STABILITY OF DECOMPOSITION NUMBERS

FOR FINITE CHEVALLEY GROUPS

Leonard Chastkofsky

Department of Mathematics University of Georgia Athens, GA 30602

§1 Introduction

Let K be an algebraic closure of a finite field 𝔽_{p}, p a prime. Let G be a connected, simply connected, semisimple algebraic group defined and split over 𝔽_{p}. Let Γ_{n} be the finite group of points of G over a finite field 𝔽_{q}, where q = p^{n}, 𝔽_{p} ⊆ 𝔽_{q} ⊆ K. We consider how the representation theory of Γ_{n} varies with n. The question as to how the characteristic p Cartan invariants vary with n was considered in general in [2]. We will consider here the corresponding question for the decomposition numbers.

The irreducible KΓ_{n} modules are indexed by the p^{n}-restricted weights of G. If Γ_{n} ⊂ Γ_{m}, and λ is a restricted weight then the irreducible KΓ_{n}-module L_{λ} can be considered as an irreducible KΓ_{m}-module. Denote the projective cover of L_{λ}, considered as a KΓ_{n} module by U(n, λ), and its character by Φ(n, λ). The way that U(n, λ) varies with n was considered in [2]. In particular, some results on how the Cartan invariants vary was obtained. An important technique in obtaining these results is the construction of certain other projective characters, denoted Ψ(n, λ). (These can be considered either as formal characters or Brauer characters. We make no distinction in the notation, leaving it to the context to make clear as to how they are to be considered.) Generally, these are equal to the Φ(n, λ), and a formula for the way they decompose in general is given in [1] and [4].

The ordinary irreducible characters can be obtained by decomposing the Deligne-Lusztig characters. These can be denoted, up to sign, as characters R_{w}(n, μ), w in the Weyl group, μ in the weight lattice, as explained in [4]. The characters Ψ(n, λ), when considered as Brauer characters can be decomposed into characters R_{w}(n, μ). The purpose of this paper is to show that for large enough p and for fixed λ and μ, the decomposition number given by the multiplicity of R_{w}(n, μ) in Ψ(n, λ) becomes stable as n becomes large. Since “generically”, the R_{w}(n, μ) are irreducible ordinary characters and the Ψ(n, λ) are projective indecomposable characters, this says that decomposition numbers are generically stable as n → ∞.

For simplicity of notation, we will assume throughout that Γ_{n} is of untwisted type, though the results can be easily extended to the twisted type case.

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§2 Preliminaries

2.1 We discuss here the background needed for our results. In particular, we recall how the projective indecomposable modules can be determined, and how they can be decomposed into ordinary irreducibles.

Let G be as in the introduction. Let T be a maximal torus in G, defined and split over 𝔽_{p}. Let X = X(T) be the group of characters on T. Let R be the set of roots of G with respect to T, and B a basis for R. For α ∈ S, let ω_{α} be the corresponding fundamental weight and let ρ = ∑_{α∈B} ω_{α}, the half-sum of the positive

roots. Let X^{+} be the set of dominant weights and for any positive integer n let X_{n} = { ∑_{α∈B} n_{α}ω_{α} : 0 ≤ n_{α} < p^{n} for all α ∈ B}. Let W be the Weyl group of G

with respect to T, and w_{0} the longest element of W. Let α_{0} be the highest short root of R. We denote the root dual to α in R by α^{∨}.

Let ℤ[X] be the group ring with canonical basis e(λ), λ ∈ X. For λ ∈ X, let s(λ) = ∑_{μ∈W λ} e(μ). For λ ∈ X^{+}, let L(λ) be the irreducible KG-module with

highest weight λ, and let χ_{p}(λ) be its character. The set {L(λ) : λ ∈ X_{n}} is the set of irreducible KΓ_{n}-modules. We can consider χ_{p}(λ) as the Brauer character of L(λ), considered as a KΓ_{n}-module.

