Decomposition Numbers of Chevalley Groups

Leonard Chastkofsky Department Of Mathematics University of Georgia Athens, GA 30602

Introduction

Consider a finite Chevalley group G(p^{n}) defined over a field of p^{n} elements, p a prime. The projective indecomposable characters in characteristic p are indexed by the p^{n} - restricted weights. If a weight λ is p-restricted, it is also p^{n}-restricted. If Φ_{λ,n} is the corresponding projective indecomposable character of G(p^{n}), we can ask how the decomposition of Φ_{λ,n} into ordinary irreducible characters varies with n.

The ordinary irreducible characters are obtained from the Deligne-Lusztig characters. In this paper we will show how to write the Deligne-Lusztig characters in terms of what we call a p-adic coding, so that we can answer the above question. We will show how to associate with each component of the p-adic coding a matrix, so that the trace of the product of the corresponding matrices gives the multiplicity of the corresponding character in Φ_{λ,n} .

We shall assume here that we are in the generic case, so that all the Deligne-Lusztig characters occurring in the decomposition of Φ_{λ,n} are irreducible. We shall also assume that Φ_{λ,n} can be expressed as a tensor product of G(p) - projective characters raised to a power of the Frobenius. Indeed, our method allows us to decompose, more generally, G(p^{n}) - projectives which can be written this way. The results apply as well to the twisted groups, and illustrate the relationship between the decomposition numbers in the twisted and untwisted case. The methods can be extended to the non-generic case, which we shall deal with elsewhere.

We apply our method to determine the decomposition of Φ_{λ,n} for SL_{3}(p^{n}) and SU_{3}(p^{n}), λ a generic weight for SL_{3}(p). We also show how generic Cartan invariants of G(p^{n}) can be calculated using our methods, and again apply this to SL_{3}(p^{n}) and SU_{3}(p^{n}).

Our method works when p is a large enough prime, although it can be extended to work for all primes.

§1. Preliminaries

Let G be a semi-simple simply connected algebraic group, defined over k, the algebraic closure of a finite field 𝔽_{p} of characteristic p. Denote the Frobenius au-

Typeset by AMS-TEX

tomorphism by Fr. Let π be an automorphism of the Dynkin diagram, and let F = F r^{n} ∘ π be the induced automorphism of G. Denote the finite group of fixed points of F on G by G(p^{n})_{π}.

Let T be a maximal torus of G, split and defined over 𝔽_{p}. Let X = X(T) be the weight lattice. Let X_{p^{n}} be the subset of p^{n} - restricted weights, consisting of those elements of X whose coordinates in terms of the fundamental dominant weights lie between 0 and p^{n} − 1. Let ρ be half the sum of the positive roots.

The irreducible p-modular characters of G(p^{n})_{π} are indexed by X_{p^{n}}. For λ ∈ X_{p^{n}}, let χ_{p}(λ) be the corresponding p-modular irreducible, and let Φ_{λ,n} be the character of its projective cover. In the generic case, these characters have a nice tensor product property: (see [1] and [2]). If λ is written p-adically, λ = ∑ p^{k}λ_{k}, with each λ_{k} p - restricted, then Φ_{λ} = Φ_{λ,n} = ∏_{k} Φ_{λk}^{F r^{k}}, where Φ_{λ_{k}} = Φ_{λ_{k},1} is the

projective cover of χ_{p}(λ_{k}) as a kG(p) character. Moreover, each Φ_{λ_{k}} is divisible by the Steinberg character for G(p), which we denote by St. Thus we can write

Φ_{λ_{k}} = St·φ(λ_{k}) and Φ_{λ,n} = St_{n}·∏_{k} φ(λ_{k})^{F r^{k}}, where St_{n} = ∏_{k} St^{F r^{k}} is the Steinberg

character for G(p^{n})_{π}.

Let W be the Weyl group of G. For λ ∈ X, let s(λ) = ∑_{λ′} e(λ′) the sum over

distinct λ′ in the W -orbit of λ.

In [1] , a formula is given for expressing φ(λ) as a linear combination of the s(λ), modulo a knowledge of the characters χ_{p}(λ). Thus in the generic case, we can write

a projective indecomposable character of G(p^{n})_{π} as Φ = St_{n}· ∏_{k=0}^{n−1} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}},

where a_{k} is a function from the p-restricted weight lattice X_{p} to ℤ.

The ordinary irreducible characters are obtained from the Deligne-Lusztig characters. We shall use Jantzen’s notation for these characters (see [4]). Let w ∈ W and let n_{w} ∈ N_{g}(T) be a representative of the coset wT. There is an element g_{w} ∈ G such that g_{w}^{−1}F g_{w} = n_{w}. Let T_{w} = g_{w}T g_{w}^{−1}. If λ ∈ X, we have a homomorphism from T_{w} ∩ G(p^{n})_{π} → k^{∗} given by t → λ(g_{w}^{−1}tg_{w}), and consequently one also from T_{w} ∩ G(p^{n})_{π} to ℂ^{∗}, which we denote by θ. (We use the same embedding of roots of unity as we do in relating Brauer characters to ordinary characters.) This gives rise to a Deligne-Lusztig character R_{T_{w}}(θ) of G(p^{n})_{π}, and we denote ϵ_{G}ϵ_{T_{w}}R_{T_{w}}(θ) by R(w, λ).(R(w, λ) depends on n and π but we shall suppress this in the notation.)

We have an action of W × X on itself given by (w, λ)^{(v,ν)} → (vwπv^{−1}π^{−1}, λ + (p^{n} − vwπv^{−1})ν). We have R(w, λ) = R(y, μ) if and only if (w, λ) and (y, μ) are conjugate under this action.

§2. P -adic Coding of Deligne-Lusztig Characters

To analyze the decomposition numbers, we will be considering Deligne-Lusztig

characters R(w, λ) with λ = ∑_{k=0}^{n−1} p^{k}w_{k}λ_{k} , λ_{k} ∈ X_{p}. It turns out that what is

important for the analysis is how w_{k} is related to w_{k+1}. We therefore introduce an association between n-tuples ([μ_{0}, z_{n−1}]; [μ_{n−1}, z_{n−2}]; ⋯; [μ_{1}, z_{0}]) with μ_{i} ∈ X^{+} and z_{i} ∈ W and pairs (w, λ), with w ∈ W and λ ∈ X, which we call a p-adic coding.

In the following definition, subscripts are to be read modulo n, so a subscript of −1 is understood to be the same as n − 1.

