClasses(W);
    UnipotentClasses( A3 )
    gap> uc.classes;
    [ UnipotentClass(1111), UnipotentClass(211), UnipotentClass(22),
      UnipotentClass(31), UnipotentClass(4) ]
    gap> PrintRec(uc.classes[3]);
    rec(
      name      := 22,
      Au        := CoxeterGroup("A",1),
      dimBu     := 2,
      dimunip   := 4,
      dimred    := 3,
      parameter := [ 2, 2 ],
      balacarter:= [ 1, 3 ],
      dynkin    := [ 0, 2, 0 ],
      red       := ReflectionSubgroup(CoxeterGroup("A",1), [ 1 ]),
      AuAction  := A1,
      operations:= UnipotentClassOps )
    gap> uc.orderClasses;
    [ [ 2 ], [ 3 ], [ 4 ], [ 5 ], [  ] ]
    gap> uc.springerSeries;
    [ rec(
          relgroup := A3,
          Z := [ 1 ],
          levi := [  ],
          locsys := [ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ] )
        , rec(
          relgroup := A1,
          Z := [ -1 ],
          levi := [ 1, 3 ],
          locsys := [ [ 3, 1 ], [ 5, 3 ] ] ), rec(
          relgroup := .,
          Z := [ E(4) ],
          levi := [ 1, 2, 3 ],
          locsys := [ [ 5, 2 ] ] ), rec(
          relgroup := .,
          Z := [ -E(4) ],
          levi := [ 1, 2, 3 ],
          locsys := [ [ 5, 4 ] ] ) ]

The Display and Format functions for unipotent classes accept all the options of FormatTable, CharNames. Giving the option mizuno (resp. shoji) uses the names given by Mizuno (resp. Shoji) for unipotent classes. Moreover, there is also an option fourier which gives the correspondence tensored with the sign character of each relative Weyl group, which is the correspondence obtained via a Fourier-Deligne transform (here we assume that p is very good, so that there is a nondegenerate invariant bilinear form on the Lie algebra, and also one can identify nilpotent orbits with unipotent classes).

Here is how to display only the ordinary Springer correspondence of the unipotent classes of E6 using the notations of Mizuno for the classes and those of Frame for the characters of the Weyl group and of Spaltenstein for the characters of G2 (this is convenient for checking our data with the original paper of Spaltenstein):

    gap> uc:=UnipotentClasses(CoxeterGroup("E",6));;
    gap> Display(uc,rec(columns:=[1..5],mizuno:=true,frame:=true,
    > spaltenstein:=true));
    1<A1<2A1<3A1<A2<A2+A1<A2+2A1<2A2+A1<A3+A1<D4(a1)<D4<D5(a1)<A5+A1<D5<E6\ 
    (a1)<E6
    A2+A1<2A2<2A2+A1
    A2+2A1<A3<A3+A1
    D4(a1)<A4<A4+A1<A5<A5+A1
    A4+A1<D5(a1)
         u |   D-R dBu    B-C            C(u)                 E6()
    _______________________________________________________________
    E6     |222222   0 222222             q^6                   1p
    E6(a1) |222022   1 222022             q^8                   6p
    D5     |220202   2 22222.       q^9.(q-1)                  20p
    A5+A1  |200202   3 200202         q^12.Z2         11:15p 2:30p
    A5     |211012   4 2.2222         q^11.A1                  15q
    D5(a1) |121011   4 22202.      q^13.(q-1)                  64p
    A4+A1  |111011   5 2222.2      q^15.(q-1)                  60p
    D4     |020200   6 .2222.         q^10.A2                  24p
    A4     |220002   6 2222..   q^14.A1.(q-1)                  81p
    D4(a1) |000200   7 .2202. q^18.(q-1)^2.S3 111:20s 3:80s 21:90s
    A3+A1  |011010   8 22.22.   q^18.A1.(q-1)                  60s
    2A2+A1 |100101   9 222.22         q^21.A1                  10s
    A3     |120001  10 2.22..   q^15.B2.(q-1)                 81p'
    A2+2A1 |001010  11 222.2.   q^24.A1.(q-1)                 60p'
    2A2    |200002  12 2.2.22         q^16.G2                 24p'
    A2+A1  |110001  13 222...   q^23.A2.(q-1)                 64p'
    A2     |020000  15 2.2...    q^20.(A2xA2)       11:15p' 2:30p'
    3A1    |000100  16 22..2.      q^27.A2xA1                 15q'
    2A1    |100001  20 22....   q^24.B3.(q-1)                 20p'
    A1     |010000  25 2.....         q^21.A5                  6p'
    1      |000000  36 ......              E6                  1p'

