Unipotent classes of reductive groups
Chevie.Ucl
Chevie.Ucl.GreenTable
Chevie.Ucl.ICCTable
Chevie.Ucl.UnipotentClass
Chevie.Ucl.UnipotentClasses
Chevie.Ucl.XTable
Chevie.Ucl.UnipotentValues
Chevie.Ucl.distinguished_parabolics
Chevie.Ucl.induced_linear_form
Chevie.Ucl.special_pieces
Chevie.Ucl
â ModuleThis module gives 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.
We give the following information for a unipotent element u
of each class:
the centralizer $C_ð (u)$, that we describe by the reductive part of $C_ð (u)^0$, by the group of components $A(u):=C_ð (u)/C_ð (u)^0$, and by the dimension of its radical.
in good characteristic, the 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 $ð€$. By the Jacobson-Morozov theorem there exists an $ð°ð©â$ subalgebra of $ð€$ containing $e$ as the element $\begin{pmatrix}1&0\\0&1 \end{pmatrix}$. Let $ð$ be the torus $\begin{pmatrix}h&0\\0&h^{-1} \end{pmatrix}$ of $SLâ$ and let $ð$ be a maximal torus containing $ð$, so that $ð$ is the image of a one-parameter subgroup $Ïâ Y(ð)$. Consider the root decomposition $ð€=â_{αâΣ}ð€_α$ given by $ð$; then $αâŠâšÏ,αâ©$ defines a linear form on $Σ$, determined by its value on simple roots. It is possible to choose a system of simple roots $Î $ so that $âšÏ,αâ©â¥ 0$ for $αâÎ $, and then $âšÏ,αâ©â{0,1,2}$ for any $αâÎ $. The Dynkin diagram of $Î $ 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 $ð€$-orbit.
Let $â¬$ be the variety of all Borel subgroups and let $â¬áµ€$ be the subvariety of Borel subgroups containing the unipotent element u
. Then $dim C_ð(u)=rank ð + 2 dim â¬_u$ 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 $âšÏ,αâ©â{0,1}$.
We describe now the Springer correspondence. Indecomposable locally constant $ð$-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 $ð$-conjugacy) cuspidal pair of a Levi subgroup $ð$ of $ð$ and a cuspidal local system on an unipotent class of $ð$, such that the set of local systems associated to a given cuspidal pair is parameterized by the characters of the relative Weyl group $W_ð (ð):=N_ð (ð)/ð$. 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 $ðœ_q$ and that $F$ is the Frobenius attached to an $ðœ_q$-structure of $ð$. Let $C$ be an $F$-stable unipotent class and let $uâ C^F$; we call $C$ the geometric class of $u$ and the $ð^F$-classes inside $C^F$ 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 $Ï$ of $A(u)$ we associate the characteristic function of the corresponding local system (actually associated to an extension $ÏÌ$ of $Ï$ to $A(u).F$); it is a class function $Y_{u,Ï}$ on $ð^F$ which can be normalized so that: $Y_{u,Ï}(uâ)=ÏÌ(cF)$ if $uâ$ is geometrically conjugate to $u$ and its $ð^F$-class is parameterized by the $F$-conjugacy class $cF$ of $A(u)$, otherwise $Y_{u,Ï}(uâ)=0$. If the pair $u,Ï$ corresponds via the Springer correspondence to the character $Ï$ of $W_ð(ð)$, then $Y_{u,Ï}$ 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 $CÌ$. To such a complex of sheaves is associated its characteristic function, a class function of $ð^F$ obtained by taking the alternating trace of the Frobenius acting on the stalks of the cohomology sheaves. If $Y_Ï$ is the characteristic function of a local system, the characteristic function of the corresponding intersection cohomology complex is denoted by $X_Ï$. This function is supported on $CÌ$, and Lusztig has shown that $X_Ï=âᵩ P_{Ï,Ï} Yᵪ$ where $P_{Ï,Ï}$ are integer polynomials in $q$ and $Yᵪ$ are attached to local systems on classes lying in $CÌ$.