\underline{Definition 2.1}: Let [μ_{k}, z_{k−1}] ∈ X_{p} × W for k = 0, …, n − 1. For 0 ≤ k ≤ n − 2, let w_{k} = z_{n−2}z_{n−3} ⋯ z_{k}. Let y = z_{n−2}z_{n−3} ⋯ z_{0}z_{n}. Let λ = p^{n−1}μ_{n−1} +

∑_{k=0}^{n−2} p^{k}w_{k}μ_{k}. Define χ([μ_{0}, z_{n−1}]; [μ_{n−1}, z_{n−2}]; ⋯; [μ_{1}, z_{0}]) = R(y, λ). We call

χ([μ_{0}, z_{n−1}]; ⋯; [μ_{1}, z_{0}]) a p-adic coding for R(y, λ).

The following proposition shows how to get a p-adic coding, given a Deligne-Lusztig character.

\underline{Proposition 2.2}: Let λ ∈ X and suppose we can write λ = ∑_{k=0}^{n−1} p^{k}w_{k}μ_{k} with

μ_{k} ∈ X_{p}. For 0 ≤ k ≤ n − 2, let z_{k} = w_{k+1}^{−1}w_{k}, and let z_{n−1} = w_{0}^{−1}yπw_{n−1}π^{−1}. Then R(y, λ) = χ([μ_{0}, z_{n−1}]; [μ_{n−1}, z_{n−2}], ⋯; [μ_{1}, z_{0}]).

\underline{Proof}: By definition,

χ([μ_{0}, z_{n−1}]; ⋯; [μ_{1}, z_{0}]) = R(v, p^{n−1}μ_{n−1} + ∑ p^{k}u_{k}μ_{k})

where

u_{k} = z_{n−2} ⋯ z_{k}
= (w_{n−1}^{−1}w_{n−2})(w_{n−2}^{−1}w_{n−3}) ⋯ (w_{k+1}^{−1}w_{k})
= w_{n−1}^{−1}w_{k} , for 0 ≤ k ≤ n − 2,

and v = z_{n−2} ⋯ z_{0}z_{n−1} = w_{n−1}^{−1}yπw_{n−1}π^{−1}. Now apply (w_{n}, 0) to (v, p^{n−1}μ_{n−1} + ∑ p^{k}u_{k}μ_{k}), via the action described at the end of section 1. We get R(v, p^{n−1}μ_{n−1}+ ∑ p^{k}u_{k}μ_{k}) = R(w_{n−1}vπw_{n−1}^{−1}π^{−1}, p^{n−1}w_{n−1}μ_{n−1}+∑ p^{k}w_{n−1}u_{k}μ_{k}). But w_{n−1}vπw_{n−1}^{−1}π^{−1} = y and w_{n}u_{k} = w_{k} for 0 ≤ k ≤ n − 2, so the latter is equal to R(y, λ), as claimed.

For purposes of analyzing how R(w, λ) behaves in characteristic p, it is also useful to write λ as ∑ p^{k}w_{k}μ_{k}, with μ_{k} “small”. Accordingly, we introduce the following notation.

Let W̃_{a,p} be the extended affine Weyl group, i.e., the semi-direct product of W in its action on ℝ⊗X with translations by elements of pX. Let C̃_{p} be the fundamental domain for W̃_{a,p} in this action which is contained in X^{+} and contains the origin. Then every element of λ ∈ X can be written as λ = pλ′ + τλ′′ with λ′′ ∈ C̃_{p} , λ′ ∈ X, and τ ∈ W.

\underline{Proposition 2.3}:

Let λ = ∑_{k=0}^{n−1} p^{k}y_{k}λ_{k}, with each λ_{k} ∈ X_{p}, and write λ_{k} = pλ′_{k} +τ_{k}λ′′_{k} as above. Let

y ∈ W. For 0 ≤ k ≤ n−2, let w_{k} = τ_{k+1}^{−1}y_{k+1}^{−1}y_{k}, and let w_{n−1} = τ_{0}^{−1}y_{0}^{−1}yπy_{n−1}π^{−1}. Let μ_{0}, …, μ_{n−1} ∈ C̃_{p} and z_{0}, …, z_{n−1} ∈ W. Assume λ′′_{k+1} + w_{k}λ′_{k} is conjugate under the Weyl group to an element of C̃_{p} for all k. Then R(yλ) has p-adic coding χ([μ_{n−1}, z_{n−1}] ⋯; [μ_{0}, z_{0}]) if and only if there exist v_{0}, v_{1}, …, v_{n−1} ∈ W such that for 0 ≤ k ≤ n − 2, z_{k} = v_{k}^{−1}w_{k}τ_{k}v_{k−1} and μ_{k} = v_{k}^{−1}(λ′′_{k+1} + w_{k}λ′_{k}), while z_{n−1} = v_{n−1}^{−1}w_{n−1}πτ_{n−1}v_{n−2}π^{−1} and μ_{n−1} = v_{n−1}^{−1}(λ′′_{0} + w_{n−1}πλ′_{n−1}) .

The subscripts, as usual, are to be read modulo n.

\underline{Proof}: R(y, ∑_{k=0}^{n−1} p^{k}y_{k}λ_{k})

= R(y, ∑_{k=0}^{n−1} p^{k}y_{k}(pλ′_{k} + τ_{k}λ′′_{k})

= R(y, p^{n}y_{n−1}λ′_{n−1} + ∑_{k=1}^{n−1} p^{k}y_{k}τ_{k}(λ′′_{k} + τ_{k}^{−1}y_{k}^{−1}y_{k−1}λ′_{k−1}) + y_{0}τ_{0}λ′′_{o})

= R(y, p^{n}y_{n−1}λ′_{n−1} + ∑_{k=1}^{n−1} p^{k}y_{k}τ_{k}(λ′′_{k} + w_{k−1}λ′_{k−1}) + y_{0}τ_{0}λ′′_{0} + (p^{n} − yπ)(−y_{n−1}λ′_{n−1})

= R(y, ∑_{k+1}^{n−1} p^{k}y_{k}τ_{k}(λ′′_{k} + w_{k−1}λ′_{k−1}) + y_{o}τ_{0}(λ′′_{0} + τ_{0}^{−1}y_{0}^{−1}yπy_{n−1}λ′_{n−1}))

= R(y, ∑_{k=1}^{n−1} p^{k}y_{k}τ_{k}(λ′′_{k} + w_{k−1}λ′_{k−1}) + y_{0}τ_{0}(λ′′_{0} + w_{n−1}πλ′_{n−1}))

For 0 ≤ k ≤ n − 2, let μ_{k} be the element of X^{+} conjugate to λ′′_{k+1} + w_{k}λ′_{k} and let v_{k} be an element of W such that v_{k}μ_{k} = λ′′_{k+1} + w_{k}λ′_{k}. Let μ_{n−1} be the element of X^{+} conjugate to λ′′_{0} + w_{n−1}πλ′_{n−1} and let v_{n−1} be an element of W such that v_{n−1}μ_{n−1} = λ′′_{0} + w_{n−1}πλ′_{n−1}

Then R(y, λ) = R(y, ∑_{k=1}^{n−1} p^{k}y_{k}τ_{k}v_{k−1} + y_{0}τ_{0}v_{n−1}μ_{n−1}).