Let F r be the Frobenius endomorphism of G. We have L(p^{m}λ_{1}+λ_{2}) ≅ L(λ_{1})^{F r^{m}}⊗ L(λ_{2}) as KG-modules for all λ_{1}, λ_{2} ∈ X^{+} and positive integers m. We also have L(p^{n}λ) ≅ L(λ)^{F r^{n}} ≅ L(λ) as KΓ_{n}-modules. Denote the Steinberg module L((p^{n} − 1)ρ) by St_{n} and its character by st_{n}.

ℤ[X]^{W} is the set of elements of ℤ[X] which are invariant under W. Note that {s(λ)^{F r^{n}}X_{p}(μ) : μ ∈ X_{n}, λ ∈ X^{+}} is a basis for ℤ[X]^{W}.

There is an automorphism x → \overline{x} of ℤ[X] with \overline{e(μ)} = e(−μ) for μ ∈ X.

2.2 Projectives.

For λ ∈ X_{n}, let U(n, λ) be the projective cover of the irreducible KΓ_{n}-module L(λ), and let Φ(n, λ) be its character. Then Φ(n, λ) is divisible in the ring of class functions on p′-elements of Γ_{n} by st_{n}.

For each λ ∈ X_{n} there is a formal character Ψ(n, λ) = st_{n} · ψ(n, λ) which can be characterized by the following property:

For every μ ∈ X_{n}, when \overline{ψ(n, λ)} χ_{p}(μ) is expressed in terms of the basis s(ν_{1})^{F r}χ_{p}(ν_{2}), ν_{1} ∈ X^{+}, ν_{2} ∈ X_{n}, of ℤ[X]^{W}, the coefficient of s(ν)^{F r}st_{n} is 0 unless λ = μ and ν = 0 when it is 1.

These characters can be constructed in a purely formal manner (see [1]), or alternatively, they can be constructed as characters of projective indecomposables in Jantzen’s category of u_{n} − T modules (see [4]).

2.3 Decomposition of ψ(n, λ) into s(μ).

There are formulas in [1] and [4] showing how to decompose ψ(n, λ) as a linear combination of characters s(μ) with positive integer coefficients. For our purposes, we only need the following description of the s(μ) which appear in the decomposition:

Lemma. If s(μ) occurs in ψ(n, λ) then μ = p^{n}ζ + w((p^{n} − 1)ρ + w_{0}ν) for some ζ ∈ X^{+}, w ∈ W and ν ∈ X_{p^{n}} is of the form ν = y(λ + ρ) − ρ + p^{n}η for some y in W and η in X^{+} (so that ν and λ are conjugate under an affine Weyl group). Moreover, we have the inequality −(p^{n} −1)ρ ≤ zμ ≤ (p^{n} −1)ρ for all z in W, where the partial order is defined by: θ_{1} ≤ θ_{2} if and only if θ_{2} − θ_{1} is a rational linear combination of positive roots.

The lemma follows immediately from the formula cited above.

2.4 Tensor product property.

A key tool in examining what happens as we extend the field is the following tensor product property of the ψ(n, λ):

If λ_{0}, …, λ_{n−1} ∈ X_{p} then

ψ(n, ∑_{i=0}^{n−1} p^{i}λ_{i}) = ∏_{i=0}^{n−1} ψ(1, λ_{i})^{F r^{i}}

(see [1, Lemma 9]).

2.5 Indecomposability.

There are formulas in [1] and [4] which show how to decompose the Ψ(n, λ) into sums of Φ(n, μ). These formulas show that if λ is far enough away from the walls of a p^{n}-alcove to which it belongs, then Ψ(n, λ) = Φ(n, λ).

2.6 Deligne-Lusztig Characters.

For w ∈ W = N_{G}(T)/T, choose a representative n_{w} ∈ N_{G}(T). By Lang’s theorem, there is an element g_{w} ∈ G with g_{w}^{−1}F r^{n}g_{w} = n_{w}. Let T_{w} = g_{w}T g_{w}^{−1}, an F r^{n}-stable maximal torus in G. For μ ∈ X, define a character θ(μ) : T_{w} → ℂ^{∗}, by first mapping t → μ(g_{w}^{−1}tg_{w}) to get an element of K^{∗}, and then composing with the map K^{∗} → ℂ^{∗} chosen when Brauer characters are defined. Define

R_{w}(n, μ) = det(w)R_{T_{w}}^{θ(μ)} ,

where the latter is the Deligne-Lusztig character.

