A combinatorial character formula for some highest weight modules. Olivier Mathieu and Georges Papadopoulo*

Abstract: We give a combinatorial formula for the weight multiplicities of some infinite dimensional highest weight 𝔤𝔩(n)-modules. Our proof, which does not rely on Kazdhan-Lusztig combinatorics, uses a reduction to finite characteristics. The character formula for the corresponding modular representations, which has been computed in [MP], is based on a dual pair which has no obvious counterpart in characteristic zero.

Introduction: Set 𝔤=𝔤𝔩(n, K), where K is an algebraically closed field of characteristic zero. Although the characters of highest weight 𝔤-modules L(λ) are determined by the Kazdhan-Lusztig polynomials, there are no general combinatorial formulas. For finite dimensional 𝔤-modules, such formulas have been provided by the work of Littlewood and Richardson [LR], which is based on semi-standard tableaux (see also [L]). In this paper, we will show that the combinatorics of semi-standard tableaux applies as well for a certain class of infinite dimensional highest weight representations. Indeed, the result is a corollary of some character formulas for modular representations (see [MP]), and is strongly connected with the combinatorics of Verlinde’s formula for modular representations. Our proof uses a reduction to finite characteristics. However, we believe that there should be a natural purely characteristic zero proof, based on representation theory of the loop algebra.

Let m ≥ 0. In the paper, we will determine a combinatorial formula for all highest weight modules L(λ), where λ is m-cospecial. By definition, a non-zero weight λ is called m-cospecial if there are three integers i, s, j, with i ≤ s ≤ j, such that λ = ∑_{i≤l≤j} a_{l}ω_{l} (where ω_{l} is the l^{th} fundamental weight), a_{l} ≥ 0 for all l ≠ s and j − i ≤ m, where m = − ∑_{i≤l≤j} a_{l}. Typical examples of cospecial weights are the negative multiple of the fundamental weights, i.e. the weight −mω_{s} is m-cospecial. For simplicity, we will only explain the character formula for L(−mω_{s}) in the introduction.

Let Y_{∞} be the semi-infinite Young diagram Z_{≤0} × {1, …, … s} of height s. We can draw this semi-infinite Young diagram as follows:

\overline{| | | | | | |}
\overline{| | | |}
\overline{| |}
\overline{|}
\overline{|}
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By convention, the first line is the top line, and the last line is the s^{th} line which is at bottom. A semi-standard tableau of shape Y_{∞} is a filling of the boxes of Y by indices

* Research supported by U.A. 1 du C.N.R. S. running from 1 to n, with the usual convention that the indices are increasing from top to bottom, non decreasing from left to right and with the special requirement that on the i^{th} line almost all labels are i. Here is an example of semi-standard tableau of shape Y_{∞}:

\overline{… | 1 | 1 | 1 … … 1| 2 | 2 | 4 |} \overline{… | 2 | 2 | 2 … … 2| 3 | 5 | 6 |} \overline{… | 3 | 3 … … | 7 |}

\overline{… |}
\overline{|}

As the Young diagram Y_{∞} is infinite, we cannot define the weight of the tableau T as usual. However it is easy to renormalize the usual definition, which allows to define its relative weight rw(T). Denote by L_{1} < L_{2} … < L_{s} the indices on the last column (i.e. the rightmost column) of the tableau T, and denote by P the set of all semi-standard tableaux such that L_{s−m} ≤ s (by convention, this condition is automatically satisfied if s − m ≤ 0).

THEOREM We have: ch(L(−m ω_{s})) = e^{−mω_{s}} ∑_{T ∈P} e^{rw(T)}.

For general m-cospecial weights, the combinatorics is slighty more complicated and it involves a pair consisting of a semi-infinite Young diagram Y_{∞} and an ordinary Young diagram Y_{f}.