By 2.2, this has p-adic coding

χ([μ_{n−1}, z_{n−1}]; [μ_{n−2}, z_{n−2}]; ⋯; [μ_{0}, z_{0}])

where z_{k} = v_{k}^{−1}τ_{k+1}^{−1}y_{k+1}^{−1}y_{k}τ_{k}v_{k−1} = v_{k}^{−1}w_{k}τ_{k}v_{k−1}, for 1 ≤ k ≤ n − 2 , while z_{0} = v_{0}^{−1}τ_{1}^{−1}y_{1}^{−1}y_{0}τ_{0}v_{n−1} = v_{0}^{−1}w_{0}τ_{0}v_{n−1} and z_{n−1} = v_{n−1}^{−1}τ_{0}^{−1}y_{0}^{−1}yπy_{n−1}τ_{n−1}v_{n−2}π^{−1} = v_{n−1}^{−1}w_{n−1}πτ_{n−1}v_{n−2}π^{−1}.

Thus R(y, λ) has the p-adic coding given in the statement of the proposition, if and only if the values of μ_{k} and z_{k} are as stated there. This proves the proposition.

\underline{Remark}: A p-adic coding for a given R(y, λ) is certainly not unique. For example, if the W -stabilizer of μ_{k} is non-trivial, then there is a choice of v_{k} in the proof of 2.3, so there is a corresponding choice for z_{k}.

§3. Decomposition of Projective Characters

The basis for decomposing the projective characters is the following theorem of Humphreys [3]:

\underline{Theorem 3.1} (Humphreys): Let λ ∈ X. Then |Stab_{W} (λ)|St_{n} ·s(λ) = ∑_{w∈w} R(w, λ)

We now assume that we have a projective character Φ which can be written as

Φ = St_{n} · ∏_{k=0}^{n−1} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}}, where a_{k} is a function from X_{p} to ℤ. Assume that

Stab_{W} (λ) is trivial for every λ for which a_{k}(λ) is non-zero for some k.

We will now define the matrices, the product of whose traces will give the decomposition numbers.

\underline{Definition 3.2}: Let μ ∈ X^{+}, z ∈ W. Suppose a : X_{p} → ℤ is a function from X_{p} to the integers. Define a matrix M_{a}(μ, z) indexed by W × X^{+} as follows. The (v, λ′′) − (y, ν′′) entry of M_{a}(μ, z) is ∑ a(λ), where the sum is over λ ∈ X_{p} such that λ = pλ′ + τλ′′ for some λ′ ∈ X^{+}, τ ∈ W, and there exists w ∈ W with μ = y^{−1}(ν′′ + wλ′) and z = y^{−1}wτv.

Our results will apply when for each λ for which a_{k}(λ) is non-zero and λ = pλ′+τλ′′ with λ′′ ∈ C̃_{p}, λ′′ is far enough away from the upper boundary of C̃_{p}. We state this in the following condition:

\underline{Condition 3.3}: Φ = St_{n} · ∏_{k} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}} such that 1) if a_{k}(λ) is non-zero

then Stab_{W} (λ) is trivial and 2) if a_{k}(λ) and a_{k−1}(μ) are both non-zero (subscript modulo n), and we write λ = pλ′ + τ_{1}λ′′ and μ = pμ′ + τ_{2}μ′′ as in section 2 then λ′′+wμ′ is conjugate under W to an element of C̃_{p}, not lying on the upper boundary of C̃_{p}.

We can now state our main results. We first assume that we are in the untwisted case.

\underline{Theorem 3.4}: Assume π = 1. Suppose Φ is a projective character, with Φ =

St_{n}· ∏_{k=0}^{n−1} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}} satisfying condition 3.3. Also assume that every Deligne-

Lusztig character occurring in the decomposition of Φ is irreducible. Let μ_{0}, …, μ_{n−1} ∈ C̃_{p} and z_{0}, …, z_{n−1} ∈ W. Then the multiplicity of the Deligne-Lusztig character with p-adic coding χ([μ_{n−1}, z_{n−1}]; ⋯; [μ_{0}, z_{0}]) in Φ is equal to

T r( ∏_{k=0}^{n−1} M_{a_{k}}(μ_{k}, z_{k})).

\underline{Proof}: Expanding the product ∏_{k=0}^{n−1} ∑_{λ∈X^{+}} a_{k}(λ)s(λ)^{F r^{k}} gives ∑_{(λ_{0},··· ,λ_{n−1})∈(X_{p})^{n}} ∏_{k=0}^{n−1} a_{k}(λ_{k})s(p^{k}λ_{k}) =

∑_{(λ_{0},··· ,λ_{n−1})∈(X_{p})^{n}} ∏_{k=0}^{n−1} a_{k}(λ_{k})s(p^{n−1}λ_{n−1}+ ∑_{k=0}^{n−2} ∑_{y_{k}∈W} p^{k}y_{k}λ_{k}). Using Humphreys’ the-

orem we get Φ = ∑_{(λ_{0},...,λ_{n−1})∈(X_{p})^{n}} ∏_{k=0}^{n−1} a_{k}(λ_{k})St_{n}s(p^{n−1}λ_{n−1} + ∑_{k=0}^{n−2} ∑_{y_{k}∈w} p^{k}y_{k}λ_{k})

= ∑_{\begin{matrix}(λ_{0},...,λ_{n−1})∈(X_{p})^{n} \\ (y,y_{0},...,y_{n−2})∈W^{n} \end{matrix}} ∏_{k=0}^{n−1} a_{k}(λ_{k})R(y, p^{n−1}λ_{n−1} + ∑_{k=0}^{n−2} p^{k}y_{k}λ_{k}).