100.2 ICCTable

ICCTable(uc[,seriesNo[,q]])

This function gives the table of decompositions of the functions Xu,φ in terms of the functions Yu,φ. Here (u,φ) runs over the pairs where u is a unipotent element of the reductive group G and φ is a character of the group of components A(u); such a pair describes a G-equivariant local system on the class C of u. The function Yu,φ is the characteristic function of this local system and Xu,φ is the characteristic function of the corresponding intersection cohomology complex. The local systems can also be indexed by characters of the relative Weyl group occurring in the Springer correspondence, and since the coefficient of Xχ on Yψ is 0 if χ and ψ do not correspond to the same relative Weyl group (are not in the same Springer series), the table given is for a given Springer series, the series whose number is given by the argument seriesNo (if omitted this defaults to seriesNo=1 which is the principal series). The decomposition multiplicities are graded, and are given as polynomials in one variable (specified by the argument q; if not given Indeterminate(Rationals) is assumed).

    gap> W:=CoxeterGroup("A",3);;
    gap> uc:=UnipotentClasses(W);;
    gap> Display(ICCTable(uc));
    Coefficients of X_phi on Y_psi for A3

          |4 31 22 211 1111
    ________________________
    X4    |1  1  1   1    1
    X31   |0  1  1  P2   P3
    X22   |0  0  1   1   P4
    X211  |0  0  0   1   P3
    X1111 |0  0  0   0    1

In the above the multiplicities are given as products of cyclotomic polynomials to display them more compactly. However the Format or the Display of such a table can be controlled more precisely.

For instance, one can ask to not display the entries as products of cyclotomic polynomials:

    gap> Display(ICCTable(uc),rec(CycPol:=false));
    Coefficients of X_phi on Y_psi for A3

          |4 31 22 211    1111
    ___________________________
    X4    |1  1  1   1       1
    X31   |0  1  1 q+1 q^2+q+1
    X22   |0  0  1   1   q^2+1
    X211  |0  0  0   1 q^2+q+1
    X1111 |0  0  0   0       1

Since Display and Format use the function FormatTable, all the options of this function are also available. We can use this to restrict the entries displayed to a given subset of the rows and columns:

    gap> W:=CoxeterGroup("F",4);;
    gap> uc:=UnipotentClasses(W);;
    gap> show:=[13,24,22,18,14,9,11,19];;
    gap> Display(ICCTable(uc),rec(rows:=show,columns:=show));
    Coefficients of X_phi on Y_psi for F4

                |A1+~A1 A2 ~A2 A2+~A1 ~A2+A1 B2(11) B2 C3(a1)(11)
    ______________________________________________________________
    Xphi{9,10}  |     1  0   0      0      0      0  0          0
    Xphi{8,9}'' |     1  1   0      0      0      0  0          0
    Xphi{8,9}'  |     1  0   1      0      0      0  0          0
    Xphi{4,7}'' |     1  1   0      1      0      0  0          0
    Xphi{6,6}'  |    P4  1   1      1      1      0  0          0
    Xphi{4,8}   |   q^2  0   0      0      0      1  0          0
    Xphi{9,6}'' |    P4 P4   0      1      0      0  1          0
    Xphi{4,7}'  |   q^2  0  P4      0      1      0  0          1

The function ICCTable returns a record with various pieces of information which can help further computations.

.scalar:

this contains the table of multiplicities Pψ,χ of the Xψ on the Yχ.

.group:

The group W.

.relgroup:

The relative Weyl group for the Springer series.

.series:

The index of the Springer series given for W.

.dimBu:

The list of dimBu for each local system (u,φ) in the series.

.L:

The matrix of (unnormalized) scalar products of the functions Yψ with themselves, that is the (φ,ψ) entry is g∈G(𝔽q) Yφ(g) Yψ(g). This is thus a symmetric, block-diagonal matrix where the diagonal blocks correspond to geometric unipotent conjugacy classes. This matrix is obtained as a by-product of Lusztig's algorithm to compute Pψ,χ.

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gap3-jm
18 Jun 2018