Unipotent characters

Chevie.Uch — Module

Let $𝐆$ be a connected reductive group over the algebraic closure of a finite field $𝔽_q$, defined over $𝔽_q$ with corresponding Frobenius automorphism $F$. We want to study the irreducible characters of $𝐆 ^F$. More generally we consider $𝐆 ^F$ where $F$ is an isogeny of $𝐆$ such that a power is a Frobenius (this covers the Suzuki and Ree groups).

If $𝐓$ is an $F$-stable maximal torus of $𝐆$, and $𝐁$ is a (not necessarily $F$-stable) Borel subgroup containing $𝐓$, we define the Deligne-Lusztig variety $X_𝐁=\{g𝐁 ∈ 𝐆 /𝐁 ∣ g𝐁 ∩ F(g𝐁 )≠∅ \}$. This variety affords a natural action of $𝐆 ^F$ on the left, so the corresponding Deligne-Lusztig virtual module $H^*_c(X_𝐁):=∑ᵢ (-1)ⁱ Hⁱ_c(X_𝐁,ℚ̄ _ℓ)$ also. The (virtual) character of this module is the Deligne-Lusztig character $R_𝐓 ^𝐆 (1)$; the notation reflects the theorem that this character does not depend on the choice of $𝐁$. This character can be parameterized by an $F$-conjugacy class of $W$: if $𝐓₀⊂𝐁₀$ is an $F$-stable pair, there is an unique $w∈ W=N_𝐆 (𝐓₀)/𝐓₀$ such that the triple $(𝐓,𝐁,F)$ is $𝐆$-conjugate to $(𝐓₀,𝐁₀,wF)$. We will thus denote $R_w$ for $R_𝐓^𝐆 (1)$; this character depends only on the $F$-class of $w$.

The unipotent characters of $𝐆 ^F$ are the irreducible constituents of the $R_w$. In a similar way that the Jordan decomposition shows that the unipotent classes are a building block for describing the conjugacy classes of a reductive group, Lusztig has defined a Jordan decomposition of characters where the unipotent characters are the building block. The unipotent characters are parameterized by combinatorial data that Lusztig has defined just from the coset $Wφ$, where φ is the finite order automorphism of $X(𝐓₀)$ such that $F=qφ$. Thus, from our viewpoint, unipotent characters are objects combinatorially attached to a Coxeter coset.

A subset of the unipotent characters, the principal series unipotent characters, can be described in a more elementary way. They are the constituents of $R₁$, or equivalently the characters of the virtual module $H^*_c(X_{𝐁 ₀})$, where $X_{𝐁 ₀}$ is the discrete variety $(𝐆 /𝐁₀)^F$; this virtual module reduces to the actual module $ℚ̄ _ℓ[(𝐆 /𝐁₀) ^F]$. Thus the Deligne-Lusztig induction $R_{𝐓₀}^𝐆 (1)$ reduces to Harish-Chandra induction, 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 $R_𝐋^𝐆$ of a character $χ$ of $𝐋^F$ is the character $Ind_{𝐏^F}^{𝐆 ^F}χ̃$, where $χ̃$ is the lift to $𝐏^F$ of $χ$ via the quotient $𝐏^F/𝐔 ^F=𝐋^F$; Harish-Chandra induction is a particular case of Lusztig induction, which is defined when $𝐏$ is not $F$-stable using the variety $X_𝐔 =\{ g𝐔 ∈𝐆 /𝐔 ∣ g𝐔 ∩ F(g𝐔 )≠∅\}$, and gives for an $𝐋^F$-module a virtual $𝐆 ^F$-module. Like ordinary induction, these functors have adjoint functors going from representations of $𝐆 ^F$ to representations (resp. virtual representations) of $𝐋^F$ called Harish-Chandra restriction (resp. Lusztig restriction).

The commuting algebra of $𝐆^F$-endomorphisms of $R₁=R_{𝐓₀}^𝐆(1)$ is an Iwahori-Hecke algebra for $W^φ$, with parameters some powers of q; the parameters are all equal to q when $W^φ=W$. Thus principal series unipotent characters are parametrized by characters of $W^φ$.