Now by 2.3, χ([μ_{n−1}, z_{n−1}]; ⋯; [μ_{0}, z_{0}]) is a p-adic coding for R(y, p^{n−1}λ_{n−1} + ∑ p^{k}y_{k}λ_{k}) if and only if there exist v_{−1}, v_{0}, v_{1}, …, v_{n−1} ∈ W, with v_{−1} = v_{n−1} and

z_{k} = v_{k}^{−1}w_{k}τ_{k}v_{k−1}  for all k,
μ_{k} = v_{k}^{−1}(λ′′_{k+1} + w_{k}λ′_{k}) for 0 ≤ k ≤ n − 2, and
μ_{n−1} = v_{n−1}^{−1}(λ′′_{0} + w_{n−1}λ′_{n−1}),

where w_{k} = τ_{k+1}^{−1}y_{k+1}^{−1}y_{k} for 0 ≤ k ≤ n−2 and w_{n−1} = τ_{0}^{−1}y_{0}^{−1}y. Now for a given sequence λ_{0}, …, λ_{n−1}, as (y_{1}, y_{0}, …, y_{n−2}) ranges over W^{n}, so does (w_{0}, w_{1}, …, w_{n−1}). Thus the multiplicity of χ([μ_{n−1}, z_{n−1}]; ⋯; [μ_{0}, z_{0}]) in Φ is equal to ∑ ∏_{k} a_{k}(λ_{k}),

the sum over all sequences v_{−1}, v_{0}, v_{1}, …, v_{n−1} ∈ W with v_{−1} = v_{n−1} and sequences λ′′_{0}, λ′′_{1}, ⋯, λ′′_{n} with λ′′_{n} = λ′′_{0}, such that there exist λ′_{0}, λ′_{1}, ⋯, λ′_{n−1} ∈ X^{+}; τ_{0}, τ_{1}, …, τ_{n−1}; w_{0}, w_{1}, ⋯, w_{n−1} ∈ W with λ_{k} = pλ′_{k}+τ_{k}λ′′_{k} and μ_{k} = v_{k}^{−1}(λ′′_{k+1}+ w_{k}λ′_{k}); and z_{k} = v_{k}^{−1}w_{k}τ_{k}v_{k−1} for 0 ≤ k ≤ n−1. Thus by definition of M_{a_{k}}(μ_{k}, z_{k}), the multiplicity is equal to

∑ m_{α_{0},α_{1}}m_{α_{1},α_{2}} … m_{α_{n−1},α_{0}},

where the sum is over α_{0}, α_{1}, … α_{n−1} ∈ W × X^{+}, and where m_{α_{k},α_{k+1}} is the α_{k} − α_{k+1} element of M_{a_{k}}(μ_{k}, z_{k}). But this is precisely equal to the trace of the matrices as stated.

We now generalize the theorem so that it applies to twisted groups as well. We need to define another matrix to account for the twisting by π.

\underline{Definition 3.5}: Let M_{π} be the permutation matrix indexed by W × X^{+} whose (w, ν_{1}) − (y, ν_{2}) entry is 1 if y = πwπ^{−1} and ν_{2} = πν_{1}, and 0 otherwise.

\underline{Theorem 3.6}: Suppose Φ = St_{n} ∏_{k=0}^{n−1} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}} is a projective character

of G(p^{n})_{π} satisfying the same conditions as in 3.4 . Let μ_{0}, …, μ_{n−1} ∈ C̃_{p} and z_{0}, …, z_{n−1} ∈ W. The multiplicity of the Deligne-Lusztig character with p-adic

coding χ([μ_{n−1}, z_{n−1}]; ⋯; [μ_{0}, z_{0}]) is T r( ∏_{k=0}^{n−2} M_{a_{k}}(μ_{k}, z_{k})M_{π}M_{a_{n−1}}(μ_{n−1}, z_{n−1})).

\underline{Proof}: As in the proof of 3.4, we have

Φ = ∑_{\begin{matrix}(λ_{0},··· ,λ_{n−1})∈X^{n} \\ (y,y_{0},...,y_{n−2})∈W^{N} \end{matrix}} ∏_{k=0}^{n−1} a_{k}(λ_{k})R(y, p^{n−1}λ_{n−1} + ∑_{k=0}^{n−2} p^{k}y_{k}λ_{k}).

By 2.3, χ([μ_{n−1}, z_{n−1}]; ⋯; [μ_{0}, z_{0}]) is a p-adic coding for R(y, p^{n−1}λ_{n−1}+∑ p^{k}y_{k}λ_{k}) if and only if there exist v_{−1}, v_{0}, ⋯, v_{n−1} ∈ W with v_{−1} = v_{n−1} and for 0 ≤ k ≤ n − 2

z_{k} = v_{k}^{−1}w_{k}τ_{k}v_{k−1}

and μ_{k} = v_{k}^{−1}(λ′′_{k+1} + w_{k}λ′_{k}), while

z_{n−1} = v_{n−1}^{−1}w_{n−1}πτ_{n−1}v_{n−2}π^{−1}

and μ_{n−1} = λ′′_{0} + w_{n−1}πλ′_{n−1},

where w_{k} = τ_{k+1}^{−1}y_{k+1}^{−1}y_{k} for 0 ≤ k ≤ n − 2

and w_{n−1} = τ_{0}^{−1}y_{0}^{−1}y. Thus the multiplicity of χ([μ_{n−1}, z_{n−1}]; ⋯, [μ_{0}, z_{0}]) in Φ is equal to ∑ ∏_{k} a_{k}(λ_{k}) where the sum is over sequences v_{−1}, v_{0}, …, v_{n−1}, v_{π} ∈ W with

v_{−1} = v_{n−1}, and sequences λ′′_{0}, λ′′_{1}, …, λ′′_{n−1}, λ′′_{π} ∈ X^{+} with λ′′_{n} = λ′′_{0}, such that there exist λ′_{0}, …, λ′_{n−1}, λ′_{π} ∈ X^{+}, τ_{o}, …, τ_{n−1}τ_{π}w_{0}, …, w_{n−1} ∈ W with λ_{k} = pλ′_{k} +τ_{k}λ′′_{k} for 0 ≤ k ≤ n − 1

and μ_{k} = v_{k}^{−1}(λ′′_{k+1} + w_{k}λ′_{k}),

z_{k} = v_{k}^{−1}w_{k}τ_{k}v_{k+1} for 0 ≤ k ≤ n − 2;

μ_{n−1} = v_{n−1}^{−1}(λ′′_{n} + w_{n−1}λ′_{π}),

z_{n−1} = v_{n−1}^{−1}w_{n−1}τ_{π}v_{π},

τ_{π} = πτ_{n−1}π^{−1}, v_{π} = πv_{n−2}π^{−1}, λ′_{π} = πλ_{n−1}, λ′′_{π} = πλ′′_{n−1}.

Thus by definition of M_{a_{k}}(μ_{k}, z_{k}), the multiplicity is equal to

∑ m_{α_{0},α_{1}}m_{α_{1},α_{2}} … m_{α_{n−2},α_{π}}m_{α_{π},α_{n−1}}m_{α_{n−1},α_{0}},

where the sum is over α_{0}, α_{1}, … α_{n−2}α_{π}α_{n−1} ∈ W × X^{+}, and where m_{α,β} is the α − β element of a matrix M indexed by W × X^{+}. But this is equal to the trace of the matrices as stated.