1. A semi-continuity principle. Roughly speaking, a semi-continuity principle states that a ”finite statement” which holds in characteristic p >> 0 also holds in characteristic 0. For any algebraically closed field k, set 𝔤_{k}=𝔤𝔩(n, k), let P be the lattice of integral weights and let H(k) be the torus of GL_{n}(k), i.e. the subgroup of diagonal matrices. Denote by ϵ_{1}, …, ϵ_{n} the natural basis of P, by α_{1} = ϵ_{1} − ϵ_{2}, …, α_{n−1} = ϵ_{n−1} − ϵ_{n} the simple roots and by h_{1}, …, h_{n−1} ∈ Hom(P, Z) the corresponding simple coroots (with our definition, coroots are not elements in the Cartan subalgebra). Denote by Q^{+} the monoid generated by the roots α_{i} and set Q^{−} = −Q^{+}.

For any λ ∈ P, denote by L_{k}(λ) the simple module with highest weight λ. As we are only interested by its character, we will set L_{k}(λ) = L(λ) if the characteristic of k is zero, and we will set L_{k}(λ) = L_{p}(λ) if the characteristic of the field is p ≠ 0. When k is a field of characteristic p, L_{k}(λ) is a restricted 𝔤_{k}-H(k) module, i.e. it is a restricted 𝔤_{k}-module with a compatible action of H(k).

Let χ(p) = ∑_{λ∈P} m_{λ}(p) e^{λ} be a sequence of characters indexed by all prime numbers p, and let χ = ∑_{λ∈P} m_{λ} e^{λ} be a character. We say that the sequence (χ(p)) converges to the character χ if for all λ ∈ P, we have m_{λ}(p) = m_{λ} for p big enough, i.e. for p > N(λ). In such case, we set lim_{p→∞} χ(p) = χ.

LEMMA 1 Let λ ∈ P.

(i) For any prime number p, we have ch(L_{p}(λ)) ≤ ch(L(λ)).

(ii) We have lim_{p→∞}ch(L_{p}(λ)) = ch(L(λ)).

Proof: Let k be any algebraically closed field. Let 𝔤_{k} =𝔫_{k}^{−}⊕𝔥_{k}⊕𝔫_{k}^{+} be the triangular decomposition of 𝔤_{k}. Let U_{k}, U_{k}^{±}, A_{k} be the enveloping algebras of 𝔤_{k}, 𝔫_{k}^{±}, 𝔥_{k}. Let T : U_{k} → A_{k} be the Harish-Chandra projector, which is uniquely defined by T (u) = u if u ∈ A_{k}, T (u) = 0 if u ∈𝔫_{k}^{−}.U_{k} +U_{k}.𝔫_{k}^{+}. For any weight ν ∈ Q^{−}, one defines the Shapovalov form B_{k}^{ν} : (U_{k}^{+})_{−ν} × (U_{k}^{−})_{ν} → k by B_{k}^{ν}(u^{+}, u^{−}) = \overline{λ}(T (u^{+}.u^{−})), where \overline{λ} : A_{k} → k is the algebra morphism extending λ. As we are only interested by the rank of the Shapovalov forms, we will set B_{k}^{ν} = B^{ν} if k is a field of characteristic zero and we will set B_{p}^{ν} = B_{k}^{ν} if k is a field of characteristic p ≠ 0.

The Shapovalov form B_{k}^{ν} is naturally defined over Z. Hence we have rk(B^{ν}) ≥ rk(B_{p}^{ν}) for any prime number p and rk(B^{ν}) = rk(B_{p}^{ν}) for p big enough (where rk denotes the rank). As the dimension of L_{k}(λ)_{λ+ν} is the rank of B_{k}^{ν}, the assertions (i) and (ii) are proved. Q.E.D.

Let λ ∈ P. Define a weight ρ_{λ} ∈ P by the following requirements:

(i) ρ_{λ}(h_{i}) = 0 if λ(h_{i}) ≥ 0, (ii) ρ_{λ}(h_{i}) = 1 if λ(h_{i}) < 0. For any prime number p, set λ_{p} = λ + p ρ_{λ}. COROLLARY 2

(i) For p big enough, λ_{p} is dominant and restricted. (ii) We have ch(L(λ)) = lim_{p→∞} e^{−pρ_{λ}} ch(L_{p}(λ_{p})). Proof: For p big enough, we have |λ(h_{i})| < p, for all 1 ≤ i ≤ n. Hence the first

assertion follows. We have λ = λ_{p} − p ρ_{λ}. Hence for any field k of characteristic p, the 𝔤_{k}- H(k)-module L_{k}(λ) is isomorphic to L_{k}(λ_{p}) ⊗ k_{−p ρ_{λ}}, where k_{−p ρ_{λ}} is the one dimensional H(k)-module of weight −p ρ_{λ} with a trivial 𝔤_{k}-action. So the formula follows from Lemma 1. Q.E.D.

