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ON SIMPLE POLYNOMIAL G T-MODULES r CHRISTIAN DRENKHAHN Abstract. Using the general framework of polynomial representations defined by Doty and generalizing the definition given by 6 Doty, Nakano and Peters for G = GLn, we consider polynomial 1 representations of GrT for an arbitrary closed reductive subgroup 0 2 scheme G ⊆ GLn and a maximal torus T of G in positive characteristic. We give sufficient conditions on G making a classification n of simple polynomial GrT-modules similar to the case G = GLn a J possibleandapplythistorecoverthecorrespondingresultforGLn withadifferentproof,extendingittosymplecticsimilitudegroups, 4 1 Levi subgroups of GLn and, in a weaker form, to odd orthogonal similitudegroups. WealsoconsiderorbitsoftheaffineWeylgroup ] andgivea condition forequivalence of blocksof polynomialrepre- T R sentations for GrT in the case G=GLn. . h t a m 1. Introduction [ When considering rational representations of G = GLn over an in- 1 finite field, one can reduce many questions to polynomial representa- v tions. These representations correspond to representations of certain 4 3 finite-dimensional algebras, the so-called Schur algebras (see for exam- 6 ple [6, 8]). Letting T be a maximal torus of G and k be an infinite 3 0 field of positive characteristic, one can also study the category of GrT- . modules to get information on the representation theory of G (see [7, 1 0 II.9]). In [4], Doty, Nakano and Peters combined these approaches and 6 considered polynomial representations of G T for G = GL . They 1 r n : show that these representations correspond to representations of the v so-called infinitesimal Schur algebras which are subalgebras of the or- i X dinary Schur algebras. In [1], Doty defined a general notion of poly- r a nomial representations and analogues of Schur algebras for algebraic groups contained in GL . Using his definition, one can study polyno- n mialrepresentations ofG T forother algebraicgroupsG. If a reductive r group G ⊆ GL admits a graded polynomial representation theory in n the sense of [1], it is natural to ask for which character λ contained in the character group X(T) of T the simple G T-module L (λ) is a r r polynomial G T-module. This problem was solved for G = GL in [4], r n but in contrast to most results of that paper, the proof doescnot carry over to other reductive groups. Let D = T be the closure of T in the algebraic monoid of (n×n)-matrices, P(D) ⊆ X(T) the set of polynomial weights, i.e. the character monoid of D, and X (T) ⊆ X(T) the r 1 2 CHRISTIANDRENKHAHN set of pr-restricted weights. Then our main result can be summarized as follows: Theorem. (a) Let G = GSp be the symplectic dilation group, T ⊆ G 2l the torus of diagonal matrices. Then a complete set of representatives forthe isomorphismclassesof simplepolynomialG T-modules r is given by {L (λ)|λ ∈ P (D) + prP(D)}, where P (D) = {λ ∈ r r r P(D)∩X (T)|λ−prd ∈/ P(D)} and d is the character of T given r by t 7→ t11tnn.c (b) Let G = GO be the connected component of the odd orthog- 2l+1 onal dilation group, T the torus of diagonal matrices in G. If λ ∈ X (T) + prX(T) and all weights of the module L (λ) are r r polynomial, then λ ∈ P (D) + prP(D), where P (D) = {λ ∈ r r P(D)∩Xr(T)|λ−prd ∈/ P(D)} and d is the character ocf T given by t 7→ t . l+1 (c) Let n ∈ N such that n = l n and embed G = GL ×...