d-Harish-Chandra series

d-Harish-Chandra series describe unipotent l-blocks of a finite reductive group $𝐆(𝔽_q)$ for $l|Φ_d(q)$ (at least, when l is not too small which means mostly not a bad prime for 𝐆). Some of the facts stated below are still partly conjectural, we do not try to distinguish precisely what has been established and what is still conjectural.

If (𝐋,λ) is a d-cuspidal pair then the constituents of the Lusztig induced $R_𝐋^𝐆(λ)$ are called a d-Harish-Chandra series; they form the unipotent part of an l-block of $𝐆^F$. It is conjectured (and proven in some cases) that the $𝐆^F$-endomorphism algebra of the l-adic cohomology of the variety 𝐗 which defines the Lusztig induction is a d-cyclotomic Hecke algebra $H_𝐆(𝐋,λ)$ for the group $W_𝐆(𝐋,λ):=N_𝐆(𝐋,λ)/𝐋$, which is a complex reflection group –- here d-cyclotomic means that the parameters of $H_𝐆(𝐋,λ)$ are monomials in q and that $H_𝐆(𝐋,λ)$ specializes to the algebra of $W_𝐆(𝐋,λ)$ for $q↦ζ_d$.

It follows that the decomposition of the Lusztig induction is of the form $R_𝐋^𝐆(λ)=∑_{ϕ∈Irr(W_𝐆(𝐋,λ))}(-1)^{nᵩ} ϕ(1)γᵩ,$ where γᵩ is a unipotent character of 𝐆^F attached to ϕ and where nᵩ is the degree $H^{nᵩ}_c(𝐗)$ where γᵩ occurss; and further for any ϕ we have $R_𝐋^𝐆(λ)(1)= (-1)^{nᵩ} γᵩ(1)Sᵩ$ where Sᵩ is the Schur element of the character of $H_𝐆(𝐋,λ)$ which deforms to ϕ. The function |Series| allows to explore a d-Harish-Chandra series.

julia> W=rootdatum("3D4")
³D₄

julia> l=cuspidal_data(W,3)
2-element Vector{NamedTuple{(:levi, :cuspidal, :d), Tuple{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}, Int64, Root1}}}:
 (levi = ³D₄, cuspidal = 8, d = ζ₃)
 (levi = ³D₄₍₎=Φ₃², cuspidal = 1, d = ζ₃)

julia> Series(W,l[2]...)
ζ₃-series R^³D₄_{³D₄₍₎=Φ₃²}(λ==Id)  H_G(L,λ)==hecke(G₄,Vector{Mvp{Cyc{Int64}, Int64}}[[ζ₃q², ζ₃, ζ₃q]])
 │    γᵩ    φ  ε family #
─┼────────────────────────
1│  φ₁‚₀ φ₁‚₀  1        1
2│  φ₁‚₆ φ₁‚₄  1        2
3│  φ₂‚₂ φ₁‚₈ -1        5
6│ φ″₁‚₃ φ₂‚₅  1        4
5│ φ′₁‚₃ φ₂‚₃ -1        3
7│  φ₂‚₁ φ₂‚₁ -1        5
4│³D₄[1] φ₃‚₂  1        5

Above we explore the 3-series corresponding to $R_𝐓^𝐆(Id)$ where 𝐆 is the triality group and 𝐓 is the torus of type (q²+q+1)². The group $W_𝐆(𝐓)$ is the complex reflection group G₄. The displays shows in the column 'γᵩ' the name of the unipotent characters constituents of $R_𝐓^𝐆(Id)$, and in the first column the number of these characters in the list of unipotent characters. In the column 'φ' the name of the character of $W_𝐆(𝐓)$ corresponding to the unipotent character γᵩ is shown; in the column 'ε' we show the sign $(-1)^{nᵩ}$. Finally in the last column we show in which family of unipotent characters is γᵩ.

The theory of d-Harish-Chandra series can be generalized to spetsial complex reflection groups using some axioms. We show below an example.

julia> W=complex_reflection_group(4)
G₄

julia> l=cuspidal_data(W,3)
5-element Vector{NamedTuple{(:levi, :cuspidal, :d), Tuple{Spets{PRSG{Cyc{Rational{Int64}}, Int16}}, Int64, Root1}}}:
 (levi = G₄, cuspidal = 3, d = ζ₃)
 (levi = G₄, cuspidal = 6, d = ζ₃)
 (levi = G₄, cuspidal = 7, d = ζ₃)
 (levi = G₄, cuspidal = 10, d = ζ₃)
 (levi = G₄₍₎=Φ₁Φ′₃, cuspidal = 1, d = ζ₃)

julia> Series(W,l[5]...)
ζ₃-series R^G₄_{G₄₍₎=Φ₁Φ′₃}(λ==Id)  W_G(L,λ)==Z₆
 │   γᵩ φ(mod 3)  ε parameter family #
─┼─────────────────────────────────────
1│ φ₁‚₀        1  1      ζ₃q²        1
5│ φ₂‚₃       ζ₆  1      -ζ₃q        2
2│ φ₁‚₄       ζ₃ -1        ζ₃        4
8│ Z₃:2       -1 -1     -ζ₃²q        2
9│Z₃:11      ζ₃² -1       ζ₃²        4
4│ φ₂‚₅      ζ₆⁵ -1       -ζ₃        4