\underline{Remark}: We need the condition of smallness on the λ′′_{k} (condition 3.3) because a Deligne-Lusztig character may be represented by more than one p-adic coding. If μ_{k} = v_{k}^{−1}(λ′′_{k} + w_{k}λ′_{k}) lies in an alcove above C̃_{p}, the corresponding Deligne-Lusztig character may be equal to one whose p-adic coding has the corresponding component inside C̃_{p}. The multiplicity would not be given correctly. If μ_{k} lies on a lower boundary, we can also have different codings representing the same character. (See the remark at the end of section 2.) However, each coding is accounted for, since v_{k} is allowed to range over all of Stab_{w}(μ_{k}) in the proof of the theorem. Thus the multiplicity is correctly given in this case. The only problem which could arise would be if the stabilizer of every μ_{k} were non-trivial, when there would be a choice of v_{k} for every component. However, the corresponding Deligne-Lusztig character would not be irreducible, and we are assuming it is. We will discuss this situation elsewhere when we consider non-generic cases.

Note that the theorem applies to the situation described in the introduction. Suppose p is large enough and ν is deep enough in the interior of a W̃_{a,p} alcove of X_{p}, and let Φ_{ν} be the projective cover of the corresponding irreducible G(p^{n})_{π} charac-

ter. Then by the results of [1] and [2] we can write Φ_{ν} = St_{n} ∏_{k=0}^{n−1} ∑_{λ∈X_{p}} a_{k}(λ)s(λ)^{F r^{k}},

where a_{k}(λ) = a(λ) is the same for all k > 0(St ∑ a_{0}(λ)s(λ)+(resp.St ∑ a_{k}(λ)s(λ), k >

1) is the restriction to G(p) of the projective cover of the u_{n} − T module with highest weight λ (resp. trivial u_{n} − T module).). Also, λ′′ is small whenever a_{k}(λ) is non-zero. For k = 0 we can ensure this by taking λ deep enough in the interior of a W̃_{a,p} alcove. For k > 1, this follows from the fact that λ′′ is either conjugate under the affine Weyl group W_{a,p} to ρ, or to ρ + wζ where w ∈ w and ζ has all components in terms of the fundamental weights equal to or less than ( see [1]).

\underline{Corollary 3.7} Assume p is large enough and that ν is deep enough in the interior of a W̃_{a,p} alcove contained in X_{p}. Let Φ_{ν,n} be the character of the projective cover

of the corresponding irreducible G(p^{n})_{π} module. Write Φ_{ν,n} = St_{n}φ_{ν} ∏_{k=1}^{n−1} φ_{0}^{F r^{k}},

where φ_{ν} = ∑_{λ∈X_{p}} a_{0}(λ)s(λ) and φ_{0} = ∑_{λ∈X_{p}} a(λ)s(λ) . Then the multiplicity of

χ([μ_{0}, z_{0}]; ⋯; [μ_{n−1}, z_{n−1}]) in Φ_{ν,n} is equal to T r(M_{a_{0}}(μ_{0}, z_{0}) ∏_{k=1}^{n−2} M_{a}(μ_{k}, z_{k})M_{π}M_{a}(μ_{n−1}, z_{n−1})).

§4. Decomposition numbers of SL_{3}(p^{n}) and SU_{3}(p^{n}).

We will apply the results of the previous section to the groups SL_{3}(p^{n}) and SU_{3}(p^{n}) for large enough p. We will consider a weight sufficiently deep in the interior of the W̃_{a,p} alcove C̃_{p} of SL_{3}(p) and determine the decomposition of the projective indecomposable Φ_{ν} of SL_{3}(p^{n}) and SU_{3}(p^{n}). Similar computations apply to weights in other W̃_{a,p} alcoves.

The weight lattice X is isomorphic to ℤ^{+}×ℤ^{+}. The Weyl group W is isomorphic to S_{3} and is generated by σ_{1} and σ_{2} where these elements act on X via σ_{1}(r, s) = (−r, r + s) and σ_{2}(r, s) = (r + s, −s).

Let W_{a,p} be the ordinary affine Weyl group, that is, the semidirect product of W with translations by elements of p times the root lattice. There are two W_{a}-alcoves in X_{p}. The top alcove consists of (r, s) in X_{p} with r + s ≥ p − 1, while the bottom alcove consists of those with r + s < p − 1.

If ν is in the bottom W_{a,p} alcove of X_{p}, then φ_{ν} = (s(ν + ρ) + s(pρ + σ_{0}(ν +

ρ)) ∏_{k=1}^{n−1}(s(ρ)+s(pρ+σ_{0}ρ))^{F r^{k}}. Thus we have φ_{ν} = ∏_{k} ∑_{λ} a_{k}(λ)s(λ)^{F r^{k}} where a_{0}(λ) =

1 if λ = ν +ρ or pρ+σ_{0}(ν +ρ), and is equal to 0 otherwise, and for k > 1, a_{k}(λ) = 1 if λ = σ or pρ + σ_{0}ρ, and is equal to 0 otherwise.

Thus for k > 1, if a_{k}(λ_{k}) is non-zero then λ_{k} = λ′_{k} + τ_{k}λ′′_{k} with λ′_{k} = 0, τ_{k} = 1, λ′′_{k} = ρ or λ′_{k} = ρ, τ_{k} = σ_{0}, λ′′_{k} = ρ . If a_{0}(λ_{0}) is non zero then λ_{0} = λ′_{0} + τ_{0}λ′′_{0} with λ′_{0} = 0, τ_{0} = 1, λ′′ = ν + ρ or λ′_{0} = ρ, τ_{o} = σ_{0}, λ′′_{0} = ν + ρ.

We will now determine the matrices M_{a_{k}}(μ, z). We will see that the only entries which contribute to a non-zero trace are those with indices whose W component corresponds to 1, σ_{1}, or σ_{2}. The only non-zero rows correspond to indices with X^{+} - component ρ for k > 0 and ν +ρ for k = 0. The only non-zero columns correspond to indices with X^{+} component ρ for 0 ≤ k < n − 2 and ν + ρ for k = n − 1. We will only give the portion of M_{a_{k}}(μ, z) corresponding to these rows and columns. The order will correspond to 1, σ_{1}, σ_{2}.

We first consider M_{a_{0}}(μ, z).

\underline{Proposition 4.1}: With the above notation,

M_{a_{0}}(ν + ρ, w) ↔ ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ),

M_{a_{0}}(ν + ρ + wρ, wσ_{0}) ↔ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ),

M_{a_{0}}(ν + ρ + wρ, wσ_{1}σ_{2}) ↔ ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

and M_{a_{0}}(ν + ρ + wρ, wσ_{2}σ_{1}) ↔ ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ), for all w ∈ W. M_{a_{0}}(μ, z) is 0 for

all other μ ∈ X^{+}, z ∈ W.