×GL i i=1 i n1 nl into GL as a Levi subgroup in the obvious way. Let T be the torus n P of diagonalmatrices in G, M = {n +...+n +1,...,n +...+n } i 1 i−1 1 i and d = e for 1 ≤ i ≤ l, where e ,...,e is the standard i j∈Mi j 1 n basis of X(T) ∼= Zn. Then a complete set of representatives for P the isomorphism classes of simple polynomial G T-modules is given r by {L (λ)|λ ∈ P (D) + prP(D)}, where P (D) = {λ ∈ X (T) ∩ r r r r P(D)|λ−prd ∈/ P(D) for 1 ≤ i ≤ l}. i c We derive this as a corollary of a more general and technical result giving a sufficient condition on G for a classification of simple polynomial G T-modules. Our result also gives a new proof for the case r G = GL originally treated in [4] and allows all cases to be treated n in a uniform way. By studying the intersection of orbits of the affine Weyl group with suitable sets of polynomial weights, we also show that certain shift functors induce Morita equivalences between blocks of polynomial representations of G T for G = GL . r n 2. Notation and prerequisites In this section, we fix notation and recall some basic results on polynomial representations. For algebraic groups and G T, we follow the r notation from [7]. For an introduction to algebraic monoids, we refer the reader to [10]. Let k be aninfinite perfect field of characteristic p > 0 andMat bethe n monoid scheme of (n×n)-matrices over k. Let d ∈ N and denote by 0 A(n,d) the space generated by all homogeneous polynomials of degree d in the coordinate ring k[Mat ] = k[X |1 ≤ i,j ≤ n] of Mat . Then n ij n k[Mat ] = A(n,d) is a graded k-bialgebra with comultiplication n d≥0 ∆ : k[Mat ] → k[Mat ]⊗ k[Mat ] andcounit ǫ : k[Mat ] → k given by n n k n n Ln ∆(X ) = X ⊗ X ,ǫ(X ) = δ and each A(n,d) is a subcoalge- ij l=1 il k lj ij ij bra. As GL is dense in Mat , the canonical map k[Mat ] → k[GL ] is n n n n P ON SIMPLE POLYNOMIAL G T-MODULES 3 r injective, so that k[Mat ] and each A(n,d) can be viewed as a subcoal- n gebra of k[GL ]. Now let G be a closed subgroup scheme of GL and n n π : k[GL ] → k[G] the canonical projection. Set A(G) = π(k[Mat ]) n n and A (G) = π(A(n,d)) as well as S (G) = A (G)∗. Since π is a d d d homomorphism of Hopf algebras, A(G) is a subbialgebra of k[G] and each A (G) is a subcoalgebra of A(G). As A (G) is finite dimensional, d d S (G) is a finite dimensional associative algebra in a natural way. For d G = GL and T a maximal torus of G, the algebra S (G) is the or- n d dinary Schur algebra, while S (G T) is the infinitesimal Schur algebra d r defined in [4]. Following [1], we