We have the following criterion for equality of the R_{w}(n, λ):

R_{w}(n, λ) = R_{w′}(n, λ′) if and only if there exist y ∈ W and μ ∈ X such that λ′ = yλ + (p^{n} − ywy^{−1})μ, w′ = ywy^{−1} (see [4, 3.1]).

This can also be expressed as follows:

For (μ, y) ∈ X × W, define a map J(n, μ, λ) : X × W → X × W by

J(n, μ, y)(λ, w) = (yλ + (p^{n} − ywy^{−1})μ, ywy^{−1}) .

Then pairs (λ, w), (λ′, w′) give rise to the same R_{w}(n, λ) if and only if they are conjugate under this action.

2.7 Decomposing Ψ(n, λ) into Deligne-Lusztig Characters.

There is a formula in [3], expressing st_{n} · s(λ) (considered as a Brauer character) in terms of the R_{w}(n, λ). Namely, we have

st_{n} · s(λ) = (1/|Stab_{W} (λ)|) ∑_{w∈W} R_{w}(n, λ) .

Combining this with the formula for Ψ(n, λ) in terms of st_{n} · s(μ) allows us to decompose the Ψ(n, λ) into sums of R_{w}(n, μ).

§3 Statement and proof of theorem

We now state our main result.

3.1 Theorem. Suppose that λ is small enough. Then every Deligne-Lusztig character R_{w}(n, μ) which occurs in the decomposition of Ψ(n, λ) is equal to one for which μ is in C_{p^{n}}. Moreover the number of times that such a R_{w}(n, μ) occurs in the decomposition of Ψ(n, λ) is equal to the number of times that R_{w}(n+1, μ) occurs in that of Ψ(n + 1, λ).

By λ being small enough we mean that it can be written as λ = p^{n−1}λ_{1} + λ_{2} where λ_{2} ∈ X_{p^{n−1}} and ⟨λ_{1} + ρ, α_{0}^{v}⟩ < p − (h − 1) (1 + \frac{p}{p^{n}+1}). (Here, h = ⟨ρ, α_{0}^{∨}⟩ + 1 is the Coxeter number.) (If λ is also far enough away from the walls of C_{p^{n}}, Ψ(n, λ) is indecomposable. The multiplicities described in the theorem are then decomposition numbers.)

Corollary. For λ, μ ∈ X^{+}, the multiplicity of R_{w}(n, μ) in Ψ(n, λ) becomes stable as n → ∞.

The corollary follows since for large enough n, both λ and μ belong to C_{p^{n}}. The rest of this paper is devoted to proving the theorem.

3.2 Decomposition of Ψ(n, λ).

We assume that λ = p^{n−1}λ_{1} + λ_{2} is as stated in the theorem. By Humphreys’ formula (2.7), the number of times that R_{w}(n, μ) occurs in Ψ(n, λ) is equal to the number of times that s(μ) occurs in ψ(n, λ), times a constant, depending on λ and μ, but independent of n. (This constant depends on the size of the stabilizer in W of λ, and the number of v ∈ W for which R_{v}(n, μ) = R_{w}(n, μ).) By (2.3), if s(μ) occurs in ψ(n, λ), there exist w and y ∈ W and ζ and η ∈ X^{+} with μ = p^{n}ζ + wp^{n}(ρ + w_{0}η) + y(λ + ρ). Combining this with the tensor product property of Ψ(n, λ) (2.4), we have the following.

If s(μ) occurs with non-zero multiplicity in ψ(n, λ) then μ = μ_{1} + yμ_{2} with

μ_{1} = p^{n−1}(pζ_{1} + w_{1}p(ρ + w_{0}η_{1}) + y_{1}(λ_{1} + ρ))

and

μ_{2} = p^{n−1}ζ_{2} + w_{2}p^{n−1}(ρ + w_{0}η_{2}) + y_{2}(λ_{2} + ρ)

with ζ_{1}, ζ_{2}, η_{1}, η_{2} ∈ X^{+} and w_{1}, w_{2}, y_{1}, y_{2}, y ∈ W.

