Let G be a connected reductive group defined over the algebraic closure of a finite field Fq, with corresponding Frobenius automorphism F, ore more generally let F be an isogeny of G such that a power is a Frobenius (to cover the Suzuki and Ree groups).
If T is an F-stable maximal torus of G, and B is a (usually not F-stable) Borel subgroup containing T, we define the Deligne-Lusztig variety XB={ gB∈G/B | gB∩ F(gB) ≠∅}. This variety has a natural action of GF on the left, so the corresponding Deligne-Lusztig virtual module ∑i (-1)i Hic(XB,Ql) also. The character of this virtual module is the Deligne-Lusztig character RTG(1); the notation reflects the fact that one can prove that this character does not depend on the choice of B. Actually, this character is parameterized by an F-conjugacy class of W: if T0⊂B0 is an F-stable pair, there is an unique w∈ W=NG(T0)/T0 such that the triple (T,B,F) is conjugate to (T0,B0,wF). Thus we will denote by Rw this Deligne-Lusztig characters. It depends only on the F-class of w.
The unipotent characters of GF are the irreducible constituents of the Rw. In a similar way that the unipotent classes are a building block for describing the conjugacy classes of a reductive group, the unipotent characters are a building block for the irreducible characters of a reductive group. They can be parameterized by combinatorial data that Lusztig has attached just to the coset Wφ, where φ is the finite order automorphism of X(T0) such that F=qφ. Thus, from the viewpoint of CHEVIE, they are object combinatorially attached to a Coxeter coset.
A subset of the unipotent characters, the principal series unipotent characters, can be described in an elementary way. They are the constituents of R1, or equivalently the characters in the Deligne-Lusztig virtual module attached to T0. In that case the Deligne-Lusztig variety XB0 reduces to the discrete variety (G/B0)F, so the attached virtual module reduces to the actual module Ql[(G/B0)F]. This is the same as the constituents of the Harish-Chandra induced RT0G(1) where Harish-Chandra induction is defined as follows: let P=U⋊L be an F-stable Levi decomposition of an F-stable parabolic subgroup of G. Then the Harish-Chandra induced of a character χ of LF is the character IndPFGF~χ, where ~χ is the lift to PF of χ via the quotient PF/UF=LF; Harish-Chandra induction is a particular case of Lusztig induction, which is defined when P is not F-stable using the variety XU={ gU∈G/U | gU∩ F(gU) ≠∅}, and gives for an LF-module a virtual GF-module. Like ordinary induction, these functors have adjoint functors going from representations of GF to representations (resp. virtual representations) of LF called Harish-Chandra restriction (resp. Lusztig restriction).
The commuting algebra of GF-endomorphisms of Ql[(G/B0)F] is an Iwahori-Hecke algebra for Wφ (with some parameters which are powers of q; they are all equal to q when Wφ=W). Thus principal series unipotent characters correspond to characters of Wφ.
To understand the decomposition of Deligne-Lusztig characters, and thus unipotent characters, is is useful to introduce another set of class functions which are parameterized by irreducible characters of the coset Wφ. If χ is such a character, we define the associated almost character by: Rχ=|W|-1∑w∈ Wχ(wφ) Rw. The reason to the name is that these class function are close to irreducible characters: they satisfy 〈 Rχ, Rψ〉GF=δχ,ψ; for the linear and unitary group they are actually unipotent characters (up to sign in the latter case). They are in general sum (with rational coefficients) of a small number of unipotent characters in the same Lusztig family. The degree of Rχ is a polynomial in q equal to the fake degree of the character χ of Wφ (see Functions for Reflection cosets).
We now describe the parameterization of unipotent characters when Wφ=W, thus when the coset Wφ identifies with W (the situation is similar but a bit more difficult to describe in general). The characters of Wφ=W and the unipotent characters are divided in Lusztig families. For the characters of W a family F corresponds to a block of the Hecke algebra over a ring called the Rouquier ring. Each almost character Rχ for χ∈ F decomposes over the same subset of unipotent characters, disjoint from the subsets needed for Rψ with ψ∉ F. Further, to F Lusztig associates a small group Γ (not bigger than (Z/2)n, or Si for i ≤ 5) such that the unipotent characters corresponding to F are parameterized by the pairs (x,θ) taken up to Γ-conjugacy, where x∈Γ and θ is an irreducible character of CΓ(x). Further, the elements of F themselves are parameterized by a subset of such pairs, and their is a symmetric unitary matrix call the Lusztig Fourier transform in the vector space spanned by these pairs, which has the property that its (x,θ),(x1,θ1)-coefficient describes the scalar product between an Rχ where χ is attached to (x,θ) and a unipotent character attached to the pair (x1,θ1).