Above we explore the 3-series corresponding to the trivial character of the torus of type (q-1)(q-ζ₃). For cyclic groups $W_𝐆(𝐋,λ)$ we display the parameters in the table since they are associated to characters of $W_𝐆(𝐋,λ)$. Finally the mention '(mod 3)' which appears in the 'φ' column means that in this case the axioms leave an ambiguity in the correspondence between unipotent characters γᵩ and characters ϕ (as well as with parameters): the correspondence is known only up to a translation by 3 (in this case, the same as a global multiplication of all ϕ by -1).

Finally, we should note that if the reflection group or coset W is not defined over the integers, what counts is not cyclotomic polynomials but factors of them over the field of definition of W. In this case, one should not give as argument an integer d representing $ζ_d$ but specify a root of unity. For instance, in the above case we get a different answer with:

julia> cuspidal_data(W,E(3,2))
5-element Vector{NamedTuple{(:levi, :cuspidal, :d), Tuple{Spets{PRSG{Cyc{Rational{Int64}}, Int16}}, Int64, Root1}}}:
 (levi = G₄, cuspidal = 2, d = ζ₃²)
 (levi = G₄, cuspidal = 5, d = ζ₃²)
 (levi = G₄, cuspidal = 7, d = ζ₃²)
 (levi = G₄, cuspidal = 10, d = ζ₃²)
 (levi = G₄₍₎=Φ₁Φ″₃, cuspidal = 1, d = ζ₃²)

cuspidal_data(W[,d[,ad]];proper=false,all=false)

returns named tuples (levi=LF,cuspidal=λ,d=d) where LF is a d-split Levi (with d-center of dimension ad if ad is given) and λ is a d-cuspidal character of LF. If d=1 this returns ordinary cuspidal characters. The character λ is given as its index in the list of unipotent characters. If d was given as an integer, it is returned as a Root1 representing E(d).

If the keyword proper=true is given, only the data where LF!=W (or equivalently ad>0) are returned.

If d is omitted, data for all d orders of eigenvalues of elements of W is returned. If in addition the keyword argument all=true is given, data for all eigenvalues of elements of W is returned.

julia> cuspidal_data(coxgroup(:F,4),1)
9-element Vector{NamedTuple{(:levi, :cuspidal, :d), Tuple{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}, Int64, Root1}}}:
 (levi = F₄, cuspidal = 31, d = 1)
 (levi = F₄, cuspidal = 32, d = 1)
 (levi = F₄, cuspidal = 33, d = 1)
 (levi = F₄, cuspidal = 34, d = 1)
 (levi = F₄, cuspidal = 35, d = 1)
 (levi = F₄, cuspidal = 36, d = 1)
 (levi = F₄, cuspidal = 37, d = 1)
 (levi = F₄₍₃₂₎=B₂₍₂₁₎Φ₁², cuspidal = 6, d = 1)
 (levi = F₄₍₎=Φ₁⁴, cuspidal = 1, d = 1)

julia> cuspidal_data(complex_reflection_group(4),3)
5-element Vector{NamedTuple{(:levi, :cuspidal, :d), Tuple{Spets{PRSG{Cyc{Rational{Int64}}, Int16}}, Int64, Root1}}}:
 (levi = G₄, cuspidal = 3, d = ζ₃)
 (levi = G₄, cuspidal = 6, d = ζ₃)
 (levi = G₄, cuspidal = 7, d = ζ₃)
 (levi = G₄, cuspidal = 10, d = ζ₃)
 (levi = G₄₍₎=Φ₁Φ′₃, cuspidal = 1, d = ζ₃)

Series(W, L, cuspidal, d)

If the reflection coset or group W corresponds to the algebraic group 𝐆 and cuspidal to the d-cuspidal unipotent character λ of 𝐋, constructs the d-series corresponding to $R_𝐋^𝐆(λ)$. The result s it is a record with the following fields and functions:

s.spets: the reflection group or coset W.

s.levi: the subcoset L.

s.cuspidal: the index of λ in UnipotentCharacters(L).

s.d: the value of d (a Root1).

relative_group(s): the group $W_𝐆(𝐋,λ)$.

dSeries.RLG(s): the UnipotentCharacter given by $R_𝐋^𝐆(λ)$.

dSeries.eps(s): for each character φ of relative_group(s) the sign $(-1)^{n_φ}$ in the cohomology of the variety defining RLG(s) of the corresponding constituent γᵩ of RLG(s).

degree(s): the generic degree of RLG(s), as a CycPol.

charnumbers(s): the indices in UnipotentCharacters(W) of the constituents of RLG(s).

hecke(s): the hecke algebra $H_𝐆(𝐋,λ)$.