Lusztig and Shoji have given an algorithm to compute the matrix $P_{Ï,Ï}$, which is implemented in Chevie. The relationship with characters of $ð(ðœ_q)$, taking to simplify the ordinary Springer correspondence, is that the restriction to the unipotent elements of the almost character $R_Ï$ is equal to $q^{bᵪ} Xᵪ$, where $bᵪ$ is $dim â¬áµ€$ for an element u
of the class C
such that the support of Ï
is $CÌ$. The restriction of the Deligne-Lusztig characters $R_w$ 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:
julia> UnipotentClasses(rootdatum(:sl,4))
UnipotentClasses(slâ)
1111<211<22<31<4
uâD-R dBu B-C C(u) Aâ(Aâââ=Ίâ³) Aâ(Aâââââ=AâÃAâΊâ)/-1 .(Aâ)/ζâ
âââââŒââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
4 â222 0 222 q³.Zâ 1:4 -1:2 ζâ:Id
31 â202 1 22. qâŽ.Aâââ=Ίâ Id:31
22 â020 2 2.2 qâŽ.Aâ.Zâ 2:22 11:11
211 â101 3 2.. qâµ.Aââââ=AâΊâ Id:211
1111â000 6 ... Aâ Id:1111
uâ.(Aâ)/ζâ³
âââââŒââââââââââ
4 â ζâ³:Id
31 â
22 â
211 â
1111â
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 $⬠ᵀ$. The column B-C
gives the Bala-Carter classification of $u$, that is in the case of $slâ$ 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 $C_ð(u)$: a power $qáµ$ describes that the unipotent radical of $C_ð(u)$ has dimension $d$ (thus $qáµ$ rational points); then follows a description of the reductive part of the neutral component of $C_ð(u)$, given by the name of its root datum. Then if $C_ð(u)$ 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 $C_ð(u)^0$ unipotent of dimension 3
and $A(u)$ equal to Z4
, the cyclic group of order 4. The class 22
has $C_G(u)$ 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 $C_ð(u)$ connected. For the class 31
the reductive part of $C_G(u)$ 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 $ð$. 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 $ð$; 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:
julia> UnipotentClasses(coxgroup(:A,3))
UnipotentClasses(Aâ)
1111<211<22<31<4
uâD-R dBu B-C C(u) Aâ(Aâââ=Ίâ³)
âââââŒâââââââââââââââââââââââââââââââââââââââ
4 â222 0 222 q³ Id:4
31 â202 1 22. qâŽ.Aâââ=Ίâ Id:31
22 â020 2 2.2 qâŽ.Aâ Id:22
211 â101 3 2.. qâµ.Aââââ=AâΊâ Id:211
1111â000 6 ... Aâ Id:1111
Here is another example:
julia> UnipotentClasses(coxgroup(:G,2))
UnipotentClasses(Gâ)
1<Aâ<AÌâ<Gâ(aâ)<Gâ
uâD-R dBu B-C C(u) Gâ(Gâââ=Ίâ²) .(Gâ)
âââââââŒâââââââââââââââââââââââââââââââââââââââââ
Gâ â 22 0 22 q² Id:Ïâââ
Gâ(aâ)â 20 1 20 qâŽ.Sâ 21:Ïâ²âââ 3:Ïâââ 111:Id
AÌâ â 01 2 .2 q³.Aâ Id:Ïâââ
Aâ â 10 3 2. qâµ.Aâ Id:Ïâ³âââ
1 â 00 6 .. Gâ Id:Ïâââ
which illustrates that on class Gâ(aâ)
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
:
julia> UnipotentClasses(coxgroup(:G,2),3)
UnipotentClasses(Gâ,3)
1<Aâ,(AÌâ)â<AÌâ<Gâ(aâ)<Gâ
uâdBu B-C C(u) Gâ(Gâââ=Ίâ²) .(Gâ) .(Gâ) .(Gâ)
âââââââŒââââââââââââââââââââââââââââââââââââââââââââââ
Gâ â 0 22 q².Zâ 1:Ïâââ ζâ:Id ζâ²:Id
Gâ(aâ)â 1 20 qâŽ.Zâ 2:Ïâââ 11:Id
AÌâ â 2 .2 qâ¶ Id:Ïâââ
Aâ â 3 2. qâµ.Aâ Id:Ïâ³âââ
(AÌâ)â â 3 ?? qâµ.Aâ Id:Ïâ²âââ
1 â 6 .. Gâ Id:Ïâââ
The function ICCTable
gives the transition matrix between the functions $Xᵪ$ and $Y_Ï$.
julia> uc=UnipotentClasses(coxgroup(:G,2));
julia> t=ICCTable(uc)
Coefficients of Xᵪ on Yᵩ for series L=Gâââ=Ίâ² W_G(L)=Gâ
âGâ Gâ(aâ)âœÂ²Â¹âŸ Gâ(aâ) AÌâ Aâ 1
âââââââŒââââââââââââââââââââââââââââââ
XÏâââ â 1 0 1 1 1 1
XÏâ²ââââ 0 1 0 1 0 q²
XÏâââ â 0 0 1 1 1 Ίâ
XÏâââ â 0 0 0 1 1 Ίâ
XÏâ³ââââ 0 0 0 0 1 1
XÏâââ â 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 $A(u)=Aâ$.