A second parameterization of unipotent character is via Harish-Chandra series. A character is called cuspidal if all its proper Harish-Chandra restrictions vanish. There are few cuspidal unipotent characters (none in linear groups, and at most one in other classical groups). The GF-endomorphism algebra of an Harish-Chandra induced RLFGFλ, where λ is a cuspidal unipotent character turns out to be a Hecke algebra associated to the group WGF(LF):=NGF(L)/L, which turns out to be a Coxeter group. Thus another parameterization is by triples (L,λ,φ), where λ is a cuspidal unipotent character of LF and φ is an irreducible character of the relative group WGF(LF). Such characters are said to belong to the Harish-Chandra series determined by (L,λ).
A final piece of information attached to unipotent characters is the eigenvalues of Frobenius. Let Fδ be the smallest power of the isogeny F which is a split Frobenius (that is, Fδ is a Frobenius and φδ=1). Then Fδ acts naturally on Deligne-Lusztig varieties and thus on the corresponding virtual modules, and commutes to the action of GF; thus for a given unipotent character ρ, a submodule of the virtual module which affords ρ affords a single eigenvalue μ of Fδ. Results of Lusztig and Digne-Michel show that this eigenvalue is of the form qaδλρ where 2a∈Z and λρ is a root of unity which depends only on ρ and not the considered module. This λρ is called the eigenvalue of Frobenius attached to ρ. Unipotent characters in the Harish-Chandra series of a pair (L,λ) have the same eigenvalue of Frobenius as λ.
CHEVIE contains table of all this information, and can compute Harish-Chandra and Lusztig induction of unipotent characters and almost characters. We illustrate the information on some examples:
gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> uc:=UnipotentCharacters(W); UnipotentCharacters( G2 ) gap> Display(uc); Unipotent characters for G2 Name | Degree FakeDegree Eigenvalue Label ______________________________________________________________ phi{1,0} | 1 1 1 phi{1,6} | q^6 q^6 1 phi{1,3}' | (1/3)qP3P6 q^3 1 (1,r) phi{1,3}'' | (1/3)qP3P6 q^3 1 (g3,1) phi{2,1} | (1/6)qP2^2P3 qP8 1 (1,1) phi{2,2} | (1/2)qP2^2P6 q^2P4 1 (g2,1) G2[-1] | (1/2)qP1^2P3 0 -1 (g2,eps) G2[1] | (1/6)qP1^2P6 0 1 (1,eps) G2[E3] |(1/3)qP1^2P2^2 0 E3 (g3,theta) G2[E3^2] |(1/3)qP1^2P2^2 0 E3^2 (g3,theta^2)
The first column gives the name of the unipotent character; the first 6 are in the principal series so are named according to the corresponding characters of W. The last 4 are cuspidal, and named by the corresponding eigenvalue of Frobenius, which is displayed in the fourth column. In general the names of the unipotent characters come from their parameterization by Harish-Chandra series; in addition, for classical groups, they are associated to symbols.
The first two characters are each in a family by themselves. The last eight
One can get more information on the Lusztig Fourier matrix of the big family by asking
gap> Display(uc.families[1]); |eigen ___________________________________________________________ (1,1) | 1 1/6 1/2 1/3 1/3 1/6 1/2 1/3 1/3 (g2,1) | 1 1/2 1/2 0 0 -1/2 -1/2 0 0 (g3,1) | 1 1/3 0 2/3 -1/3 1/3 0 -1/3 -1/3 (1,r) | 1 1/3 0 -1/3 2/3 1/3 0 -1/3 -1/3 (1,eps) | 1 1/6 -1/2 1/3 1/3 1/6 -1/2 1/3 1/3 (g2,eps) | -1 1/2 -1/2 0 0 -1/2 1/2 0 0 (g3,theta) | E3 1/3 0 -1/3 -1/3 1/3 0 2/3 -1/3 (g3,theta^2) | E3^2 1/3 0 -1/3 -1/3 1/3 0 -1/3 2/3
One can do computations with individual unipotent characters. Here we construct the Coxeter torus, and then the identity character of this torus as a unipotent character.
gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> T:=ReflectionCoset(ReflectionSubgroup(W,[]),EltWord(W,[1,2])); (q^2-q+1) gap> u:=UnipotentCharacter(T,1); [(q^2-q+1)]=<>
Then here are two ways to construct the Deligne-Lusztig character associated to the Coxeter torus:
gap> LusztigInduction(W,u); [G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]> gap> v:=DeligneLusztigCharacter(W,[1,2]); [G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]> gap> Degree(v); q^6 + q^5 - q^4 - 2*q^3 - q^2 + q + 1 gap> v*v; 6
The last two lines ask for the degree of v, then for the scalar product of v with itself.