\underline{Proof}: If M_{a_{0}}(μ, z) is non-zero then μ = y^{−1}(λ′′ + wλ′) and z = y^{−1}wτv for some w ∈ W, where λ′′ = ν + ρ and either λ′ = 0, τ = 1, or λ′ = ρ, τ = σ_{0}. We then get an entry in the row corresponding to v and column corresponding to y. Since we are assuming that ν +ρ is deep enough in C̃_{p}, y must be the identity, so all columns but the first are 0. If λ′ = 0 and τ = 1, then μ = ν + ρ and z = wv. For any v, as w ranges over W, so does z. This gives the values for M_{a_{0}}(ν + ρ, z). If λ′ = ρ and τ = σ_{0}, we have μ = ν + ρ + wρ and z = wσ_{0}ν. Taking v = 1, σ_{1} and σ_{2} gives the remaining matrices.

We now consider M_{a_{k}}(μ, z) for k > 0. \underline{Proposition 4.2}: With the above notation, for k > 0 we have

M_{a_{k}}(2ρ, σ_{0}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}(2ρ, σ_{1}σ_{2}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}(2ρ, σ_{2}σ_{1}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}((0, 3)σ_{2}σ_{1}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )

M_{a_{k}}((0, 3), σ_{1}σ_{2}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}((0, 3), σ_{0}) ⟷ ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}((0, 3), σ_{2}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}((3, 0), σ_{1}σ_{2}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}((3, 0), σ_{2}σ_{1}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )

M_{a_{k}}((3, 0), σ_{0}) ⟷ ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

M_{a_{k}}((3, 0), σ_{1}) ⟷ ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}(ρ, σ_{1}σ_{2}) ⟷ ( \begin{matrix}1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}(ρ, σ_{2}σ_{1}) ⟷ ( \begin{matrix}1 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}(ρ, σ_{0}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix} )

M_{a_{k}}(ρ, σ_{1}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix} )

M_{a_{k}}(ρ, σ_{2}) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}(ρ, 1) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

M_{a_{k}}(0, 1) ⟷ ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

M_{a_{k}}(μ, z) is 0 for all other non-zero μ ∈ X^{+}, z ∈ W. There are also non-zero M_{a_{k}}(0, w) for other w ∈ W, whose values we omit.

\underline{Proof}: If M_{a_{k}}(μ, z) is non-zero then

μ = y^{−1}(λ′′ + wλ′)

and z = y^{−1}wτv for some w ∈ W, where λ′′ = ρ and either λ′ = 0, τ = 1 or λ′ = ρ, τ = σ_{0}. We then get an entry in the column corresponding to y and row corresponding to v.

If λ′ = 0, τ = 1, then y = 1 , μ = ρ and z = wv. For any v, we get all z ∈ W as w ranges over W. This gives an entry of 1 in the first column of M_{a_{k}}(ρ, z) for each z ∈ W.

If λ′ = ρ, τ = σ_{0} then μ = y^{−1}(ρ + wρ) and z = y^{−1}wσ_{0}v. The following table gives the possibilities (blank entries signify the same value as in the previous row);

\underline{w}  \underline{ρ + wρ}  \underline{y}  \underline{μ}  \underline{v}  \underline{z}
1  2ρ  1  2ρ  1  σ_{0}
σ_{1}  σ_{1}σ_{2}
σ_{2}  σ_{2}σ_{1}
σ_{1} (0, 3) 1 (0, 3) 1 σ_{2}σ_{1}
σ_{1}  σ_{2}
σ_{2}  σ_{0}
σ_{1} (0, 3) 1 σ_{0}
σ_{1}  σ_{1}σ_{2}
σ_{2}  σ_{2}σ_{1}
σ_{2} (3, 0) 1 (3, 0) 1 σ_{1}σ_{2}
σ_{1}  σ_{0}
σ_{2}  σ_{1}
σ_{2} (3, 0) 1 σ_{0}
σ_{1}  σ_{1}σ_{2}
σ_{2}  σ_{2}σ_{1}
σ_{1}σ_{2} (−1, 2) σ_{1} ρ 1 σ_{1}σ_{2}
σ_{1}  σ_{0}
σ_{2}  σ_{1}
σ_{2}σ_{1} (2, −1) σ_{2} ρ 1 σ_{2}σ_{1}
σ_{1}  σ_{2}
σ_{2}  σ_{0}
σ_{0}  0  v^{−1}  0  v  1

In the last row, y can be any element of W, but we only choose to give the one that is the same as v^{−1}, so that z is always the identity.

Putting together all of the above gives the values for M_{a_{k}}(μ, z) stated in the proposition.

From the computations in the proof of 4.2, we see that there is a non-zero entry in M_{a_{k}}(μ_{k}, z_{k}) only in a row corresponding to 1, σ_{1}, σ_{2}, except when (μ_{k}, z_{k}) = (0, 1).

Since not all components can be of this type, we are indeed justified in ignoring all rows except those corresponding to these three elements.

\underline{Proposition 4.3}: If π ≠ 1, then

M_{π} ↔ ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )

\underline{Proof}: We have 1^{π} = 1. If (r, s) ∈ X,

σ_{1}^{π}(r, s) = πσ_{1}π^{−1}(r, s) = πσ_{1}(s, r) = π(−s, r + s) = (r + s, −s) = σ_{2}(r, s),

so σ_{1}^{π} = σ_{2}

Similarly σ_{2}^{π} = σ_{1}.

We now have by 3.6

\underline{Theorem 4.4} With notation as above, the multiplicity of χ([μ_{0}, z_{0}], ⋯; [μ_{n−1}, z_{n−1}]) in Φ_{ν} is equal to

T r (∏_{k=0}^{n−2} M_{a_{k}} (μ_{k}, z_{k})M_{π}M_{a_{0}}(μ_{n−1}z_{n−1}))

where the M_{a_{k}} are given in 4.1 and 4.2 and M_{π} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) if π ≠ 1, while

M_{π} = identity matrix if π = 1.

\underline{Remark 1:} When a Deligne-Lusztig character has a p-adic coding with component μ_{k} = (0, 3) or (3, 0), it will have more than one such coding. This corresponds to the fact that there is a choice of y = 1 or σ_{1} (resp. y = 1 or σ_{2}) in the proof of 4-2, in these cases. Thus when making an inventory of the characters appear in Φ_{ν}, we must take this into account. We can choose a component of type [(0, 3), σ_{2}σ_{1}] or [(0, 3), σ_{0}], but not both; of type [(0, 3), σ_{2}] or [(0, 3), σ_{1}σ_{2}], but not both; of type [(3, 0), σ_{1}σ_{2}] or [(3, 0), σ_{0}], but not both; of type [(3, 0), σ_{1}] or [(3, 0), σ_{2}σ_{1}], but not both.