Remark: Assume that the characteristic of k is p >> 0. As λ_{p} is restricted and dominant, Steinberg proved that L_{k}(λ_{p}) is indeed the restriction to 𝔤_{k} of the simple GL(n, k)- module with highest weight λ_{p} [St]. Hence the character of any module of the category O is the limit of characters of modular representations. This method of computing the characters of modules in the category O is usually not practical, because the character of a general modular representation is usually more complicated than its counterpart in the category O (see [So]). However, we will implicitly use a dual pair which holds in finite characteristics and which has no obvious counterpart in characteristic zero (indeed, the dual pair occurs in the proof of Theorem 3, see [MP]).

2. Some results about modular theory. Let k be an algebraically closed field of characteristic p. Let Y be a Young diagram. Denote by M the number of columns of Y, and by c_{i}(Y) the number of boxes on the i^{th} column. For any subdiagram Y′ in Y, set c_{first}(Y′) = c_{1}(Y′) and c_{last}(Y′) = c_{M}(Y′). As usual, the dominant weight λ associated to Y is the weight λ = ∑_{l≤M} ω_{c_{i}(Y)}. The weight λ is called M-special if and only if c_{first}(Y) − c_{last}(Y) ≤ p − M and M < p. Denote by P(λ) the set of all semi-standard tableaux T of shape Y such that c_{first}(T [l]) − c_{last}(T [l]) ≤ p − M for all l ≤ n. For any semi-standard tableau T of shape Y, denote by w(T) its weight and set rw(T) = w(T)−λ.

THEOREM 3 Let λ be a M-special weight as before. We have:

ch(L_{p}(λ) = e^{λ} ∑_{T ∈P(λ)} e^{rw(T)}.

THEOREM 4 Let λ be a M-special. Then L_{p}(λ)|_{GL(n−1,k)} is semi-simple.

Theorem 3 is proved in [MP] (Theorem 4.3). Theorem 4 has been proved independently by Brundan, Kleshchev, Suprunenko [BKS] and the authors [MP] by very different methods. Soon after that, Brundan, Kleshchev and Suprunenko used their methods to give a new proof of Theorem 3.

3. The character formula for some highest weight modules. Let K be an algebraically closed field of characteristic zero. For a ∈ Z, we denote by ] − ∞, a] the set of all integers n ≤ a. Such a set is called a semi-infinite intervall of Z. A semiinfinite Young diagram of heigth s is a collection of s semi-infinite intervalls I_{i} such that I_{1} ⊃ I_{2} ⊃ … ⊃ I_{s}. We draw a semi-infinite Young diagram as follows:

\overline{| | | | |}

\overline{| | | |}

(\overline{| |}) / (\overline{|})

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A semi-standard tableau of shape Y is a filling of the boxes of Y by indices running from 1 to n, which is increasing from top to bottom, non decreasing from left to right and with the stabilization requirement that almost all boxes on the i^{th} line are filled with the index i, as on the introduction.

Because the semi-standard tableau T is infinite, we cannot define its weight as usual. However for any semi-infinite Young tableau there is an highest semi-standard tableau Y_{h} of shape Y for which the i^{th} line is filled only with the index i. Then the difference of the weights of T and T_{h} is well defined, and will be called the relative weight of T (and it will be denoted by rw(T)). More precisely, the relative weight rw(T) of T is defined as follows. For any box b on the i^{th} line of T, we set α(b) = ϵ_{j} − ϵ_{i} if b is filled with the index j. For almost all b, we have α(b) = 0, and α(b) is a negative root otherwise. We set rw(T) = ∑_{b} α(b).