Finally we mention that CHEVIE can also provide unipotent characters of
gap> Display(UnipotentCharacters(ComplexReflectionGroup(4))); Unipotent characters for G4 Name | Degree FakeDegree Eigenvalue Label _________________________________________________________________ phi{1,0} | 1 1 1 phi{1,4} | (-ER(-3)/6)q^4P"3P4P"6 q^4 1 1.-E3^2 phi{1,8} | (ER(-3)/6)q^4P'3P4P'6 q^8 1 -1.E3^2 phi{2,5} | (1/2)q^4P2^2P6 q^5P4 1 1.E3^2 phi{2,3} |((3+ER(-3))/6)qP"3P4P'6 q^3P4 1 1.E3^2 phi{2,1} |((3-ER(-3))/6)qP'3P4P"6 qP4 1 1.E3 phi{3,2} | q^2P3P6 q^2P3P6 1 Z3:2 | (-ER(-3)/3)qP1P2P4 0 E3^2 E3.E3^2 Z3:11 | (-ER(-3)/3)q^4P1P2P4 0 E3^2 E3.-E3 G4 | (-1/2)q^4P1^2P3 0 -1 -E3^2.-1
UnipotentCharacters(W)
W should be a Coxeter group, a Coxeter Coset or a Spetses. The function gives back a record containing information about the unipotent characters of the associated algebraic group (or Spetses). This contains the following fields:
group
:
charNames
:
charSymbols
:
harishChandra
:
levi
:l
such that L corresponds to ReflectionSubgroup(W,l)
.
cuspidalName
:
eigenvalue
:
relativeType
:
parameterExponents
:
charNumbers
:
families
:
charNumbers
:
fourierMat
:
eigenvalues
:
group
:
charLabels
:
gap> W:=CoxeterGroup("Bsym",2); CoxeterGroup("Bsym",2) gap> WF:=CoxeterCoset(W,(1,2)); 2Bsym2 gap> uc:=UnipotentCharacters(W); UnipotentCharacters( Bsym2 ) gap> Display(uc); Unipotent characters for Bsym2 Name | Degree FakeDegree Eigenvalue Label _____________________________________________ 11. | (1/2)qP4 q^2 1 +,- 1.1 |(1/2)qP2^2 qP4 1 +,+ .11 | q^4 q^4 1 2. | 1 1 1 .2 | (1/2)qP4 q^2 1 -,+ B2 |(1/2)qP1^2 0 -1 -,- gap> uc.harishChandra[1]; rec( levi := [ ], relativeType := [ rec( series :="B", indices :=[ 1, 2 ], rank :=2 ) ], eigenvalue := 1, parameterExponents := [ 1, 1 ], charNumbers := [ 1, 2, 3, 4, 5 ], cuspidalName := [ ]) gap> uc.families[2]; Family("012",[1,2,5,6]) gap> Display(uc.families[2]); label |eigen +,- +,+ -,+ -,- ________________________________ +,- | 1 1/2 1/2 -1/2 -1/2 +,+ | 1 1/2 1/2 1/2 1/2 -,+ | 1 -1/2 1/2 1/2 -1/2 -,- | -1 -1/2 1/2 -1/2 1/2
This function requires the package "chevie" (see RequirePackage).
85.2 Display for UnipotentCharacters
On can control the display of unipotent characters in various ways.
In the record controlling Display
, a field items
will specify
which columns are displayed. The possible values are
"n0"
:
"Name"
:
"Degree"
:
"FakeDegree"
:
"Eigenvalue"
:
"Symbol"
:
"Family"
:
"Delta1"
:
The default value is
items:=["Name","Degree","FakeDegree","Eigenvalue","Family"]
This can be changed by setting the variable UnipotentCharactersOps.items
which holds this default value. In addition if the field byFamily
is set,
the characters are displayed family by family instead of in index order.