To understand the decomposition of more general $R_w$, and thus parameterize 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: $Rᵪ=|W|⁻¹∑_{w∈ W}χ(wφ) R_w$. The name reflects that these class function are close to irreducible characters. They satisfy $⟨Rᵪ, R_ψ⟩_{𝐆^F}=δ_{χ,ψ}$; for the linear and unitary group they are actually unipotent characters (up to sign in the latter case). They are in general the sum (with rational coefficients) of a small number of unipotent characters in the same Lusztig family, see Families. The degree of $Rᵪ$ is a polynomial in $q$ equal to the fake degree of the character $χ$ of $Wφ$ (see fakedegree).

We now describe the parameterization of unipotent characters when $W^φ=W$, thus when the coset $Wφ$ identifies with $W$ (the situation is similar but a bit more difficult to describe in general). The (rectangular) matrix of scalar products $⟨ρ, Rᵪ⟩_{𝐆 ^F}$, when characters of $W$ and unipotent characters are arranged in the right order, is block-diagonal with rather small blocks which are called 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. To 𝓕 Lusztig associates a small group $Γ$ (not bigger than $(ℤ/2)ⁿ$, or $𝔖ᵢ$ for $i≤5$) such that the unipotent characters in the family are parameterized by the pairs $(x,θ)$ taken up to $Γ$-conjugacy, where $x∈Γ$ and $θ$ is an irreducible character of $C_Γ(x)$. Further, the elements of 𝓕 themselves are parameterized by a subset of such pairs, and Lusztig defines a pairing between such pairs which computes the scalar product $⟨ρ, Rᵪ⟩_{𝐆^F}$, called the Lusztig Fourier matrix. For more details see drinfeld_double.

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 $GLₙ$ for $n>1$, and at most one in other classical groups). The $𝐆^F$-endomorphism algebra of an Harish-Chandra induced $R_{𝐋^F}^{𝐆^F}λ$, where $λ$ is a cuspidal unipotent character turns out to be a Hecke algebra associated to the group $W_{𝐆^F}(𝐋^F):=N_{𝐆^F}(𝐋)/𝐋$, 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 $𝐋^F$ and $φ$ is an irreducible character of the relative group $W_{𝐆^F}(𝐋^F)$. 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 $𝐆^F$; 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 $λ$.

This package contains tables of all this information, and can compute Harish-Chandra and Lusztig induction of unipotent characters and almost characters. We illustrate this on some examples:

julia> W=coxgroup(:G,2)
G₂

julia> uc=UnipotentCharacters(W)
UnipotentCharacters(G₂)
┌───────┬────────────────────────────────────────────────────┐
│γ      │n₀    Deg(γ)  Feg              Symbol Fr(γ)    label│
├───────┼────────────────────────────────────────────────────┤
│φ₁‚₀   │ 1         1    1       (0,0,0,0,0,2)     1         │
│φ₁‚₆   │ 2        q⁶   q⁶ (01,01,01,01,01,12)     1         │
│φ′₁‚₃  │ 3   qΦ₃Φ₆/3   q³            (0,0,1+)     1    (1,ρ)│
│φ″₁‚₃  │ 4   qΦ₃Φ₆/3   q³            (0,0,1-)     1   (g₃,1)│
│φ₂‚₁   │ 5  qΦ₂²Φ₃/6  qΦ₈       (0,0,0,0,1,1)     1    (1,1)│
│φ₂‚₂   │ 6  qΦ₂²Φ₆/2 q²Φ₄       (0,0,0,1,0,1)     1   (g₂,1)│
│G₂[-1] │ 7  qΦ₁²Φ₃/2    0       (01,0,01,,0,)    -1   (g₂,ε)│
│G₂[1]  │ 8  qΦ₁²Φ₆/6    0       (01,01,0,,,0)     1    (1,ε)│
│G₂[ζ₃] │ 9 qΦ₁²Φ₂²/3    0       (01,0,0,01,,)    ζ₃  (g₃,ζ₃)│
│G₂[ζ₃²]│10 qΦ₁²Φ₂²/3    0       (01,01,,0,0,)   ζ₃² (g₃,ζ₃²)│
└───────┴────────────────────────────────────────────────────┘

The first column gives the name of the unipotent character, derived from its Harish-Chandra classification; the first 6 characters are in the principal series so are named by characters of W. The last 4 are cuspidal, and named by the corresponding eigenvalue of Frobenius, which is displayed in the fourth column. For classical groups, the Harish-Chandra data can be synthesized combinatorially to give a symbol.