Similarly, we get multiple representations of a Deligne-Lusztig character with a component of type [0, w]. As noted above, we will always represent such a character with a component of [0, 1].

\underline{Remark 2:} It is not hard to see that 4.3 holds, not only for ν ∈ C̃_{p}, but for any ν deep enough in the interior of any W̃_{a,p} alcove contained in the bottom W_{a,p} - alcove of X_{p}. For in the definition of the decomposition of λ_{0} = λ′_{0} + τ_{0}λ′′_{0}, we can relax the requirement that λ′′_{0} ∈ C̃_{p} and allow it to be in any particular W̃_{a,p} alcove contained in the bottom W_{a,p} - alcove, and all the computations above will be valid.

Similar computations apply to ν in the upper W_{a,p}-alcove.

\underline{Theorem 4.5:} Suppose ν is in the top W_{a,p} - alcove of X_{p}, and deep enough in the interior of a W̃_{a,p} - alcove. Then the multiplicity of χ([μ_{0}, z_{0}]; ⋯; [μ_{n−1}, z_{n−1}]) in Φ_{ν} considered as a character of G(p^{n})_{π}, is T r(∏_{k=0}^{n−2} M_{a_{k}}(μ_{k}, z_{k})M_{π}M_{a_{n−1}}(μ_{n−1}, z_{n−1})), where M_{a_{k}}(μ_{k}, z_{k}) are as in 4.1 and 4.2 for 1 ≤ k ≤ n − 1, while M_{a_{0}}(μ_{0}, z_{0}) =

( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) if μ_{0} = ν + ρ and z is any element of W, and is 0 otherwise.

§5. Cartan Invariants

We will show that our methods can also be used to calculate Cartan Invariants. To do so we need to assume that we have resolved any ambiguities due to the fact that p-adic codings of Deligne - Lusztig characters may not be unique. We shall assume that we have a complete listing of all the irreducible ordinary characters occurring in a projective indecomposable character Φ, and each such character is represented exactly once in the p-adic coding.

We now assume that we have 2 projective indecomposable characters Φ_{1} and Φ_{2}, and that for each k we have a subset S_{k} of X^{+}×W such that {χ([μ_{0}, z_{0}], …, [μ_{n−1}, z_{n−1}]) : [μ_{k}, z_{k}] ∈ S_{k}} gives a complete simultaneous listing, without repetition, of the Deligne-Lusztig characters appearing in both Φ_{1} and Φ_{2}. With the assumptions on Φ_{1} and Φ_{2} in section 3, the multiplicity of the character χ([μ_{0}, z_{0}]; …; [μ_{n−1}z_{n−1}])

in Φ_{i} for i = 1, 2, can be written as T r( ∏_{k=0}^{n−2} M_{a_{i},k}(μ_{k}, z_{k})M_{π}M_{a_{i,n−1}}(μ_{n−1}, z_{n−1})),

where Φ_{i} = St_{n} ∏ ∑ a_{i,k}(λ)s(λ)^{F r^{k}}. Write M_{i,k}(μ, z) = M_{a_{i,k}}(μ, z) for k = 0, ⋯, n − 2 and M_{i,n−1}(μ, z) = M_{π}M_{a_{i,n−1}}(μ, z).

Let c(Φ_{1}, Φ_{2}) be the Cartan invariant corresponding to Φ_{1} and Φ_{2}, i.e., the number of times the irreducible kG(p^{n})_{π} character corresponding to Φ_{1} occurs in Φ_{2}. We have the following formula for c(Φ_{1}, Φ_{2}):

\underline{Theorem 5.1}: With the above notation, c(Φ_{1}, Φ_{2}) = T r( ∏_{k=0}^{n−1} ∑_{(μ,z)∈S_{k}} M_{1,k}(μ, z) ⊗

M_{2,k}(μ, z))

\underline{Proof:} The Cartan invariant is equal to the inner product (Φ_{1}, Φ_{2}), so is given by the sum of products of decomposition numbers. We have

Φ_{i} = ∑_{\begin{matrix}(μ_{0},z_{0}),··· ,(μ_{n−1},z_{n−1}) \\ (μ_{k},z_{k})∈S_{k} \end{matrix}} T r( ∏ M_{i,k}(μ_{k}, z_{k}))χ([μ_{0}, z_{0}]; ⋯; [μ_{n−1}, z_{n−1}])

Thus c(Φ_{1}, Φ_{2})

= ∑_{\begin{matrix}(μ_{0},z_{0});···;(μ_{n−1},z_{n−1}) \\ (μ_{k},z_{k})∈S_{k} \end{matrix}} T r (∏_{k} M_{1,k}(μ_{k}, z_{k}))T r(∏_{k} M_{2,k}(μ_{k}, z_{k}))

= T r (∏_{k} ∑_{\begin{matrix}(μ,z) \\ ∈S_{k} \end{matrix}} M_{1,k}(μ, z)) T r (∏_{k} ∑_{\begin{matrix}(μ,z) \\ ∈S_{k} \end{matrix}} M_{2,k}(μ, z))

= T r (∏_{k} ∑_{(μ,z)∈S_{k}} M_{1,k}(μ, z) ⊗ M_{2,k}(μ, z)) as required.

We apply this result to compute the generic Cartan invariants c(Φ_{ν}, Φ_{ν}) for ν ∈ X_{p}, for SL_{3}(p^{n}) and SU_{3}(p^{n}).

\underline{Theorem 5.2:} Suppose ν is in the bottom alcove of SL_{3}(p) and sufficiently deep in the interior of a W̃_{a,p} - alcove. Then the Cartan invariant c(Φ_{ν,n}, Φ_{ν,n}) of SL_{3}(p^{n}) or SU_{3}(p^{n}) is equal to the coefficient of x^{n} in the series expansion of 12x(1 − 4x)/(48x^{2} − 18x + 1).

\underline{Proof:} By 4.1, the matrices ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , and ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) ,

each occur as M_{a_{0}}(μ, z) for 6 pairs (μ, z). Thus ∑_{(μ,z)∈S_{n−1}} M_{a_{0}}(μ, z) ⊗ M_{a_{0}}(μ, z)

= ( \begin{matrix}12 & · & · & · & · & · & · & · & · \\ 6 \\ 6 \\ 6 \\ 12 \\ 6 \\ 6 \\ 6 \\ 12 & · & · & · & · & · & · & · & · \end{matrix} )

where omitted entries in the 9 × 9 matrix are 0. Call this matrix M_{n−1}^{(2)}.

By 4.2, the following 12 matrices occur as M_{a_{k}}(μ, z) for 1 ≤ k ≤ n − 1:

( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix} ) ,

( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ,

( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ).