Finally, the field chars
can be set, indicating which characters are to be
displayed in which order.
gap> W:=CoxeterGroup("B",2); CoxeterGroup("B",2) gap> uc:=UnipotentCharacters(W); UnipotentCharacters( B2 ) gap> Display(uc); Unipotent characters for B2 Name | Degree FakeDegree Eigenvalue Label _____________________________________________ 11. | (1/2)qP4 q^2 1 +,- 1.1 |(1/2)qP2^2 qP4 1 +,+ .11 | q^4 q^4 1 2. | 1 1 1 .2 | (1/2)qP4 q^2 1 -,+ B2 |(1/2)qP1^2 0 -1 -,- gap> Display(uc,rec(byFamily:=true)); Unipotent characters for B2 Name | Degree FakeDegree Eigenvalue Label _____________________________________________ *.11 | q^4 q^4 1 _____________________________________________ 11. | (1/2)qP4 q^2 1 +,- *1.1 |(1/2)qP2^2 qP4 1 +,+ .2 | (1/2)qP4 q^2 1 -,+ B2 |(1/2)qP1^2 0 -1 -,- _____________________________________________ *2. | 1 1 1 gap> Display(uc,rec(items:=["n0","Name","Symbol"])); Unipotent characters for B2 n0 |Name Symbol __________________ 1 | 11. (12,0) 2 | 1.1 (02,1) 3 | .11 (012,12) 4 | 2. (2,) 5 | .2 (01,2) 6 | B2 (012,)
This function requires the package "chevie" (see RequirePackage).
UnipotentCharacter(W,l)
Constructs an object representing the unipotent character of the algebraic group associated to the Coxeter group or Coxeter coset W which is specified by l. There are 3 possibilities for l: if it is an integer, the l-th unipotent character of W is returned. If it is a string, the unipotent character of W whose name is l is returned. Finally, l can be a list of length the number of unipotent characters of W, which specifies the coefficient to give to each.
gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> u:=UnipotentCharacter(W,7); [G2]=<G2[-1]> gap> v:=UnipotentCharacter(W,"G2[E3]"); [G2]=<G2[E3]> gap> w:=UnipotentCharacter(W,[1,0,0,-1,0,0,2,0,0,1]); [G2]=<phi{1,0}>-<phi{1,3}''>+2<G2[-1]>+<G2[E3^2]>
This function requires the package "chevie" (see RequirePackage).
85.4 Operations for Unipotent Characters
+
:
-
:
*
:
gap> u+v; [G2]=<G2[-1]>+<G2[E3]> gap> w-2*u; [G2]=<phi{1,0}>-<phi{1,3}''>+<G2[E3^2]> gap> w*w; 7
Degree
:
gap> Degree(w); q^5 - q^4 - q^3 - q^2 + q + 1 gap> Degree(u+v); (5/6)*q^5 + (-1/2)*q^4 + (-2/3)*q^3 + (-1/2)*q^2 + (5/6)*q
String
and Print
:CHEVIE.PrintUniChars
. It is a record; if the field
short
is bound (the default) they are printed in a compact form. If the
field long
is bound, they are printed one character per line:
gap> CHEVIE.PrintUniChars:=rec(long:=true); rec( long := true ) gap> w; [G2]= <phi{1,0}> 1 <phi{1,6}> 0 <phi{1,3}'> 0 <phi{1,3}''> -1 <phi{2,1}> 0 <phi{2,2}> 0 <G2[-1]> 2 <G2[1]> 0 <G2[E3]> 0 <G2[E3^2]> 1 gap> CHEVIE.PrintUniChars:=rec(short:=true);;
This function requires the package "chevie" (see RequirePackage).
DeligneLusztigCharacter(W,w)
This function returns the Deligne-Lusztig character RTG(1) of the algebraic group G associated to the Coxeter group or Coxeter coset W.
gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> DeligneLusztigCharacter(W,3); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''> gap> DeligneLusztigCharacter(W,W.1); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''> gap> DeligneLusztigCharacter(W,[1]); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''> gap> DeligneLusztigCharacter(W,[1,2]); [G2]=<phi{1,0}>+<phi{1,6}>-<phi{2,1}>+<G2[-1]>+<G2[E3]>+<G2[E3^2]>
This function requires the package "chevie" (see RequirePackage).
LusztigInduction(W,u)
u should be a unipotent character of a parabolic subcoset of the Coxeter coset W. It represents a unipotent character λ of a Levi L of the algebraic group G attached to W. The program returns the Lusztig induced RLG(λ).
gap> W:=CoxeterGroup("G",2);; gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.1); (q-1)(q+1) gap> u:=UnipotentCharacter(T,1); [(q-1)(q+1)]=<> gap> LusztigInduction(CoxeterCoset(W),u); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''> gap> DeligneLusztigCharacter(W,W.1); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''>
This function requires the package "chevie" (see RequirePackage).