Chevie.Ucl.UnipotentClasses
â TypeUnipotentClasses(W[,p])
W
should be a CoxeterGroup
record for a Weyl group or RootDatum
describing a reductive algebraic group ð
. The function returns a record containing information about the unipotent classes of ð
in characteristic p
(if omitted, p
is assumed to be any good characteristic for ð
). 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 $CÌⱌâ CÌáµ¢$ and there is no class $Câ$ such that $CÌⱌâ CÌââ CÌáµ¢$.
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 ð
.
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 $C_ð(u)$.
dimBu
: the dimension of the variety ðáµ€
.
The records for classes contain additional fields for certain groups: for instance, the names given to classes by Mizuno in Eâ, Eâ, Eâ
or by Shoji in Fâ
. See the help for UnipotentClass
for more details.
The records describing individual Springer series have the following fields:
levi
:the indices of the reflections corresponding to the Levi subgroup ð
where lives the cuspidal local system ι
from which the Springer series is induced.
relgroup
: The relative Weyl group $N_ð(ð,ι)/ð$. The first series is the principal series for which .levi=[]
and .relgroup=W
.
locsys
: a list of length nconjugacy_classes(.relgroup)
, holding in i
-th position a pair describing which local system corresponds to the i
-th character of $N_ð(ð,ι)$. 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 center.
julia> W=rootdatum(:sl,4)
slâ
julia> uc=UnipotentClasses(W);
julia> uc.classes
5-element Vector{UnipotentClass}:
UnipotentClass(1111)
UnipotentClass(211)
UnipotentClass(22)
UnipotentClass(31)
UnipotentClass(4)
The show
function for unipotent classes accepts all the options of formatTable
and of 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 Springer 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 the non-cuspidal part of the Springer correspondence of the unipotent classes of Eâ
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 Gâ
(this is convenient for checking our data with the original paper of Spaltenstein):
julia> uc=UnipotentClasses(rootdatum(:E6sc));
julia> xdisplay(uc;cols=[5,6,7],spaltenstein=true,frame=true,mizuno=true,
order=false)
UnipotentClasses(Eâsc)
uâ Eâ(Eâââ) Gâ(Eâââââ
ââ=AâÃAâ)/ζâ Gâ(Eâââââ
ââ=AâÃAâ)/ζâ²
âââââââŒââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Eâ â 1:1â ζâ:1 ζâ²:1
Eâ(aâ)â 1:6â ζâ:ε_c ζâ²:ε_c
Dâ
â Id:20â
Aâ
+Aâ â -1:15â 1:30â ζâ:ΞⲠζâ²:Ξâ²
Aâ
â 1:15_q ζâ:ΞⳠζâ²:Ξâ³
Dâ
(aâ)â Id:64â
Aâ+Aâ â Id:60â
Dâ â Id:24â
Aâ â Id:81â
Dâ(aâ)â111:20â 3:80â 21:90â
Aâ+Aâ â Id:60â
2Aâ+Aââ 1:10â ζâ:εâ ζâ²:εâ
Aâ â Id:81ââ²
Aâ+2Aââ Id:60ââ²
2Aâ â 1:24âⲠζâ:ε ζâ²:ε
Aâ+Aâ â Id:64ââ²
Aâ â 11:15ââ² 2:30ââ²
3Aâ â Id:15_qâ²
2Aâ â Id:20ââ²
Aâ â Id:6ââ²
1 â Id:1ââ²
Chevie.Ucl.UnipotentClass
â TypeA struct UnipotentClass
representing the class C
of a unipotent element u
of the reductive group ð
with Weyl group W
, contains always the following information
.name
The name ofC
.parameter
A parameter describingC
. Sometimes the same as.name
; a partition describing the Jordan form, for classical groups..dimBu
The dimension of the variety of Borel subgroups containingu
.
For some types there is a field .mizuno
or .shoji
giving alternate names used in the literature.
