92 Unipotent classes of reductive groups

CHEVIE contains information about the unipotent conjugacy classes of a connected reductive group over an algebraically closed field k. 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 also contains information on invariants attached to the classes (the groups of components of their centralizers, and in good characteristic their Dynkin-Richardson diagram) and on the Springer correspondence .

There is a certain number of invariants attached to, and characterizing, unipotent classes (or equivalently, nilpotent orbits of the Lie algebra). The Jacobson-Morozov theorem states, given e∈g nilpotent, there exists an sl2 subalgebra of g such that e corresponds to the element (
01
00
)
in sl2. Let S be the torus (
h0
0h-1
)
of SL2, let T be a maximal torus containing S, and consider the associated root decomposition g=∑αgα. Let σ∈ Y(T) be the one-parameter subgroup whose image is S. Then α→⟨σ,α⟩ defines a linear form on Σ, thus is determined by its value on simple roots. The result of Dynkin and Richardson is that 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 decorated by these values 0,1,2 is called the Dynkin-Richardson diagram of the considered nilpotent orbit, and is a complete invariant of the nilpotent orbit in good characteristic.

Let B be the variety of all Borel subgroups and let Bu be the subvariety of Borel subgroups containing the unipotent element u. Let C be the class of u; then dim C/2=dimB-dimBu and both dim C and dim Bu can be computed from the Dynkin-Richardson diagram of the associated nilpotent orbit: dim C is the total number of roots such that ⟨σ,α⟩∉{0,1}.

Another important invariant is the group A(u)=CG(u)/CG(u)0, which is involved in the Springer correspondence. Indecomposable locally constant G-equivariant sheaves on C, called local systems, are parameterized by irreducible characters of A(u). The 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, that is those parameterized by the trivial character of A(u) for each u. More generally, the generalized Springer correspondence associates to each local system a cuspidal local system on a class u of a Levi subgroup L of G (there only a few such cuspidal systems), such that the set of local systems associated to the same cuspidal pair is parameterized by the characters of the relative Weyl group NG(L)/L.

The Springer correspondence is related to the character values of a finite reductive groups as follows: assume that k is the algebraic closure of a finite field Fq; we assume for the moment that G is split over Fq (the more general situation corresponding to a Coxeter coset is not yet implemented), and even that the Frobenius automorphism acts trivially on the fundamental group (or equivalently on the group of components of the center of G). Let u be a unipotent element of G(Fq). In our situation the Frobenius automorphism acts trivially on A(u); the G(Fq)-classes of unipotent elements of G(Fq) conjugate by G(k) to u (such elements form the geometric class of u) are parameterized by the conjugacy classes of A(u). To a character φ of A(u) we associate the characteristic function of the corresponding local system, a class function Yu,φ on G(Fq) defined by: Yu,φ(u1)=φ(c) if u1 is geometrically conjugate to u and its G(Fq)-class is parameterized by the conjugacy class c of A(u), otherwise Yu,φ(u1)=0. If the pair u,φ corresponds via the Springer correspondence to the character χ of W, then Yu,φ is also denoted Yχ. There is another important set of functions indexed by local systems whose relation to the Yχ is related to the character table of G(Fq). To a local system ι on the 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 a function on G(Fq) called the characteristic function, by taking the alternating trace of the Frobenius acting on the stalks of the cohomology sheaves at points of G(Fq). If Yχ is associated to the local system ι, the characteristic function of the intersection cohomology complex of ι is denoted by Xχ. This function is supported on C, and Lusztig has shown that it is a linear combination with integer polynomials in q of functions Yψ where ψ corresponds to local systems on some class in C.

Lusztig and Shoji have given an algorithm to compute the transition matrix Pψ,χ between Xψ and Yχ, which is implemented in CHEVIE. The relationship with characters of G(Fq) is that the restriction to the unipotent elements of the almost character Rχ is equal to qbχ Xχ. 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 in cases where known may depend on the congruence class of q modulo some small primes like 3).

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 |diagram Dim Bu A(u)  A3() A1(2A1)/-1 .(A3)/I .(A3)/-I
    ____________________________________________________________
    4    |    222      0   Z4   1:4       -1:2      I:      -I:
    31   |    202      1    .    31
    22   |    020      2   A1  2:22      11:11
    211  |    101      3    .   211
    1111 |    000      6    .  1111

Here 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 second column of the table, displayed only in good characteristic, gives the Dynkin-Richardson diagram for each class; the next column gives the dimension of the variety Bu, and the next one describes the group A(u). 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 |diagram Dim Bu A(u) A3()
    _______________________________
    4    |    222      0    .    4
    31   |    202      1    .   31
    22   |    020      2    .   22
    211  |    101      3    .  211
    1111 |    000      6    . 1111