say that G admits a graded polynomial representation theory if the sum A (G) is direct and we say that d≥0 d a rational G-module V is a polynomial G-module if the corresponding P comodule map V → V ⊗ k[G] factors through V ⊗ A(G). If the comodule map factors through V ⊗A (G) for some d ∈ N , we say that d 0 V is homogeneous of degree d. Clearly, every A (G)-comodule is an d A(G)-comodule and every A(G)-comodule is a G-module in a natural way. We record some other properties of polynomial representations from [1, 1.2] in the following proposition. Proposition 2.1. (1) Suppose G admits a graded polynomial representation theory and V is a polynomial G-module. Then V = V , where each V is homogeneous of degree d. Furthermore, d≥0 d d the category of homogeneous modules of degree d is equivalent to L the category of S (G)-modules. d (2) If G contains the center Z(GL ) of GL , then G admits a graded n n polynomial representation theory. As A(G) is a factor bialgebra of k[Mat ], it corresponds to a closed n submonoid M of Mat . Since A(G) ⊆ k[G] is a subbialgebra, G ⊆ M n is a dense subscheme, so that M is the closure of G in Mat . Hence n polynomial representations of G can be regarded as rational representations of the algebraic monoid scheme M = G. If G = GL , a comparison of coordinate rings shows G T = M D, n r r where M D = (Fr)−1(D) with Fr the r-th iteration of the Frobenius r homomorphism, see [4]. This can fail for general G: If G is the full orthogonal similitude group in dimension two and T the torus of diagonal matrices in G, then G T = T, but M D is strictly larger than r r D = T. However, we always have G T ⊆ M D. We do not know r r whether M D = G T e.g. for connected G. r r 3. Simple polynomial G T-modules r In this section, let G ≤ GL be a closed connected reductive sub- n groupscheme containing Z(GL ), T ⊆ Gbeamaximaltoruscontained n in a maximal torus T′ ⊆ GL , M be the closure of G in Mat , D be the n n closure ofT, D′ betheclosure ofT′ andP(D) ⊆ X(T)bethecharacter monoid of D, i.e. the set of polynomial weights, W the Weyl group of 4 CHRISTIANDRENKHAHN G. Let R be the root system of G with respect to T and S ⊆ R a set of simple roots. Denote by h−,−i : X(T)×Y(T) → Z the canonical perfect pairing of X(T) and the cocharacter group Y(T). For α ∈ R, let α∨ be the coroot of α. Let X (T) = {λ ∈ X(T)|hλ,α∨i = 0 for all 0 α ∈ R} as well as X (T) = {λ ∈ X(T)|0 ≤ hλ,α∨i ≤ pr − 1 for all r α ∈ S}. We want to determine the simple polynomial G T-modules. Recall r that the isomorphism classes of simple G T-modules are parametrized r by X(T) via λ 7→ L (λ), where L (λ) has highest weight λ, see [7, II.9]. r r As a motivation for our approach, consider the case where G = GL n and T ≤ G is thce maximal tcorus of diagonal matrices. According to [4, Section 3], the simple G T-module L (λ) is a polynomial G T- r r r module iff λ ∈ P (D) + prP(D), where P (D) = {λ ∈ P(D)|0 ≤ r r λi −λi+1 ≤ pr −1 for 1 ≤ i ≤ n−1, 0 ≤cλn ≤ pr −1}. A character λ ∈ X(T) belongs to P(D) iff λ ≥ 0 for 1 ≤ i ≤ n, that is, iff i min{λ |1 ≤ i ≤ n} ∈ N . Taking this minimum is compatible with i 0 multiplication by natural numbers, in particular with multiplication by pr. Given characters λ,µ ∈ X(T), we can of course find a permutation ∼ w ∈ W = S suchthatthisminimumisattainedatthesamecoordinate n for w.