A UnipotentClass
contains also some of the following information (all of it for some types and some characteristics but sometimes much less)
.dynkin
the Dynkin-Richardson diagram ofC
(a vector giving a weight 0, 1 or 2 to the simple roots)..dimred
the dimension of the reductive part ofC_G(u)
..red
aCoxeterCoset
recording the type of the reductive part ofC_G(u)
, with the twisting induced by the Frobenius if any..Au
the groupA_G(u)=C_G(u)/C^0_G(u)
..balacarter
encodes the Bala-Carter classification ofC
, which says thatu
is distinguished in a LeviL
(the Richardson class in a parabolicP_L
) as a vector listing the indices of the simple roots inL
, with those not inP_L
negated..rep
a list of indices for roots such that ifU=UnipotentGroup(W)
thenprod(U,u.rep)
is a representative ofC
..dimunip
the dimension of the unipotent part ofC_G(u)
..AuAction
anExtendedCoxeterGroup
recording the action ofA_G(u)
onred
.
Chevie.Ucl.ICCTable
â TypeICCTable(uc,seriesNo=1;q=Pol())
This function gives the table of decompositions of the functions $X_ι$ in terms of the functions $Y_ι$. Here ι
is a ð
-equivariant local system on the class C
of a unipotent element u
. Such a local system is parametrized by the pair (u,Ï)
of u
and a character of the group of components A(u)
of $C_ð (u)$. The function $Y_ι$ is the characteristic function of this local system and $X_ι$ is the characteristic function of the corresponding intersection cohomology complex on CÌ
. The Springer correspondence says that the local systems can also be indexed by characters of a relative Weyl group. Since the coefficient of Xᵪ
on Yᵩ
is 0
if Ï
and Ï
are not characters of the same relative Weyl group (are not in the same Springer series), the table is for one Springer series, specified by the argument 'seriesNo' (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 Pol()
is assumed).
julia> uc=UnipotentClasses(coxgroup(:A,3));t=ICCTable(uc)
Coefficients of Xᵪ on Yᵩ for series L=Aâââ=Ίâ³ W_G(L)=Aâ
â4 31 22 211 1111
ââââââŒâââââââââââââââââ
X4 â1 1 1 1 1
X31 â0 1 1 Ίâ Ίâ
X22 â0 0 1 1 Ίâ
X211 â0 0 0 1 Ίâ
X1111â0 0 0 0 1
In the above the multiplicities are given as products of cyclotomic polynomials to display them more compactly. However the format of such a table can be controlled more precisely.
For instance, one can ask to not display the entries as products of cyclotomic polynomials:
julia> xdisplay(t;cycpol=false)
Coefficients of Xᵪ on Yᵩ for A3
â4 31 22 211 1111
ââââââŒâââââââââââââââââââ
X4 â1 1 1 1 1
X31 â0 1 1 q+1 q²+q+1
X22 â0 0 1 1 q²+1
X211 â0 0 0 1 q²+q+1
X1111â0 0 0 0 1
Since show
uses the function format
for tables, 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):
julia> uc=UnipotentClasses(coxgroup(:F,4));
julia> t=ICCTable(uc);
julia> sh=[13,24,22,18,14,9,11,19];
julia> show(IOContext(stdout,:rows=>sh,:cols=>sh,:limit=>true),t);
Coefficients of Xᵪ on Yᵩ for Fâ
âAâ+AÌâ Aâ AÌâ Aâ+AÌâ AÌâ+Aâ BââœÂ¹Â¹âŸ Bâ Câ(aâ)âœÂ¹Â¹âŸ
âââââââŒâââââââââââââââââââââââââââââââââââââââââââââ
XÏâââââ 1 0 0 0 0 0 0 0
XÏâ³ââââ 1 1 0 0 0 0 0 0
XÏâ²ââââ 1 0 1 0 0 0 0 0
XÏâ³ââââ 1 1 0 1 0 0 0 0
XÏâ²ââââ Ίâ 1 1 1 1 0 0 0
XÏâââ â q² 0 0 0 0 1 0 0
XÏâ³ââââ Ίâ Ίâ 0 1 0 0 1 0
XÏâ²ââââ q² 0 Ίâ 0 1 0 0 1
The function 'ICCTable' returns an object with various pieces of information which can help further computations.
.scalar
: this contains the table of multiplicities Pᵪᵩ
of the Xᵪ
on the Yᵩ
. One should pay attention that by default, the table is not displayed in the same order as the stored |.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.