The first 10 occur once each while the last 2 each occur for two M(μ, z). As explained before, when μ = (0, 3) or (3, 0) we must make a choice for z. Here we are choosing the pairs [(0, 3), σ_{2}σ_{1}], [(0, 3), σ_{2}], [(3, 0), σ_{1}σ_{2}], [(3, 0), σ_{1}], and omitting [(0, 3), σ_{0}], [(0, 3), σ_{1}σ_{2}], [(3, 0), σ_{0}] and [(3, 0), σ_{2}σ_{1}].

We also take [0, 1] but omit [0, w] for w ≠ 1.

This gives for 1 ≤ k ≤ n − 1, ∑_{(μ,z)∈S_{k}} M_{a_{k}}(μ, z) ⊗ M_{a_{k}}(μ, z)

= ( \begin{matrix}10 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 6 & 2 & 2 & 1 & 0 & 0 & 1 & 0 & 0 \\ 6 & 2 & 2 & 1 & 0 & 0 & 1 & 0 & 0 \\ 6 & 1 & 1 & 2 & 0 & 0 & 2 & 0 & 0 \\ 8 & 1 & 1 & 1 & 2 & 0 & 1 & 0 & 2 \\ 6 & 1 & 1 & 1 & 0 & 2 & 1 & 0 & 0 \\ 6 & 1 & 1 & 2 & 0 & 0 & 2 & 0 & 0 \\ 6 & 1 & 1 & 1 & 0 & 0 & 1 & 2 & 0 \\ 8 & 1 & 1 & 2 & 0 & 0 & 1 & 0 & 2 \end{matrix} )

Call this matrix M^{(2)}.

First consider the case when π is the identity. By 5.1, c(Φ_{ν,n}, Φ_{ν,n}) = T r(M_{0}^{(2)}(M^{(2)})^{n−1}). A computation using Mathematica shows that this trace is equal to (2/11)[(33 + 5\sqrt{33})α^{n−1} + (33 − 5\sqrt{33})β^{n−1}] where α = 9 + \sqrt{33} and β = 9 − \sqrt{33} (see appendix).An elementary computation gives the generating function stated in the theorem.

Now consider the case of SU_{3}(p^{n}). Then c(Φ_{ν,n}, Φ_{ν,n}) = T r(M_{0}^{(2)}(M^{(2)})^{n−2}M_{π}^{(2)}M^{(2)}), where M_{π}^{(2)} denotes M_{π} ⊗M_{π}. Let S be the set of 12 matrices given above as occurring as M_{a_{k}}(μ, z), and let S_{0} be the set of four matrices occurring as M_{a_{0}}(μ, z). First note that multiplication of the elements of S on the left by M_{π} gives the same set of matrices as multiplying S on the right by M_{π}. Next note that multiplying on the left by M_{π} permutes S_{0}. Thus we get c(Φ_{ν,n}, Φ_{ν,n}) = T r(M_{0}^{(2)}(M^{(2))^{n−2}}M_{π}^{(2)}M^{(2)}) = T r(M_{0}^{(2)}(M^{(2)})^{n−1}M_{π}^{(2)}) = T r(M_{π}^{(2)}M_{0}^{(2)}(M^{(2)})^{n−1}) = T r(M_{0}^{(2)}(M^{(2)})^{n−1}). Thus the Cartan invariant is the same as for SL_{3}(p^{n}).

\underline{Theorem 5.3:} Suppose that ν is in the top W_{a,p} - alcove of SL_{3}(p) and sufficiently deep in the interior of a W̃_{a,p} - alcove. The Cartan invariant c(Φ_{ν,n}, Φ_{ν,n}) of SL_{3}(p^{n}) or SU_{3}(p^{n}) is equal to the coefficient of x^{n} in the series expansion of 6x(1 − 2x)/(48x^{2} − 18x + 1) .

\underline{Proof}: The matrix ( \begin{matrix}1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) occurs 6 times as M_{a_{0}}(μ, z). Thus ∑_{(μ,z)∈S_{0}} M_{a_{0}}(μ, z)⊗

M_{a_{0}}(μ, z) is the 9 × 9 matrix with entries of 6 in the first column, and all other entries 0. A similar Mathematica computation as in 5.2 gives the trace equal to \frac{1}{11}[(33 + 7\sqrt{33})α^{n−1} + (33 − 7\sqrt{33})β^{n−1}]. This gives the result.

§6. Conclusion

It is possible to get much more information about the decomposition numbers using the description we have given. In particular, we can give a formula for the number of times a given integer k occurs as a decomposition number for a projective indecomposable Φ. We will give the details elsewhere.

Appendix:

We give here the Mathematica calculation for T r(M_{0}^{(2)}(M^{(2)})^{n−1}) cited above.

It is equal to ∑_{j=1}^{9} b_{j}λ_{i}^{n}(v_{i}w_{i}^{T})_{i,j} where (b_{1}, ⋯, b_{9})^{T} is the first column of M_{0}^{(2)}; v_{i} is

the ith eigenvector of M^{(2)}, written as a column vector; λ_{i} is the i^{th} eigenvalue of M^{(2)}; and [w_{1}, ⋯, w_{9}] is the inverse of the matrix [v, ⋯, v_{9}]. r1: = {10, 1, 1, 1, 1, 0, 1, 0, 1} r2: = {6, 2, 2, 1, 0, 0, 1, 0, 0}

r3: = {6, 2, 2, 1, 0, 0, 1, 0, 0} r4: = {6, 1, 1, 2, 0, 0, 2, 0, 0} r5: = {8, 1, 1, 1, 2, 0, 1, 0, 2} r6: = {6, 1, 1, 1, 0, 2, 1, 0, 0} r7: = r3 r8: = {6, 1, 1, 1, 0, 0, 1, 2, 0} r9: = {8, 1, 1, 1, 2, 0, 1, 0, 2} m: = {r1, r2, r3, r4, r5, r6, r7, r8, r9} c: = Eigenvectors [m] d: = Inverse [Transpose[c]] b: = {12, 6, 6, 6, 12, 6, 6, 6, 12 }

Simplify [Sum[b[[j[[ * (((Eigenvalues[m]) [[i]])^{n}) ((Transpose [[c[[i]]}]. {d[i]]}) [[1, j]]), (i, 1, 9}, {j, 1, 9}]]

6(9 − \sqrt{33})^{n} − 10\sqrt{\frac{3}{11}}(9 − \sqrt{33})^{n} + 6(9 + \sqrt{33})^{n} + 10\sqrt{\frac{3}{11}}(9 + \sqrt{33})^{n} References:

[1] L. CHASTKOFSKY, Projective characters for finite Chevalley groups, J. Algebra 69(1981), 347-357.

[2] L. CHASTKOFSKY, 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 Charakter von Deligne and Lusztig, J. Algebra 70 (1981), 452-474.