λ and µ, so that min{(w.λ+ µ) |1 ≤ i ≤ n} = min{λ |1 ≤ i ≤ i i n} + min{µ |1 ≤ i ≤ n}. Writing d = det| = (1,...,1), we have i T X (T) = hdi,min{d |1 ≤ i ≤ n} = 1 and we can write P (D) = {λ ∈ 0 i r X (T) ∩ P(D)|λ − prd ∈/ P(D)}. We axiomatize these properties of r det| and the function X(T) → Z,λ 7→ min{λ |1 ≤ i ≤ n} for general T i G in the following Assumption 3.1. Suppose there exists a function ϕ : X(T) → Zl with the following properties: (1) ∀λ ∈ X(T) : ϕ(λ) ∈ Nl ⇐⇒ λ ∈ P(D), 0 (2) ∀λ ∈ X(T) : ϕ(prλ) = prϕ(λ), (3) ∀λ,λ′ ∈ X(T) : ∃w ∈ W : ϕ(w.λ+λ′) = ϕ(λ)+ϕ(λ′), (4) ϕ| : X (T) → Zl is bijective. X0(T) 0 Since W acts trivially on X (T), (1),(3) and (4) imply that the 0 restriction to X (T) of any function ϕ as in 3.1 is an isomorphism 0 mapping P(D)∩X (T) to Nl. It will turn out that 3.1 is sufficient for 0 0 a classification similar to the case G = GL . We first show that such a n function is essentially unique. Proposition 3.2. If ϕ,ψ : X(T) → Zl are functions as in 3.1, there is a permutation of coordinates σ : Zl → Zl such that ϕ = σ ◦ψ. Proof. Let e ,...,e be the standard basis of Zl and d ,...,d resp. 1 l 1 l d′,...,d′ be the preimages of the e in X (T) for ϕ resp. ψ. Then 1 l i 0 every element of X (T) ∩ P(D) can be written as a linear combina- 0 tion in the d with non-negative coefficients. Writing the d′ as such a i i ON SIMPLE POLYNOMIAL G T-MODULES 5 r linear combination and then writing the d as such a linear combina- i tion in the d′, we see that the base change matrix for the two bases i is a permutation matrix. Thus, there is σ ∈ S such that d = d′ l i σ(i) ˜ l for 1 ≤ i ≤ l. Now let λ ∈ X(T) and set λ = λ − ϕ(λ) d . i=1 i i Using (3) and that W acts trivially on X (T) for the first equality 0 P and the fact that ψ| is an isomorphism for the second equality, X0(T) ˜ l l we get ψ(λ) = ψ(λ)+ψ(− ϕ(λ) d ) = ψ(λ)− ϕ(λ) ψ(d ) = i=1 i i i=1 i i ψ(λ)− l ϕ(λ) ψ(d′ ) = ψ(λ)− l ϕ(λ) e . Applying the same i=1 i σ(i) P i=1 i σ(i) P ˜ ˜ arguments to ϕ, we get ϕ(λ) = 0 and ϕ(λ −d ) = −e for 1 ≤ i ≤ l. P P i i Thus, λ˜ ∈ P(D) and λ˜ − d = λ˜ − d′ ∈/ P(D) for 1 ≤ i ≤ l, so i σ(i) that ψ(λ˜) ∈ Nl and ψ(λ˜) − e ∈/ Nl. This shows ψ(λ˜) = 0, so that 0 i 0 ψ(λ) = l ϕ(λ) e , hence ϕ(λ) = ψ(λ) . (cid:3) i=1 i σ(i) i σ(i) DefinitPion 3.3. Suppose ϕ is a function as in 3.1 for G. Letting e ,...,e be the standard basis of Zl and d ,...,d be the preimages 1 l 1 l of the e in X (T) with respect to ϕ, we set P (D) = {λ ∈ P(D) ∩ i 0 r X (T)|∀i ∈ {1,...,l} : λ−prd ∈/ P(D)}. r i It follows from the proof of 3.2 that P (D) does not depend on the r choice of ϕ or the d . i Lemma 3.4. Let λ = λ +prλ˜ = λ′ +prλ˜′ with λ ,λ′ ∈ X (T),λ˜,λ˜′ ∈ 0 0 0 0 r X(T). Then there is µ ∈ X (T) such that λ = λ′ +prµ,λ˜ = λ˜′ +µ. 0 0 0 Proof. We have λ − λ′ = prµ, where µ = λ˜ − λ˜′. For all α ∈ R, we 0 0 have prhµ,α∨i = hλ −λ′,α∨i = hλ ,α∨i−hλ′,α∨i. As λ ,λ′ ∈ X (T), 0 0 0 0 0 0 r we have −(pr −1) ≤ hλ ,α∨i−hλ′,α∨i ≤ pr −1, forcing hµ,α∨i = 0. 