Moreover, for all w ∈ W, −(p−1)ρ ≤ wμ_{1} ≤ (p−1)ρ and −(p^{n−1} −1)ρ ≤ wμ_{2} ≤ (p^{n−1} − 1)ρ. The number of times that s(μ) occurs in ψ(n, λ) is the product of the multiplicity of s(μ_{1}) in ψ(1, λ_{1}) with the multiplicity of s(μ_{2}) in ψ(n − 1, λ_{2}).

Now apply the map

J(n, −y_{1}^{−1}(ζ_{1} + w_{1}(ρ + w_{0}η_{1})), y_{1}^{−1})

to the pair (μ, w). We get a pair (μ′, w′) with

μ′ = p^{n−1}[(λ_{1} + ρ) + y_{1}^{−1}y(ζ_{2} + w_{2}(ρ + w_{0}η_{2}))]
+ y_{1}^{−1}y(y_{2}(λ_{2} + ρ) + w(ζ_{1} + w_{1}(ρ + w_{0}η_{1})))

and w′ = y_{1}^{−1}wy_{1}.

We have ⟨λ_{1} + ρ, α_{0}^{∨}⟩ < p − (h − 1) (1 + \frac{p}{p^{n}+1}) and

⟨y_{1}^{−1}y[p^{n−1}(ζ_{2} + w_{2}(ρ + w_{0}η_{2})) + y_{2}(λ_{2} + ρ)], α_{0}^{∨}⟩
< (p^{n−1} − 1)⟨ρ, α_{0}^{∨}⟩ = (p^{n−1} − 1)(h − 1),

while

⟨w(ζ_{1} + w_{1}(ρ + w_{0}η_{1})), α_{0}^{∨}⟩

≤ \frac{1}{p}[(p − 1)⟨ρ, α_{0}^{∨}⟩ + ⟨λ_{1} + ρ, α_{0}^{∨}⟩]

= h − \frac{2}{p}(h − 1) − \frac{h − 1}{p^{n} + 1}

This gives

⟨μ′ + ρ, α_{0}^{∨}⟩ < p^{n} − p^{n−1}(h − 1) (1 + \frac{p}{p^{n} + 1}) + (p^{n−1} − 1)(h − 1)

+ h − \frac{2}{p}(h − 1) − \frac{h − 1}{p^{n} + 1} + (h − 1)

= p^{n} + 1 − \frac{2}{p}(h − 1).

It follows that ⟨μ′ + ρ, α_{0}^{∨}⟩ ≤ p^{n}, and that μ′ ∈ C_{p^{n}}, as stated in the theorem.

3.3 Decomposition of Ψ(n + 1, λ).

We now examine what happens when we consider the character Ψ(n + 1, λ) in Γ_{n+1}, for the same λ in C_{p^{n}}. We will show that R_{w′}(n + 1, μ′) occurs in Ψ(n + 1, λ) for the same w′ and μ′ and with the same multiplicity that R_{w′}(n, μ′) occurs in Ψ(n, λ). Again, by (2.7), the multiplicity of R_{w′}(n + 1, μ′) in Ψ(n + 1, λ) is equal to the number of times that s(μ′) occurs in ψ(n + 1, λ) times a constant independent of n.

As before from 2.3, 2.4, we get the following:

If s(ν) occurs with non-zero multiplicity in the decomposition of ψ(n + 1, λ), then ν = p^{n}ν_{0} + zp^{n−1}ν_{1} + z′ν_{2} with