The function Series has another form:

Series(<W> [,<d> [,<ad>]];k...)

where it returns a vector of Series corresponding to the cuspidal data described by the arguments and the keywords (see the help for cuspidal_data).

julia> W=complex_reflection_group(4)
G₄

julia> Series(W,3;proper=true)
1-element Vector{Series}:
 ζ₃-series R^G₄_{G₄₍₎=Φ₁Φ′₃}(λ==Id)  W_G(L,λ)==Z₆

julia> s=Series(W,3,1)[1]
ζ₃-series R^G₄_{G₄₍₎=Φ₁Φ′₃}(λ==Id)  W_G(L,λ)==Z₆
 │   γᵩ φ(mod 3)  ε parameter family #
─┼─────────────────────────────────────
1│ φ₁‚₀        1  1      ζ₃q²        1
5│ φ₂‚₃       ζ₆  1      -ζ₃q        2
2│ φ₁‚₄       ζ₃ -1        ζ₃        4
8│ Z₃:2       -1 -1     -ζ₃²q        2
9│Z₃:11      ζ₃² -1       ζ₃²        4
4│ φ₂‚₅      ζ₆⁵ -1       -ζ₃        4

julia> s.spets
G₄

julia> s.levi
G₄₍₎=Φ₁Φ′₃

julia> s.cuspidal
1

julia> s.d
Root1: ζ₃

julia> hecke(s)
hecke(G₆‚₁‚₁,Vector{Mvp{Cyc{Int64}, Int64}}[[ζ₃q², -ζ₃q, ζ₃, -ζ₃²q, ζ₃², -ζ₃]])

julia> degree(s)
ζ₃Φ₁Φ₂²Φ″₃Φ₄Φ₆

julia> dSeries.RLG(s)
[G₄]:<φ₁‚₀>-<φ₁‚₄>-<φ₂‚₅>+<φ₂‚₃>-<Z₃:2>-<Z₃:11>

julia> charnumbers(s)
6-element Vector{Int64}:
 1
 5
 2
 8
 9
 4

julia> dSeries.eps(s)
6-element Vector{Int64}:
  1
  1
 -1
 -1
 -1
 -1

julia> relative_group(s)
G₆‚₁‚₁

julia-repl

ennola(W)

Let W be an irreducible spetsial reflection group or coset, and z the generator of the center of W, viewed as a root of unity. A property checked case-by case is that, for a unipotent character γ with polynomial generic degree deg γ(q) of the spets attached to W (this spets is a finite reductive group, for W a Weyl group, in which case z=-1 if -1 is in W), deg γ(zq) is equal to ±deg γ'(q) for another unipotent character γ'; ±γ' is called the Ennola transform of γ. The function returns the permutation-with-signs done by ennola on the unipotent degrees (as a permutation-with signs of 1:length(UnipotentCharacters(W))). The argument W must be irreducible.

The permutation-with-signs is not uniquely determined by the degrees since two of them may be equal, but is uniquely determined by some additional axioms that we do not recall here.

julia> dSeries.ennola(rootdatum("3D4"))
SPerm{Int64}: (3,-4)(5,-5)(6,-6)(7,-8)

julia> dSeries.ennola(complex_reflection_group(14))
SPerm{Int64}: (2,43,-14,16,41,34)(3,35,40,18,-11,42)(4,-37,25,-17,-26,-36)(5,-6,-79)(7,-7)(8,-74)(9,-73)(10,-52,13,31,-50,29)(12,53,15,32,-51,-30)(19,71,70,21,67,68,20,69,72)(22,-39,27,-33,-28,-38)(23,24,-66,-23,-24,66)(44,46,49,-44,-46,-49)(45,48,47,-45,-48,-47)(54,-63,-55,-57,62,-56)(58,-65,-59,-61,64,-60)(75,-77)(76,-76)(78,-78)

The last example shows that it may happen that the order of z-Ennola (here 18) is greater than the order of z (here 6); this is related to the presence of irrationalities in the character table of the spetsial Hecke algebra of W.