0 0 Thus, µ ∈ X (T) and the result follows. (cid:3) 0 Theorem 3.5. Suppose 3.1 holds for G. If λ ∈ X (T)+prX(T), then r there is a unique decomposition λ = λ + prλ˜ with λ ∈ P (D),λ˜ ∈ 0 0 r X(T). Furthermore, if all weights of the simple G T-module L (λ) r r belong to P(D), then λ ∈ P (D)+prP(D). r c Proof. Letλ = λ +prλ˜ withλ ∈ X (T),λ˜ ∈ X(T). Asd ∈ X (T), we 0 0 r i 0 get that for each a ∈ Zl, λ +pr l a d +pr(λ˜− l a d ) is another 0 i=1 i i i=1 i i decomposition such that λ + pr l a d ∈ X (T). Since d ,...,d 0 P i=1 i i rP 1 l generate X (T), 3.4 shows that every decomposition of λ arises in this 0 P fashion. By (1),(3),(4) and since W acts trivially on X (T), there is 0 a unique a ∈ Zl such that λ + pr l a d ∈ P (D). Thus, there 0 i=1 i i r is a unique decomposition λ = λ + prλ˜ such that λ ∈ P (D). We 0 0 r P show by contraposition that if all weights of L (λ) are polynomial, r then λ˜ ∈ P(D). Suppose that λ˜ ∈/ P(D), so that ϕ(λ˜) ∈/ Nl. Then 0 there is i ∈ {1,...,l} such that ϕ(λ˜) < 0. Sincce λ − prd ∈/ P(D), i 0 i (1),(3),(4) yield ϕ(λ ) < pr. Using (2) and (3), we see that there 0 i is w ∈ W such that ϕ(w.λ + prλ˜) = ϕ(λ ) + prϕ(λ˜) < 0, so that 0 i 0 i i 6 CHRISTIANDRENKHAHN w.λ +prλ˜ ∈/ P(D) by (1). Since L (λ ) lifts to a simple G-module by 0 r 0 [7, II.9.6.(g)], its character is W-invariant, so that w.λ is a weight of 0 L (λ ) and w.λ +prλ˜ is a weightcof L (λ +prλ˜) ∼= L (λ )⊗ prλ˜, so r 0 0 r 0 r 0 k that not all weights of L (λ) are polynomial. (cid:3) r c c c For G = GL , it was shown in the Appendix of [9] that a G T- n r c module such that all of its weights are polynomial is a polynomial G T-module. It is not known for which general G a similar statement r holds. However, if M is normal as a variety, we can at least prove this for some simple modules. Proposition 3.6. Suppose that M is a normal variety. If λ = λ + 0 prλ˜ ∈ (X (T) ∩ P(D)) + prP(D), the simple G T-module L (λ) is a r r r polynomial G T-module. r b Proof. We have L (λ) ∼= L (λ ) ⊗ prλ˜ and L (λ ) lifts to G by [7, r r 0 k r 0 II.9.6]. Since M is normal and λ is a dominant weight contained in 0 P(D), L (λ ) liftsbto M byc[2, 3.5], so that L (λc) is a polynomial G T- r 0 r 0 r module. Asλ˜ isaD-module, prλ˜ isanF−r(D)-module, where Fr isthe r-th itecration of the Frobenius morphism, abnd thus a polynomial G T- r module by restriction. Hence L (λ) is a polynomial G T-module. (cid:3) r r Corollary 3.7. Let M be normal and suppose that 3.1 holds for G. b If λ ∈ X (T)+prX(T), then the simple module L (λ) is a polynomial r r G T-module iff λ ∈ P (D)+prP(D). r r Proof. This follows directly from 3.5 and 3.6. c (cid:3) Corollary 