.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 $dimâ¬áµ€$ for each local system (u,Ï)
in the series.
:L
: The matrix of (unnormalized) scalar products of the functions $Yᵩ$ with themselves, that is the $(Ï,Ï)$ entry is $â_{gâð(ðœ_q)} Yᵩ(g) Yᵪ(g)$. 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 $Pᵩᵪ$.
Chevie.Ucl.XTable
â TypeXTable(uc;classes=false)
This function presents in a different way the information obtained from ICCTable
. Let $XÌ_{u,Ï}=q^{1/2(codim C-dim Z(ð ))}X_{u,Ï}$ where C
is the class of u
and Z(ð )
is the center of Levi subgroup on which lives the cuspidal local system attached to the local system (u,Ï)
.
Then XTable
gives the decomposition of the functions $XÌ_{u,Ï}$ on local systems, by default. If classes==true
, it gives the values of the functions $XÌ_{u,Ï}$ on unipotent classes.
julia> W=coxgroup(:G,2)
Gâ
julia> XTable(UnipotentClasses(W))
Values of character sheaves XÌᵪ on local systems Ï
XÌᵪ|Ïâ 1 Aâ AÌâ Gâ(aâ)âœÂ¹Â¹Â¹âŸ Gâ(aâ)âœÂ²Â¹âŸ Gâ(aâ) Gâ
âââââââââââŒââââââââââââââââââââââââââââââââââââââââââââ
X_Ïâââ^Gâ â 1 1 1 0 0 1 1
X_Ïâââ^Gâ â qâ¶ 0 0 0 0 0 0
X_Ïâ²âââ^Gââ q³ 0 q 0 q 0 0
X_Ïâ³âââ^Gââ q³ q³ 0 0 0 0 0
X_Ïâââ^Gâ â qΊâ q q 0 0 q 0
X_Ïâââ^Gâ âq²Ίâ q² q² 0 0 0 0
X_Id^. â 0 0 0 q² 0 0 0
The functions XÌ
in the first column are decorated by putting as an exponent the relative groups $W_ð (ð)$.
julia> XTable(UnipotentClasses(W);classes=true)
Values of character sheaves XÌᵪ on unipotent classes
XÌᵪ|classâ 1 Aâ AÌâ Gâ(aâ) Gâ(aâ)ââââ Gâ(aâ)âââ Gâ
âââââââââââŒââââââââââââââââââââââââââââââââââââââââââ
X_Ïâââ^Gâ â 1 1 1 1 1 1 1
X_Ïâââ^Gâ â qâ¶ 0 0 0 0 0 0
X_Ïâ²âââ^Gââ q³ 0 q 2q 0 -q 0
X_Ïâ³âââ^Gââ q³ q³ 0 0 0 0 0
X_Ïâââ^Gâ â qΊâ q q q q q 0
X_Ïâââ^Gâ âq²Ίâ q² q² 0 0 0 0
X_Id^. â 0 0 0 q² -q² q² 0
julia> XTable(UnipotentClasses(W,2))
Values of character sheaves XÌᵪ on local systems Ï
XÌᵪ|Ïâ 1 Aâ AÌâ Gâ(aâ)âœÂ¹Â¹Â¹âŸ Gâ(aâ)âœÂ²Â¹âŸ Gâ(aâ) GââœÂ¹Â¹âŸ Gâ
âââââââââââŒâââââââââââââââââââââââââââââââââââââââââââââââââââ
X_Ïâââ^Gâ â 1 1 1 0 0 1 0 1
X_Ïâââ^Gâ â qâ¶ 0 0 0 0 0 0 0
X_Ïâ²âââ^Gââ q³ 0 q 0 q 0 0 0
X_Ïâ³âââ^Gââ q³ q³ 0 0 0 0 0 0
X_Ïâââ^Gâ â qΊâ q q 0 0 q 0 0
X_Ïâââ^Gâ âq²Ίâ q² q² 0 0 0 0 0
X_Id^. â 0 0 0 q² 0 0 0 0
