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 (
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. 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 trival): 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).
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
: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
:
springerSeries
:The records describing individual unipotent classes have the following fields:
name
:
parameter
:
Au
:
dynkin
: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; [ 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).
87.2 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.
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4