3.8. Let the derived subgroup of G be simply connected and M be normal. Suppose 3.1 holds for G. Then a system of representatives of isomorphism classes of simple polynomial G T-modules is given r by {L (λ)|λ ∈ P (D)+prP(D)}. r r Proof. Let λ ∈ X(T). Since the derived subgroup of G is simply con- c nected, X (T) contains a set of representatives of X(T)/prX(T), so r that we have λ ∈ X (T) + prX(T). By 3.5, all weights of L (λ) are r r polynomial iff λ ∈ P (D)+ prP(D) and by 3.6, L (λ) lifts to G T in r r r this case. Since all weights of polynomial G T-modules are poclynomial r and a polynomial module is a simple polynomialcGrT-module iff it is simple as a G T-module, the result follows. (cid:3) r We now give an explicit construction for a function as in 3.1 under certain assumptions. Theorem 3.9. Suppose there is an action of W on X(T′) such that the canonical projection π : X(T′) → X(T) is W-equivariant and there are b ,...,b ∈ π−1(X (T))∩P(D′) such that 1 s 0 (a) ∀i ∈ {1,...,s},j ∈ {1,...,n} : (b ) ∈ {0,1}, i j ON SIMPLE POLYNOMIAL G T-MODULES 7 r (b) the sets M = {j ∈ {1,...,n}|(b ) = 1} form a partition of i i j {1,...,n}, (c) the image of W in SX(T′) is contained in the image of W′ ∼= Sn and contains the subgroup S ×...×S , M1 Ms (d) ∃d ,...,d ∈ {b ,...,b } : π(d ),...,π(d ) is a basis of X (T) 1 l 1 s 1 l 0 such that each π(b ) is a linear combination of the π(d ) with non- i i negative coefficients. Then there is a function ϕ : X(T) → Zl with the properties of 3.1. l Proof. For every i ∈ {1,...,s}, write π(b ) = n π(d ) with n ∈ i j=1 ij j ij N . Define ϕ : X(T′) → Zl, λ 7→ l s min{λ |a ∈ M }n e . We 0 j=1 i=1 P a i ij j show that ϕ induces a funtion ϕ¯ : X(T) → Zl with the properties of P P 3.1. For this, we show the following statements (i)−(v). (i) ∀λ ∈ X(T′),µ ∈ ker(π) : ϕ(λ+µ) = ϕ(λ) : Let µ ∈ ker(π). Since the sets M are pairwise disjoint, we have i µ′ = µ− s min{µ |a ∈ M }b ∈ P(D′) and for every i, there i=1 a i i is a ∈ M such that µ′ = 0. By 3.10 below and (a), we get i a µ′ = 0 forPall l ∈ M . As the M form a partition of {1,...,n}, we l i i get µ′ = 0, so that µ = s min{µ |a ∈ M }b . Consequently, i=1 a i i ϕ(λ+µ) = ϕ(λ)+ϕ(µ) for all λ ∈ X(T′). As µ ∈ ker(π), we have P s 0 = π(µ) = min{µ |a ∈ M }π(b ) a i i i=1 X l s = n min{µ |a ∈ M } π(d ), ij a i j ! j=1 i=1 X X s so that n min{µ |a ∈ M } = 0 for every j and ϕ(µ) = 0. i=1 ij a i Hence (i) holds. (ii) ∀λ ∈ X(PT′) : ϕ(prλ) = prϕ(λ) : Clear by definition. (iii) ∀λ ∈ X(T′) : ϕ(λ) ∈ Nl ⇐⇒ λ ∈ P(D)+ker(π),: 0 Suppose ϕ(λ) ∈ Nl. We have λ − s min{λ |a ∈ M }b ∈ 0 i=1 a i i P(D′), so that π(λ) − s min{λ |a ∈ M }π(b ) ∈ π(P(D′)). i=1 a P i i Hence π(λ) ∈ π(P(D′)) + l s min{λ |a ∈ M }n π(d ). P j=1 i=1 a i ij j By assumption, the coefficients in the linear combination are non- negative,sothatthissetiscoPntainePdinπ(P(D′))andλ ∈ P(D′)+ ker(π). Now let λ = λ′+µ ∈ P(D′)+ker(π). Since ϕ(λ) = ϕ(λ′) by (i) and since λ′ has no negative coordinates and n > 0, we ij have ϕ(λ) ∈ Nl. 0 (iv) ∀λ,λ′ ∈ X(T′) : ∃w ∈ W : ϕ(w.