Let ω_{i} = ϵ_{1} + … + ϵ_{i} be the i^{th} fundamental weight of GL(n). Let m ≥ 0. A mcospecial weight with support [i, j] and singular node s ∈ [i, j] is a weight λ such that such that λ = ∑_{i≤l≤j} a_{l}ω_{l}, and we assume:

(i) a_{l} ≥ 0 for any l ≠ s,

(ii) ∑_{i≤l≤j} a_{l} = −m and j − i ≤ m, (iii) a_{s} < 0, or equivalently λ ≠ 0. For a m-cospecial weight λ as before, we associate a pair (Y_{∞}(λ), Y_{f}(λ)) consisting

of a semi-infinite Young diagram Y_{∞}(λ) and a ordinary Young diagram Y_{f}(λ) defined as follows:

(i) Y_{∞}(λ) is the semi-infinite Young diagram of height s defined by the semi-infinite intervalls I_{1} ⊃ I_{2} ⊃ … I_{s} such that I_{l} =] − ∞, ∑_{max(l,i)≤k≤s} a_{k}],

(ii) Y_{f}(λ) is the Young diagram associated with the dominant weight ∑_{l>s} a_{l}ω_{l}. Let T_{∞}, T_{f} be semi-standard standard tableaux of shape respectively Y_{∞}(λ) and

Y_{f}(λ). Let L be the last column of T_{∞} and let F be the first column of T_{f}. For l ≤ n, denote by T_{f}[l] (respectively T_{∞}[l]) the Young subdiagram of Y_{f}(λ) (respectively of the semi-infinite Young diagram of Y_{∞}(λ)) consisting of all boxes with label ≤ l. We set c_{first}(T_{f}[l]) = card(T_{f}[l] ∩ F), c_{last}(T_{∞}[l]) = card(T_{∞}[l] ∩ L). However, when s = j, the Young diagram Y_{f}(λ) is empty, and the definition of c_{first}(T_{f}[l]) is slighty different. In such a case, we set c_{first}(T_{f}[l]) = min(l, s).

Let T the set of all pair (T_{∞}, T_{f}) of semi-standard tableaux of shape respectively Y_{∞}(λ) and Y_{f}(λ) such that:

(i) for any l ≤ s, all boxes of l^{th} line of Y_{f}(λ) are filled with the index l,

(ii) for any l ≤ n, c_{first}(T_{f}[l]) − c_{last}(T_{∞}[l]) ≤ m.

THEOREM 5 Let λ be a m-cospecial weight. We have

ch(L(λ)) = e^{λ} ∑_{(T_{∞},T_{f})∈T} e^{rw(T_{∞})+rw(T_{f})}.

Proof. Let p be a prime number, and set M = p − m. For p big enough, the weight λ_{p} = λ + p ω_{s} is restricted (Corollary 2), and by definition λ_{p} is M-special. Let Y be the Young diagram of λ_{p}, and set a = ∑_{l>s} a_{l}, b = p − m − a. The first a columns of Y is a Young subdiagram Y_{1} which is identical to Y_{f}. The remaining b columns of Y are identical to the Young diagram Y_{2} consisting of the last b columns of the semi-infinite diagram Y_{∞}.

Fix a weight ν = − ∑_{1≤i≤n−1} m_{i} α_{i} ∈ Q^{−}, and set N = ∑_{1≤i≤n−1} m_{i}. In what follows, we will assume that p is big enough, i.e. p > N. All lines of Y_{2} contains more than N + 1 boxes. Denote by P_{p}^{ν} (respectively T^{ν}) the set of all semi-standard tableaux T ∈ P(λ_{p}) (respectively the pairs (T_{∞}, T_{f}) ∈ T ) such that rw(T) = ν (respectively rw(T_{∞}) + rw(T_{f}) = ν).

