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 computes the following information for the class C of a unipotent element u:
• the 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, the Dynkin-Richardson diagram of C.
• the Springer correspondence, attaching characters of the Weyl group or of 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 (
| 0 | 1 |
| 0 | 0 |
| h | 0 |
| 0 | h-1 |
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 then H1(F,A(u)) is just the conjugacy classes of A(u)). 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.
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:
p:0 for any good characteristic.
orderClasses:.orderclasses[i] is the list of j such that Cj⊃≠
Ci and there is no class Ck such that Cj⊃≠ Ck⊃≠ Ci.
classes:
springerSeries:The records describing individual unipotent classes have the following fields:
name:
parameter:
Au:
dynkin:
red:
dimBu: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:
relgroup:.levi=[] and .relgroup=W.
locsys: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:
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,
parameter := [ 2, 2 ],
balacarter:= [ 1, 3 ],
dynkin := [ 0, 2, 0 ],
red := ReflectionSubgroup(CoxeterGroup("A",1), [ 1 ]),
AuAction := A1,
dimunip := 4,
dimred := 3,
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'
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 sublist of the rows and columns (here the
indices correspond to the number in CHEVIE of the corresponding character
of the relative Weyl group of the given Springer series):
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, which is
in order in CHEVIE of the characters in the relative Weyl group; the
table is transposed, then lines and rows are sorted by dimBu,class no,index of character in A(u)while displayed.
XTable(uc[,rec(classes:=true)])
This function presents in a different way the information obtained from
ICCTable. Let ~ Xu,φ=q1/2(codim C-dim Z(L)) where C is
the class of u and Z(L) is the center of Levi subgroup on which lives
the cuspidal local system attached to the local system (u,φ).
Then `XTable` gives the decomposition of the functions ~ Xu,φ
on local systems, by default. If the argument rec(classes:=true) is
given, it gives the values of the functions ~ Xu,φ on
unipotent classes.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> Display(XTable(UnipotentClasses(W)));
Values of character sheaves tilde X_iota on local systems
X | 1 A1 ~A1 G2(a1)(111) G2(a1)(21) G2(a1) G2
________________________________________________________________
X^G2_phi{1,0} | 1 1 1 0 0 1 1
X^G2_phi{1,6} | q^6 0 0 0 0 0 0
X^G2_phi{1,3}' | q^3 0 q 0 q 0 0
X^G2_phi{1,3}'' | q^3 q^3 0 0 0 0 0
X^G2_phi{2,1} | qP8 q q 0 0 q 0
X^G2_phi{2,2} |q^2P4 q^2 q^2 0 0 0 0
X^._ | 0 0 0 q^2 0 0 0
The functions ~ X in the first column are decorated by putting as an exponent the relative groups WG(L).
gap> Display(XTable(UnipotentClasses(W),rec(classes:=true)));
Values of character sheaves tilde X_iota on unipotent classes
X | 1 A1 ~A1 G2(a1) G2(a1)_21 G2(a1)_3 G2
____________________________________________________________
X^G2_phi{1,0} | 1 1 1 1 1 1 1
X^G2_phi{1,6} | q^6 0 0 0 0 0 0
X^G2_phi{1,3}' | q^3 0 q 2q 0 -q 0
X^G2_phi{1,3}'' | q^3 q^3 0 0 0 0 0
X^G2_phi{2,1} | qP8 q q q q q 0
X^G2_phi{2,2} |q^2P4 q^2 q^2 0 0 0 0
X^._ | 0 0 0 q^2 -q^2 q^2 0
gap> Display(XTable(UnipotentClasses(W,2)));
Values of character sheaves tilde X_iota on local systems
X | 1 A1 ~A1 G2(a1)(111) G2(a1)(21) G2(a1) G2(11) G2
_______________________________________________________________________
X^G2_phi{1,0} | 1 1 1 0 0 1 0 1
X^G2_phi{1,6} | q^6 0 0 0 0 0 0 0
X^G2_phi{1,3}' | q^3 0 q 0 q 0 0 0
X^G2_phi{1,3}'' | q^3 q^3 0 0 0 0 0 0
X^G2_phi{2,1} | qP8 q q 0 0 q 0 0
X^G2_phi{2,2} |q^2P4 q^2 q^2 0 0 0 0 0