LusztigRestriction(R,u)
u should be a unipotent character of a parent Coxeter coset W of which R is a parabolic subcoset. It represents a unipotent character γ of the algebraic group G attached to W, while R represents a Levi subgroup L. The program returns the Lusztig restriction * RLG(γ).
gap> W:=CoxeterGroup("G",2);; gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.1); (q-1)(q+1) gap> u:=DeligneLusztigCharacter(W,W.1); [G2]=<phi{1,0}>-<phi{1,6}>-<phi{1,3}'>+<phi{1,3}''> gap> LusztigRestriction(T,u); [(q-1)(q+1)]=4<> gap> T:=CoxeterSubCoset(CoxeterCoset(W),[],W.2); (q-1)(q+1) gap> LusztigRestriction(T,u); [(q-1)(q+1)]=0
This function requires the package "chevie" (see RequirePackage).
LusztigInductionTable(R,W)
R should be a parabolic subgroup of the Coxeter group W or a parabolic
subcoset of the Coxeter coset W, in each case representing a Levi
subgroup L of the algebraic group G associated to W. The function
returns a table (modeled after InductionTable
, see InductionTable)
representing the Lusztig induction RLG between unipotent
characters.
gap> W:=CoxeterGroup("B",3);; gap> t:=Twistings(W,[1,3]); [ ~A1xA1<3>.(q-1), ~A1xA1<3>.(q+1) ] gap> Display(LusztigInductionTable(t[2],W)); Lusztig Induction from ~A1xA1<3>.(q+1) to B3 |11,11 11,2 2,11 2,2 ___________________________ 111. | 1 -1 -1 . 11.1 | -1 . 1 -1 1.11 | . . -1 . .111 | -1 . . . 21. | . . . . 1.2 | 1 -1 . 1 2.1 | . 1 . . .21 | . . . . 3. | . . . 1 .3 | . 1 1 -1 B2:2 | . . 1 -1 B2:11 | 1 -1 . .
This function requires the package "chevie" (see RequirePackage).
DeligneLusztigLefschetz(h)
Here h is an element of a Hecke algebra associated to a Coxeter group W
which itself is associated to an algebraic group G. By results of
Digne-Michel, for g∈GF, the number of fixed points of Fm on the
Deligne-Lusztig variety associated to the element wφ of the Coxeter
coset Wφ, have, for m sufficiently divisible, the form ∑χ
χqm(Twφ)Rχ(g) where χ runs over the irreducible
characters of Wφ, where Rχ is the corresponding almost
character, and where χqm is the Hecke algebra of Wφ with
parameter qm. This expression is called the Lefschetz character of the
Deligne-Lusztig variety. It can be seen as a sum of unipotent characters
with coefficients polynomials in the rootParameter
s of the Hecke algebra.
The function DeligneLusztigLefschetz
takes as argument a Hecke element
and returns the corresponding Lefschetz character. This is defined on the
whole of the Hecke algebra by linearity. The Lefschetz character of various
varieties related to Deligne-Lusztig varieties, like their completions or
desingularisation, can be obtained by taking the Lefschetz character at
various elements of the Hecke algebra.
gap> W:=CoxeterGroup("A",2);; gap> q:=X(Rationals);;q.name:="q";; gap> H:=Hecke(W,q); Hecke(A2,q) gap> T:=Basis(H,"T"); function ( arg ) ... end gap> DeligneLusztigLefschetz(T(1,2)); [A2]=<111>-q<21>+q^2<3> gap> DeligneLusztigLefschetz((T(1)+T())*(T(2)+T())); [A2]=q<21>+(q^2+2q+1)<3>
The last line shows the Lefschetz character of the Samelson-Bott desingularisation of the Coxeter element Deligne-Lusztig variety.
We now show an example with a coset (corresponding to the unitary group).
gap> H:=Hecke(CoxeterCoset(W,(1,2)),q^2); Hecke(2A2,q^2) gap> T:=Basis(H,"T"); function ( arg ) ... end gap> DeligneLusztigLefschetz(T(1)); [2A2]=-<11>-q<2A2>+q^2<2>
This function requires the package "chevie" (see RequirePackage).
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GAP 3.4.4