Let T ∈ P_{p}^{ν}. For any box b ∈ T such that α(b) ≠ 0, we also have α(b′) ≠ 0 for all b′ in the hook of b. Thus, if the hook lenght of b is ≥ N + 1, we have α(b) = 0. In particular for all boxes of the first s lines of Y_{1}, we have α(b) = 0. So, the restriction of T to Y_{1} determines a semi-standard tableau T_{f}(T) of shape Y_{1} and the l^{th} line of T_{f}(T) is only filled with the index l, for any l ≤ s. Similarly, the restriction of T to Y_{2} determines a semi-standard tableau of shape Y_{2}, and we can extend it to get a semi-standard tableau T_{∞}(T) of shape Y_{∞} by requiring that any box on the l^{th} line of Y_{∞} \ Y_{2} is filled by the index l. It is clear that the map T ↦ (T_{f}(T), T_{∞}(T)) is a bijection from P_{p}^{ν} to T^{ν}. Hence the Theorem 5 follows from Corollary 2 and Theorem 3. Q.E.D.

Remark: Assume now that s = j. In such a case, Y_{f}(λ) = ∅, and an element in T consists essentialy in one tableau T_{∞}, because T_{f} = ∅. Let T_{∞} be a semi-standard tableau of shape Y_{∞}(λ). Denote by L_{1} < … < L_{i} be the indices of the boxes on the last column L of Y_{∞}. We have c_{first}(T_{f}[l]) = min(l, s) and c_{last}(T_{∞}[l]) ≤ c_{last}(T_{∞}[l+1]) ≤ c_{last}(T_{∞}[l])+1. Hence the function l ↦ c_{first}(T_{f}[l]) − c_{last}(T_{∞}[l]) takes its maximal value at s. The condition s − c_{last}(T_{∞}[s]) ≤ m is equivalent to L_{s−m} ≤ s (and is automatically satisfied if s ≤ m). Hence T_{∞} belongs to T if and only if L_{s−m} ≤ s or s ≤ m. This proves the theorem stated in the introduction.

The combinatorics presented here is strongly connected with the Verlinde’s formula for GL(n, \overline{F}_{p}), see [GM] and [MP] for the details. Using Theorem 4 and another version of the semi-continuity principle, we can also deduce that the restriction of L(λ) to 𝔤𝔩(n−1, K) is semi-simple. However, this can be deduced directly. Denote by Λ the set of all weights μ such that there exists a pair (T_{f}, T_{∞}) ∈ T such that μ = λ + ∑_{b∈Y_{f} \T_{f} [n−1]} α(b) + ∑_{b∈Y_{∞}\T_{∞}[n−1]} α(b). For μ ∈ Λ, denote by l(μ) the 𝔤𝔩(n − 1, k)-module with highest weight μ.

COROLLARY 6 As a 𝔤𝔩(n − 1, K)-module, we have: L(λ) = ⊕_{μ∈Λ}l(μ).

Proof: The proof is identical to the proof in [MP], so it will be only sketched. Using Theorem 5, one proves that L(λ) and ⊕_{μ∈Λ}L(μ) have the same character. It follows that the multiplicity of any simple module occuring in a Jordan-Holder series of the 𝔤𝔩(n−1, K)- module L(λ) is one. It follows that L(λ) is semi-simple. Q.E.D.

Bibliography:

[BKS] J. Brundan, A. Kleshchev, I. Suprunenko, Semi-simple restrictions from GL(n) to GL(n − 1), Preprint (1997).

[GM] G. Georgiev, O. Mathieu, Catégorie de fusion pour les groupes de Chevalley. C. R. Acad. Sci. Paris, 315 (1992), 659-662.

[Li] P. Littelmann, A generalization of the Littlewood-Richardson rule. J. Algebra, 130 (1990) 328-368.

[LR] D.E. Littlewood, A.R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A, 233 (1934), 49-141.

[MP] O. Mathieu and G. Papadopoulo, A character formula for a family of simple modular representations of GL(n), Preprint (1997).

[So] W. Soergel, Conjecture de Lusztig., Séminaire Bourbaki (1994), Astérisque, 237 (1996) 75-86.

[St] R. Steinberg, Lectures on Chevalley groups. Notes, Yale University, (1967). Authors Adress: Université Louis Pasteur, IRMA, 7 rue René Descartes, 67000 Strasbourg (FRANCE). email: mathieu@math.u-strasbg.fr papadopo@math.u-strasbg.fr