X^._ | 0 0 0 q^2 0 0 0 0
X^._ | 0 0 0 0 0 0 q 0
gap> Display(XTable(UnipotentClasses(RootDatum("sl",4))));
Values of character sheaves tilde X_iota on local systems
X | 1111 211 22(11) 22 31 4 4(I) 4(-1) 4(-I)
___________________________________________________________
X^A3_1111 | q^6 0 0 0 0 0 0 0 0
X^A3_211 |q^3P3 q^3 0 0 0 0 0 0 0
X^A3_22 |q^2P4 q^2 0 q^2 0 0 0 0 0
X^A3_31 | qP3 qP2 0 q q 0 0 0 0
X^A3_4 | 1 1 0 1 1 1 0 0 0
X^A1_11 | 0 0 q^3 0 0 0 0 0 0
X^A1_2 | 0 0 q^2 0 0 0 0 q 0
X^._ | 0 0 0 0 0 0 q^(3/2) 0 0
X^._ | 0 0 0 0 0 0 0 0 q^(3/2)
A side effect of calling XTable with the argument rec(classes:=true)
is to compute the cardinal of the unipotent conjugacy classes:
gap> t:=XTable(UnipotentClasses(CoxeterGroup("G",2)),rec(classes:=true));
XTable(CoxeterGroup("G",2),rec(classes:=true))
gap> List(t.cardClass,CycPol);
[ 1, P1P2P3P6, q^2P1P2P3P6, 1/6q^2P1^2P2^2P3P6, 1/2q^2P1^2P2^2P3P6,
1/3q^2P1^2P2^2P3P6, q^4P1^2P2^2P3P6 ]
GreenTable(uc[,rec(classes:=true)])
Keeping the same notations as in the description of XTable, this function
returns a table of the functions QwF, attached to elements wF∈
WG(L). F where WG(L) are the relative weyl groups attached
to cuspidal local systems. These functions are defined by
QwF=∑u,φ ~φ(wF) ~ Xu,φ. An point to note
is that in the principal Springer series, when T is a maximal torus,
the function QwF coincides with the Deligne-Lusztig character
RGTW(1). As for XTable, by default the table gives the values
of the functions on local systems. If the argument rec(classes:=true) is
given, then it gives the values of the functions QwF on conjugacy
classes.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> Display(GreenTable(UnipotentClasses(W)));
Values of Green functions Q_wF on local systems
Q^I_wF\class | 1 A1 ~A1 G2(a1)(111) G2(a1)(21) G2(a1) G2
____________________________________________________________________________
Q^G2_A0 | P2^2P3P6 P2P3 (1+2q)P2 0 q 1+2q 1
Q^G2_~A1 | -P1P2P3P6 -P1P3 P2 0 q 1 1
Q^G2_A1 | -P1P2P3P6 P2P6 -P1 0 -q 1 1
Q^G2_G2 |P1^2P2^2P3 -P1P2^2 -P1P2 0 -q P2 1
Q^G2_A2 |P1^2P2^2P6 P1^2P2 -P1P2 0 q -P1 1
Q^G2_A1+~A1 | P1^2P3P6 -P1P6 (-1+2q)P1 0 -q 1-2q 1
Q^._ | 0 0 0 q^2 0 0 0
The functions QwF depend only on the conjugacy class of wF, so in
the first column the indices of Q are the names of the conjugacy classes
of WG(L). The exponents are the names of the groups WG(L).
gap> Display(GreenTable(UnipotentClasses(W),rec(classes:=true)));
Values of Green functions Q_wF on unipotent classes
Q^I_wF\class | 1 A1 ~A1 G2(a1) G2(a1)_21 G2(a1)_3 G2
________________________________________________________________________
Q^G2_A0 | P2^2P3P6 P2P3 (1+2q)P2 1+4q 1+2q P2 1
Q^G2_~A1 | -P1P2P3P6 -P1P3 P2 1+2q 1 -P1 1
Q^G2_A1 | -P1P2P3P6 P2P6 -P1 1-2q 1 P2 1
Q^G2_G2 |P1^2P2^2P3 -P1P2^2 -P1P2 -P1 P2 1+2q 1
Q^G2_A2 |P1^2P2^2P6 P1^2P2 -P1P2 P2 -P1 1-2q 1
Q^G2_A1+~A1 | P1^2P3P6 -P1P6 (-1+2q)P1 1-4q 1-2q -P1 1
Q^._ | 0 0 0 q^2 -q^2 q^2 0
gap> W:=RootDatum("sl",4);
RootDatum("sl",4)
gap> Display(GreenTable(UnipotentClasses(W)),rec(columns:=[1..8]));
Values of Green functions Q_{wF} on local systems
Q^I_wF\class | 1111 211 22(11) 22 31 4 4(I) 4(-1)
____________________________________________________________________________
Q^A3_1111 | P2^2P3P4 (1+2q+3q^2)P2 0 (1+2q)P2 1+3q 1 0 0
Q^A3_211 | -P1P2P3P4 1+q+q^2-q^3 0 P2 P2 1 0 0
Q^A3_22 | P1^2P3P4 -P1P4 0 1-q+2q^2 -P1 1 0 0
Q^A3_31 |P1^2P2^2P4 -P1P2 0 -P1P2 1 1 0 0
Q^A3_4 | -P1^3P2P3 P1^2P2 0 -P1 -P1 1 0 0
Q^A1_11 | 0 0 q^2P2 0 0 0 0 q
Q^A1_2 | 0 0 -q^2P1 0 0 0 0 q
Q^._ | 0 0 0 0 0 0 q^(3/2) 0
Q^._ | 0 0 0 0 0 0 0 0
UnipotentValues(uc[,rec(classes:=true)])
This function returns a table of the values of unipotent characters on
local systems (by default) or on unipotent classes (if the argument
rec(classes:=true) is given).