X_Id^. â 0 0 0 0 0 0 q 0
julia> XTable(UnipotentClasses(rootdatum(:sl,4)))
Values of character sheaves XÌᵪ on local systems Ï
XÌᵪ|Ïâ1111 211 22âœÂ¹Â¹âŸ 22 31 4 4^(ζâ) 4âœâ»Â¹âŸ 4^(ζâ³)
âââââââââŒâââââââââââââââââââââââââââââââââââââââââââââ
Xââââ^Aââ qâ¶ 0 0 0 0 0 0 0 0
Xâââ^Aâ âq³Ίâ q³ 0 0 0 0 0 0 0
Xââ^Aâ âq²Ίâ q² 0 q² 0 0 0 0 0
Xââ^Aâ â qΊâ qΊâ 0 q q 0 0 0 0
Xâ^Aâ â 1 1 0 1 1 1 0 0 0
Xââ^Aâ â 0 0 q³ 0 0 0 0 0 0
Xâ^Aâ â 0 0 q² 0 0 0 0 q 0
X_Id^. â 0 0 0 0 0 0 q³ââ 0 0
X_Id^. â 0 0 0 0 0 0 0 0 q³ââ
A side effect of calling XTable
with classes=true
is to compute the cardinal of the unipotent conjugacy classes:
julia> t=XTable(UnipotentClasses(coxgroup(:G,2));classes=true);
julia> CycPol.(t.cardClass)
7-element Vector{CycPol{Cyc{Rational{Int64}}}}:
1
ΊâΊâΊâΊâ
q²ΊâΊâΊâΊâ
q²Ίâ²Ίâ²ΊâΊâ/6
q²Ίâ²Ίâ²ΊâΊâ/2
q²Ίâ²Ίâ²ΊâΊâ/3
qâŽÎŠâ²Ίâ²ΊâΊâ
Chevie.Ucl.GreenTable
â TypeGreenTable(uc;classes=false)
Keeping the same notations as in the description of 'XTable', this function returns a table of the functions $Q_{wF}$, attached to elements $wFâ W_ð (ð)â
F$ where $W_ð (ð)$ are the relative weyl groups attached to cuspidal local systems. These functions are defined by $Q_{wF}=â_{u,Ï} ÏÌ(wF) XÌ_{u,Ï}$. An point to note is that in the principal Springer series, when ð
is a maximal torus, the function $Q_{wF}$ coincides with the Deligne-Lusztig character $R^ð _{ð_W}(1)$. As for 'XTable', by default the table gives the values of the functions on local systems. If classes=true
is given, then it gives the values of the functions $Q_{wF}$ on conjugacy classes.
julia> W=coxgroup(:G,2)
Gâ
julia> GreenTable(UnipotentClasses(W))
Values of Green functions Q_wF on local systems Ï
QᎵ_wF|Ïâ 1 Aâ AÌâ Gâ(aâ)âœÂ¹Â¹Â¹âŸ Gâ(aâ)âœÂ²Â¹âŸ Gâ(aâ) Gâ
âââââââââââŒâââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Q_Aâ^Gâ â Ίâ²ΊâΊâ ΊâΊâ (2q+1)Ίâ 0 q 2q+1 1
Q_AÌâ^Gâ â-ΊâΊâΊâΊâ -ΊâΊâ Ίâ 0 q 1 1
Q_Aâ^Gâ â-ΊâΊâΊâΊâ ΊâΊâ -Ίâ 0 -q 1 1
Q_Gâ^Gâ â Ίâ²Ίâ²Ίâ -ΊâΊâ² -ΊâΊâ 0 -q Ίâ 1
Q_Aâ^Gâ â Ίâ²Ίâ²Ίâ Ίâ²Ίâ -ΊâΊâ 0 q -Ίâ 1
Q_Aâ+AÌâ^Gââ Ίâ²ΊâΊâ -ΊâΊâ (2q-1)Ίâ 0 -q -2q+1 1
Q_^. â 0 0 0 q² 0 0 0
The functions $Q_{wF}$ 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 $W_ð(ð)$. The exponents are the names of the groups $W_ð(ð)$.