ν_{0} = p(ζ′_{0} + v_{0}(ρ + w_{0}η′_{0}) + z_{0}ρ
ν_{1} = p(ζ′_{0} + v_{1}(ρ + w_{0}η′_{0}) + z_{1}(λ_{1} + ρ)
and ν_{2} = p^{n−1}[ζ′_{2} + v_{2}(ρ + w_{0}η′_{2})] + z_{2}(λ_{2} + ρ)

with ζ′_{0}, ζ′_{1}, ζ′_{2}, η′_{0}, η′_{1}, η′_{2} ∈ X^{+} and v_{0}, v_{1}, v_{2}, z_{0}, z_{1}, z_{2} ∈ W. We have −(p − 1)ρ ≤ wν_{i} ≤ (p − 1)ρ for i = 0, 1 and −(p^{n−1} − 1)ρ ≤ wν_{2} ≤ (p^{n−1} − 1)ρ. The number of times that s(ν) occurs in ψ(n + 1, λ) is the product of the number of times that s(ν_{0}) occurs in ψ(1, 0), the number of times that s(ν_{1}) occurs in ψ(1, λ_{1}), and the number of times that s(ν_{2}) occurs in ψ(n − 1, λ_{2}).

Now apply the map J(n + 1, −z_{0}^{−1}(ζ′_{0} + v_{0}(ρ + w_{0}η′_{0})), z_{0}^{−1}) to the pair (ν, v) to get (ν′, v′) with

ν′ = p^{n}(ρ + z_{0}^{−1}zζ′_{1} + z_{0}^{−1}zv_{1}(ρ + w_{0}η′_{1}))
+ p^{n−1}(z_{0}^{−1}zz_{1}(λ_{1} + ρ) + z_{0}^{−1}z′ζ′_{2} + z_{0}^{−1}z′v_{2}(ρ + w_{0}η′_{2}))
+ z_{0}^{−1}z′z_{2}(λ_{2} + ρ) + z_{0}^{−1}v(ζ′_{0} + v_{0}(ρ + w_{0}η′_{0}))

and w′ = z_{0}^{−1}vz_{0}.

If (ν′, w′) is conjugate to any pair (ν′′, w′′) with ν′′ in C_{p^{n−1}}, we must have ρ + z_{0}^{−1}ζ′_{1} + z_{0}^{−1}v_{1}(ρ + w_{0}η′_{1}) = 0. Let ν′_{1} = ν_{1} − z_{1}λ_{1}. We have that λ_{1} and 0 belong to the same alcove with corresponding weights ν_{1} and ν′_{1} to which they are conjugate under the affine Weyl group. Therefore s(ν′_{1}) occurs in ψ(1, 0) with the same multiplicity as s(ν_{1}) occurs in ψ(1, λ_{1}). We therefore have

pz_{0}^{−1}z(ζ′_{1} + v_{1}(ρ + w_{0}η′_{1}) + z_{0}^{−1}zz_{1}(ρ) ≥ −(p − 1)ρ,

with equality only when ν_{1} = pρ + w_{0}(λ_{1} + ρ). Then v_{1} is the identity, ζ′_{1} = η′_{1} = 0 and z_{0}^{−1}z = w_{0} = z_{1}. These conditions must hold and

ν′_{1} = p^{n−1}[(λ_{1} + ρ) + z_{0}^{−1}z′(ζ′_{2} + z_{2}(ρ + w_{0}η′_{2}))]
+ z_{0}^{−1}z′z_{2}(λ_{2} + ρ) + z_{0}^{−1}v(ζ′_{0} + v_{0}(ρ + w_{0}η′_{0})).

Let z_{0} = y_{1}, z′ = y, z_{2} = y_{2} and v = w. The pair (ν′, v′) is then the same as the pair (μ′, w′) obtained before. Moreover, the multiplicity of s(ν_{0}) in ψ(1, 0) is the same as that of s(μ_{1}) in ψ(1, λ_{1}), while the multiplicity of s(ν_{1}) in ψ(1, λ_{1}) is 1. This shows that R_{w′}(n, μ′) occurs in Ψ(n, λ) the same number of times that R_{w′}(n + 1, μ′) occurs in Ψ(n + 1, λ), as claimed.

References

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2. , On the Cartan invariants of a Chevalley group over GF (p^{n}), J. Algebra 150 (1992), 388-401.

3. J. E. Humphreys, Deligne-Lusztig characters and principal indecomposable modules, J. Algebra 62 (1980), 299-303.

4. J. C. Jantzen, Zur reduktion modulo p der Charaktere von Deligne und Lusztig, J. Algebra 70 (1981), 452-474.