The first two characters are each in a Lusztig family by themselves. The last eight are in a family associated to the group Γ=𝔖₃: the last column shows the parameters (x,θ). The third column shows the degree of the unipotent characters, which is transformed by the Lusztig Fourier matrix of the third column, which gives the degree of the corresponding almost character, or equivalently the fake degree of the corresponding character of W (extended by 0 outside the principal series).

One can get more information on the Lusztig Fourier matrix of the big family by asking

julia> uc.families[1]
Family(D(𝔖 ₃),[5, 6, 4, 3, 8, 7, 9, 10])
Drinfeld double of 𝔖 ₃, Lusztig′s version
┌────────┬────────────────────────────────────────────────────┐
│label   │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     .     . -1//2 -1//2     .     .│
│(g₃,1)  │    1 1//3     .  2//3 -1//3  1//3     . -1//3 -1//3│
│(1,ρ)   │    1 1//3     . -1//3  2//3  1//3     . -1//3 -1//3│
│(1,ε)   │    1 1//6 -1//2  1//3  1//3  1//6 -1//2  1//3  1//3│
│(g₂,ε)  │   -1 1//2 -1//2     .     . -1//2  1//2     .     .│
│(g₃,ζ₃) │   ζ₃ 1//3     . -1//3 -1//3  1//3     .  2//3 -1//3│
│(g₃,ζ₃²)│  ζ₃² 1//3     . -1//3 -1//3  1//3     . -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.

julia> W=coxgroup(:G,2)
G₂

julia> T=spets(reflection_subgroup(W,Int[]),W(1,2))
G₂₍₎=Φ₆

julia> u=unipotent_character(T,1)
[G₂₍₎=Φ₆]:<Id>

To construct T one could equivalently do

julia> T=torus(W,position_class(W,W(1,2)))
G₂₍₎=Φ₆

Then here are two ways to construct the Deligne-Lusztig character associated to the Coxeter torus:

julia> lusztig_induce(W,u)
[G₂]:<φ₁‚₀>+<φ₁‚₆>-<φ₂‚₁>+<G₂[-1]>+<G₂[ζ₃]>+<G₂[ζ₃²]>

julia> v=deligne_lusztig_character(W,[1,2])
[G₂]:<φ₁‚₀>+<φ₁‚₆>-<φ₂‚₁>+<G₂[-1]>+<G₂[ζ₃]>+<G₂[ζ₃²]>

julia> degree(v)
Pol{Int64}: q⁶+q⁵-q⁴-2q³-q²+q+1

julia> 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 Spetses, as defined in BroueMalleMichel2014. An example:

julia> UnipotentCharacters(complex_reflection_group(4))
UnipotentCharacters(G₄)
┌─────┬────────────────────────────────────────┐
│γ    │n₀            Deg(γ)    Feg Fr(γ)  label│
├─────┼────────────────────────────────────────┤
│φ₁‚₀ │ 1                 1      1     1       │
│φ₁‚₄ │ 2  -√-3q⁴Φ″₃Φ₄Φ″₆/6     q⁴     1   1∧ζ₆│
│φ₁‚₈ │ 3   √-3q⁴Φ′₃Φ₄Φ′₆/6     q⁸     1 -1∧ζ₃²│
│φ₂‚₅ │ 4         q⁴Φ₂²Φ₆/2   q⁵Φ₄     1  1∧ζ₃²│
│φ₂‚₃ │ 5 -ζ₃√-3qΦ″₃Φ₄Φ′₆/3   q³Φ₄     1  1∧ζ₃²│
│φ₂‚₁ │ 6 ζ₃²√-3qΦ′₃Φ₄Φ″₆/3    qΦ₄     1   1∧ζ₃│
│φ₃‚₂ │ 7            q²Φ₃Φ₆ q²Φ₃Φ₆     1       │
│Z₃:2 │ 8     -√-3qΦ₁Φ₂Φ₄/3      0   ζ₃² ζ₃∧ζ₃²│
│Z₃:11│ 9    -√-3q⁴Φ₁Φ₂Φ₄/3      0   ζ₃² ζ₃∧ζ₆⁵│
│G₄   │10        -q⁴Φ₁²Φ₃/2      0    -1  ζ₆∧-1│
└─────┴────────────────────────────────────────┘