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> uc:=UnipotentClasses(W);
UnipotentClasses( G2 )
gap> Display(UnipotentValues(uc,rec(classes:=true)),rec(columns:=[1,4,5,6,7]));
Values of unipotent characters for G2 on unipotent classes
| 1 G2(a1) G2(a1)_21 G2(a1)_3 G2
______________________________________________________________
phi{1,0} | 1 1 1 1 1
phi{1,6} | q^6 0 0 0 0
phi{1,3}' | 1/3qP3P6 (5/3+1/3q)q -1/3qP1 1/3qP1 0
phi{1,3}'' | 1/3qP3P6 1/3qP1 -1/3qP1 (2/3+1/3q)q 0
phi{2,1} | 1/6qP2^2P3 (5/6+1/6q)q -1/6qP1 1/6qP1 0
phi{2,2} | 1/2qP2^2P6 -1/2qP1 1/2qP2 -1/2qP1 0
G2[-1] | 1/2qP1^2P3 -1/2qP1 1/2qP2 -1/2qP1 0
G2[1] | 1/6qP1^2P6 (5/6+1/6q)q -1/6qP1 1/6qP1 0
G2[E3] |1/3qP1^2P2^2 1/3qP1 -1/3qP1 (2/3+1/3q)q 0
G2[E3^2] |1/3qP1^2P2^2 1/3qP1 -1/3qP1 (2/3+1/3q)q 0
gap> uc:=UnipotentClasses(W,3);
UnipotentClasses( G2,3 )
gap> Display(UnipotentValues(uc,rec(classes:=true)),rec(columns:=[1,4,5,6,7]));
Values of unipotent characters for G2 on unipotent classes
| 1 G2(a1) G2(a1)_2 G2 G2_\zeta_3
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phi{1,0} | 1 1 1 1 1
phi{1,6} | q^6 0 0 0 0
phi{1,3}' | 1/3qP3P6 1/3qP2 -1/3qP1 -2/3q 1/3q
phi{1,3}'' | 1/3qP3P6 1/3qP2 -1/3qP1 -2/3q 1/3q
phi{2,1} | 1/6qP2^2P3 1/6qP2 -1/6qP1 2/3q -1/3q
phi{2,2} | 1/2qP2^2P6 -1/2qP1 1/2qP2 0 0
G2[-1] | 1/2qP1^2P3 -1/2qP1 1/2qP2 0 0
G2[1] | 1/6qP1^2P6 1/6qP2 -1/6qP1 2/3q -1/3q
G2[E3] |1/3qP1^2P2^2 1/3qP2 -1/3qP1 1/3q (-1-3ER(-3))/6q
G2[E3^2] |1/3qP1^2P2^2 1/3qP2 -1/3qP1 1/3q (-1+3ER(-3))/6q
SpecialPieces(uc)
The special pieces forme a partition of the unipotent variety of a reductive group G which was defined the first time in spalt82, chap. III as the fibers of d2, where d is a "duality map". Another definition is as the set of classes in the Zariski closure of a special class and not in the Zariski closure of any smaller special class, where a special class in the support of the image of a special character by the Springer correspondence.
Each piece is a union of unipotent conjugacy classes so is represented in CHEVIE as a list of class numbers. Thus the list of special pieces is returned as a list of lists of class numbers. The list is sorted by increasing piece dimension, while each piece is sorted by decreasing class dimension, so the special class is listed first.
gap> W:=CoxeterGroup("G",2);
CoxeterGroup("G",2)
gap> SpecialPieces(UnipotentClasses(W));
[ [ 1 ], [ 4, 3, 2 ], [ 5 ] ]
gap> SpecialPieces(UnipotentClasses(W,3));
[ [ 1 ], [ 4, 3, 2, 6 ], [ 5 ] ]
The example above shows that the special pieces are different in characteristic 3.
InducedLinearForm(W, K, h)
This routine can be used to find the Richardson-Dynkin diagram of the class in the algebraic group G which contains a given unipotent class of a reductive subgroup of maximum rank S of G.
It takes a linear form on the roots of K, defined by its value on the simple roots (these values can define a Dynkin-Richardson diagram); then extends this linear form to the roots of G by 0 on the orthogonal of the roots of K; and finally conjugates the resulting form by an element of the Weyl group so that it takes positive values on the simple roots.
gap> W:=CoxeterGroup("F",4);;
gap> H:=ReflectionSubgroup(W,[1,3]);;
gap> InducedLinearForm(W,H,[2,2]);
[ 0, 1, 0, 0 ]
gap> uc:=UnipotentClasses(W);;
gap> Display(uc.classes[4]);
A1+~A1: D-R0100 C=q^18.A1xA1
The example above shows that the class containing the regular class of the
Levi subgroup of type A1× ~ A1 is the class A1+~A1.
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