julia> GreenTable(UnipotentClasses(W);classes=true)
Values of Green functions Q_wF on unipotent classes
QᎵ_wF|classâ 1 Aâ AÌâ Gâ(aâ) Gâ(aâ)ââââ Gâ(aâ)âââ Gâ
ââââââââââââŒâââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Q_Aâ^Gâ â Ίâ²ΊâΊâ ΊâΊâ (2q+1)Ίâ 4q+1 2q+1 Ίâ 1
Q_AÌâ^Gâ â-ΊâΊâΊâΊâ -ΊâΊâ Ίâ 2q+1 1 -Ίâ 1
Q_Aâ^Gâ â-ΊâΊâΊâΊâ ΊâΊâ -Ίâ -2q+1 1 Ίâ 1
Q_Gâ^Gâ â Ίâ²Ίâ²Ίâ -ΊâΊâ² -ΊâΊâ -Ίâ Ίâ 2q+1 1
Q_Aâ^Gâ â Ίâ²Ίâ²Ίâ Ίâ²Ίâ -ΊâΊâ Ίâ -Ίâ -2q+1 1
Q_Aâ+AÌâ^Gâ â Ίâ²ΊâΊâ -ΊâΊâ (2q-1)Ίâ -4q+1 -2q+1 -Ίâ 1
Q_^. â 0 0 0 q² -q² q² 0
julia> GreenTable(UnipotentClasses(rootdatum(:sl,4)))
Values of Green functions Q_wF on local systems Ï
QᎵ_wF|Ïâ 1111 211 22âœÂ¹Â¹âŸ 22 31 4 4^(ζâ) 4âœâ»Â¹âŸ 4^(ζâ³)
âââââââââŒâââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Qââââ^Aââ Ίâ²ΊâΊâ (3q²+2q+1)Ίâ 0 (2q+1)Ίâ 3q+1 1 0 0 0
Qâââ^Aâ â-ΊâΊâΊâΊâ -q³+q²+q+1 0 Ίâ Ίâ 1 0 0 0
Qââ^Aâ â Ίâ²ΊâΊâ -ΊâΊâ 0 2q²-q+1 -Ίâ 1 0 0 0
Qââ^Aâ â Ίâ²Ίâ²Ίâ -ΊâΊâ 0 -ΊâΊâ 1 1 0 0 0
Qâ^Aâ â -Ίâ³ΊâΊâ Ίâ²Ίâ 0 -Ίâ -Ίâ 1 0 0 0
Qââ^Aâ â 0 0 q²Ίâ 0 0 0 0 q 0
Qâ^Aâ â 0 0 -q²Ίâ 0 0 0 0 q 0
Q_^. â 0 0 0 0 0 0 q³ââ 0 0
Q_^. â 0 0 0 0 0 0 0 0 q³ââ
Chevie.Ucl.UnipotentValues
â FunctionUnipotentValues(uc,classes=false)
This function returns a table of the values of unipotent characters on local systems (by default) or on unipotent classes (if classes=true
).
julia> W=coxgroup(:G,2)
Gâ
julia> UnipotentValues(UnipotentClasses(W);classes=true)
Values of unipotent characters for Gâ on unipotent classes
â 1 Aâ AÌâ Gâ(aâ) Gâ(aâ)ââââ Gâ(aâ)âââ Gâ
ââââââââŒââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Ïâââ â 1 1 1 1 1 1 1
Ïâââ â qâ¶ 0 0 0 0 0 0
Ïâ²âââ â qΊâΊâ/3 -qΊâΊâ/3 q (q+5)q/3 -qΊâ/3 qΊâ/3 0
Ïâ³âââ â qΊâΊâ/3 (2q²+1)q/3 0 qΊâ/3 -qΊâ/3 (q+2)q/3 0
Ïâââ â qΊâ²Ίâ/6 (2q+1)qΊâ/6 qΊâ/2 (q+5)q/6 -qΊâ/6 qΊâ/6 0
Ïâââ â qΊâ²Ίâ/2 qΊâ/2 qΊâ/2 -qΊâ/2 qΊâ/2 -qΊâ/2 0
Gâ[-1] â qΊâ²Ίâ/2 -qΊâ/2 -qΊâ/2 -qΊâ/2 qΊâ/2 -qΊâ/2 0
Gâ[1] â qΊâ²Ίâ/6 (2q-1)qΊâ/6 -qΊâ/2 (q+5)q/6 -qΊâ/6 qΊâ/6 0
Gâ[ζâ] âqΊâ²Ίâ²/3 -qΊâΊâ/3 0 qΊâ/3 -qΊâ/3 (q+2)q/3 0
Gâ[ζâ²]âqΊâ²Ίâ²/3 -qΊâΊâ/3 0 qΊâ/3 -qΊâ/3 (q+2)q/3 0
julia> UnipotentValues(UnipotentClasses(W,3);classes=true)
Values of unipotent characters for Gâ on unipotent classes
â 1 Aâ AÌâ Gâ(aâ) Gâ(aâ)âââ Gâ Gâ_(ζâ)
ââââââââŒââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââââ
Ïâââ â 1 1 1 1 1 1 1
Ïâââ â qâ¶ 0 0 0 0 0 0
Ïâ²âââ â qΊâΊâ/3 -qΊâΊâ/3 q/3 qΊâ/3 -qΊâ/3 -2q/3 q/3
Ïâ³âââ â qΊâΊâ/3 (2q²+1)q/3 q/3 qΊâ/3 -qΊâ/3 -2q/3 q/3
Ïâââ â qΊâ²Ίâ/6 (2q+1)qΊâ/6 (3q+1)q/6 qΊâ/6 -qΊâ/6 2q/3 -q/3