Here is another example:

    gap> W:=CoxeterGroup("G",2);; 
     gap> Display(UnipotentClasses(W)); 
    1<A1<~A1<G2(a1)<G2
         u |diagram Dim Bu A(u)                    G2() .(G2)
    __________________________________________________________
    G2     |     22      0    .                phi{1,0}
    G2(a1) |     20      1   A2 21:phi{1,3}' 3:phi{2,1}  111:
    ~A1    |     01      2    .                phi{2,2}
    A1     |     10      3    .              phi{1,3}''
    1      |     00      6    .                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. The characteristic 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 |Dim Bu A(u)        G2() .(G2) .(G2) .(G2)
    _________________________________________________
    G2     |     0   Z3  1:phi{1,0}         E3: E3^2:
    G2(a1) |     1   A1  2:phi{2,1}   11:
    ~A1    |     2    .    phi{2,2}
    A1     |     3    .  phi{1,3}''
    (~A1)3 |     3    .   phi{1,3}'
    1      |     6    .    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

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

This function requires the package "chevie" (see RequirePackage).

Subsections

  1. UnipotentClasses
  2. ICCTable

92.1 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 partial order induced on unipotent classes by the closure relation. This is a list whose i-th element is the list of the indices j of the classes immediately above i. That is .orderclasses[i] contains j if Cj Ci and there is no class Ck such that Cj Ck and 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.

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;
   [ rec(
	 parameter := [ [ 1, 1, 1, 1 ] ],
	 name := "1111",
	 Au := CoxeterGroup("A",0),
	 dynkin := [ 0, 0, 0 ],
	 dimBu := 6 ), rec(
	 parameter := [ [ 2, 1, 1 ] ],
	 name := "211",
	 Au := CoxeterGroup("A",0),
	 dynkin := [ 1, 0, 1 ],
	 dimBu := 3 ), rec(
	 parameter := [ [ 2, 2 ] ],
	 name := "22",
	 Au := CoxeterGroup("B",1),
	 dynkin := [ 0, 2, 0 ],
	 dimBu := 2 ), rec(
	 parameter := [ [ 3, 1 ] ],
	 name := "31",
	 Au := CoxeterGroup("A",0),
	 dynkin := [ 2, 0, 2 ],
	 dimBu := 1 ), rec(
	 parameter := [ [ 4 ] ],
	 name := "4",
	 Au := ComplexReflectionGroup(4,1,1),
	 dynkin := [ 2, 2, 2 ],
	 dimBu := 0 ) ]
   gap> uc.orderClasses;
   [ [ 2 ], [ 3 ], [ 4 ], [ 5 ], [  ] ]
   gap> uc.springerSeries;
   [ rec(
	 relgroup := CoxeterGroup("A",3),
	 levi := [  ],
	 Z := [ 1 ],
	 locsys := [ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] 
	    ] ), rec(
	 relgroup := CoxeterGroup("A",1),
	 levi := [ 1, 3 ],
	 Z := [ -1 ],
	 locsys := [ [ 3, 1 ], [ 5, 3 ] ] ), rec(
	 relgroup := CoxeterGroup("A",0),
	 levi := [ 1, 2, 3 ],
	 Z := [ E(4) ],
	 locsys := [ [ 5, 2 ] ] ), rec(
	 relgroup := CoxeterGroup("A",0),
	 levi := [ 1, 2, 3 ],
	 Z := [ -E(4) ],
	 locsys := [ [ 5, 4 ] ] ) ]

The Display and Format functions for unipotent classes accept all the options of FormatTable, CharNames. There is an additional option mizuno to use the names given by Mizuno 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 first two Springer series 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,"sc"));;
   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 |diagram Dim Bu A(u)                    E6() G2(2A2)/E3
   ______________________________________________________________
   E6     | 222222      0   Z3                   1:1_p       E3:1
   E6(a1) | 222022      1   Z3                   1:6_p   E3:eps_c
   D5     | 220202      2    .                    20_p
   A5+A1  | 200202      3   Z6          -1:15_p 1:30_p -E3:theta'
   A5     | 211012      4   Z3                  1:15_q E3:theta''
   D5(a1) | 121011      4    .                    64_p
   A4+A1  | 111011      5    .                    60_p
   D4     | 020200      6    .                    24_p
   A4     | 220002      6    .                    81_p
   D4(a1) | 000200      7   A2 111:20_s 3:80_s 21:90_s
   A3+A1  | 011010      8    .                    60_s
   2A2+A1 | 100101      9   Z3                  1:10_s   E3:eps_l
   A3     | 120001     10    .                   81_p'
   A2+2A1 | 001010     11    .                   60_p'
   2A2    | 200002     12   Z3                 1:24_p'     E3:eps
   A2+A1  | 110001     13    .                   64_p'
   A2     | 020000     15   A1        11:15_p' 2:30_p'
   3A1    | 000100     16    .                   15_q'
   2A1    | 100001     20    .                   20_p'
   A1     | 010000     25    .                    6_p'
   1      | 000000     36    .                    1_p'

This function requires the package "chevie" (see RequirePackage).

92.2 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

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

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

A by-product of Lusztig's algorithm, 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 function requires the package "chevie" (see RequirePackage).

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GAP 3.4.4
April 1997