Ïâââ â qΊâ²Ίâ/2 qΊâ/2 qΊâ/2 -qΊâ/2 qΊâ/2 0 0
Gâ[-1] â qΊâ²Ίâ/2 -qΊâ/2 -qΊâ/2 -qΊâ/2 qΊâ/2 0 0
Gâ[1] â qΊâ²Ίâ/6 (2q-1)qΊâ/6 (-3q+1)q/6 qΊâ/6 -qΊâ/6 2q/3 -q/3
Gâ[ζâ] âqΊâ²Ίâ²/3 -qΊâΊâ/3 q/3 qΊâ/3 -qΊâ/3 q/3 (-ζâ+2ζâ²)q/3
Gâ[ζâ²]âqΊâ²Ίâ²/3 -qΊâΊâ/3 q/3 qΊâ/3 -qΊâ/3 q/3 (2ζâ-ζâ²)q/3
â Gâ_(ζâ²) (AÌâ)â
ââââââââŒââââââââââââââââââââââââââ
Ïâââ â 1 1
Ïâââ â 0 0
Ïâ²âââ â q/3 (2q²+1)q/3
Ïâ³âââ â q/3 -qΊâΊâ/3
Ïâââ â -q/3 (2q+1)qΊâ/6
Ïâââ â 0 qΊâ/2
Gâ[-1] â 0 -qΊâ/2
Gâ[1] â -q/3 (2q-1)qΊâ/6
Gâ[ζâ] â (2ζâ-ζâ²)q/3 -qΊâΊâ/3
Gâ[ζâ²]â(-ζâ+2ζâ²)q/3 -qΊâΊâ/3
Chevie.Ucl.induced_linear_form
â Functioninduced_linear_form(W, K, h)
This routine can be used to find the Richardson-Dynkin diagram of the class in the algebraic group ð
which contains a given unipotent class of a reductive subgroup of maximum rank ð
of ð
.
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 ð
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.
julia> W=coxgroup(:F,4)
Fâ
julia> H=reflection_subgroup(W,[1,3])
Fâââââ=AâÃAÌâΊâ²
julia> induced_linear_form(W,H,[2,2])
4-element Vector{Int64}:
0
1
0
0
julia> uc=UnipotentClasses(W);
julia> uc.classes[4].prop
Dict{Symbol, Any} with 8 entries:
:dynkin => [0, 1, 0, 0]
:dimred => 6
:red => AâÃAâ
:Au => .
:balacarter => [1, 3]
:rep => [1, 3]
:dimunip => 18
:AuAction => AâÃAâ
julia> uc.classes[4]
UnipotentClass(Aâ+AÌâ)
The example above shows that the class containing the regular class of the Levi subgroup of type Aâà AÌâ
is the class |A1+~A1|.
Chevie.Ucl.special_pieces
â Function'special_pieces(<uc>)'
The special pieces forme a partition of the unipotent variety of a reductive group ð
which was defined the first time in Spaltenstein1982 chap. III as the fibers of d^2
, 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.
julia> W=coxgroup(:G,2)
Gâ
julia> special_pieces(UnipotentClasses(W))
3-element Vector{Vector{Int64}}:
[1]
[4, 3, 2]
[5]
julia> special_pieces(UnipotentClasses(W,3))
3-element Vector{Vector{Int64}}:
[1]
[4, 3, 2, 6]
[5]
The example above shows that the special pieces are different in characteristic 3.
Chevie.Ucl.distinguished_parabolics
â Functiondistinguished_parabolics(W)
the list of distinguished parabolic subgroups of W
in the sense of Richardson, each given as the list of the corresponding indices. The distinguished unipotent conjugacy classes of W
consist of the dense unipotent orbit in the unipotent radical of such a parabolic.
julia> W=coxgroup(:F,4)
Fâ
julia> distinguished_parabolics(W)
4-element Vector{Vector{Int64}}:
[]
[3]
[1, 3]
[1, 3, 4]