λ+λ′) = ϕ(λ)+ϕ(λ′): Using (c), we permute the coordinates in each M for λ in such a i way that the lowest coordinate for w.λ and λ′ in each M occurs i at the same place. Then clearly ϕ(w.λ+λ′) = ϕ(λ)+ϕ(λ′). 8 CHRISTIANDRENKHAHN (v) ϕ(d ) = e : i i This follows directly from (d) and the definition of ϕ. The statement (i) implies that there is a function ϕ¯ : X(T) → Zl such that ϕ = ϕ¯ ◦ π. Now (ii) − (v) imply that ϕ¯ has the properties of 3.1. (cid:3) Lemma 3.10. In the situation of 3.9, let i ∈ {1,...,s} and µ ∈ ker(π). Then µ = µ for all l,k ∈ M . k l i Proof. Let µ ∈ ker(π) and i ∈ {1,...,s}. Suppose there are k,l ∈ M i such that µ 6= µ . Let w = (k l) ∈ W be the transposition with k l respect to k and l. Then µ − w.µ = (µ − µ )(e − e ) ∈ ker(π). As k l k l the restriction of this element is zero in X(T) and X(T) is torsion-free, (e −e )| = 0, so that e | = e | . It follows that w acts trivially on k l T k T l T T, a contradiction. (cid:3) The function ϕ defined in the proof of 3.9 provides an easy way to check whether the restriction of a character λ ∈ X(T′) to T is a polynomial weight. We now apply our results to several examples. The first part of the following corollary is [5, Corollary 3.2]. For the orthogonal and symplectic similitude groups GO and GSp , we adopt the definition 2l+1 2l and notation from [1, Sections, 5, 7]. Recall that with suitable choice of defining form, the maximal torus of diagonal matrices in GSp resp. 2l GO2l+1 is given by {diag(t1,...,t2l)|tiiti′i′ = tjjtj′j′ for 1 ≤ i,j ≤ l} resp. {diag(t1,...,t2l+1)|tiiti′i′ = tjjtj′j′ for 1 ≤ i,j ≤ l +1}. Corollary 3.11. (a) ForG = GL ,T the torus of diagonalmatrices, a n complete set of representatives of simple polynomial G T-modules r is given by {L (λ)|λ ∈ P (D) + prP(D)}, where P (D) = {λ ∈ r r r P(D)|0 ≤ λ −λ ≤ pr −1 for 1 ≤ i ≤ n−1,0 ≤ λ ≤ pr −1}. i i+1 n (b) Let G = GScp2l,T the torus of diagonal matrices in G. Then a complete set of representatives of the isomorphism classes of simple polynomial G T-modules is given by {L (λ)|λ ∈ P (D)+prP(D)}, r r r where P (D) = {λ ∈ P(D)∩X (T)|λ−prd ∈/ P(D)} and d is the r r character of T given by t 7→ t11tnn. c (c) Let G = GO ,T the torus of diagonal matrices in G. If λ ∈ 2l+1 X (T)+prX(T) and all weights of the module L (λ) are polynomial, r r then λ ∈ P (D)+prP(D), where P (D) = {λ ∈ P(D)∩X (T)|λ− r r r prd ∈/ P(D)} and d is the character of T givecn by t 7→ tl+1. (d) Let n ∈ N such that n = l n and embed G = GL ×...×GL i i=1 i n1 nl into GL as a Levi subgroup in the obvious way. Let M = {n + n i 1 P ...+n +1,...,n +...+n } and d = e for 1 ≤ i ≤ l. i−1 1 i i j∈Mi j Then a complete set of representatives of isomorphism classes of P simple polynomial G T-modules is given by {L (λ)|λ ∈ P (D) + r r r c ON SIMPLE POLYNOMIAL G T-MODULES 9 r prP(D)}, where P (D) = {λ ∈ X (T)∩P(D)|λ−prd ∈/ P(D) for r r i 1 ≤ i ≤ l}. Proof. (a) Let d = det| = (1,...,1),M = {1,...,n}. Clearly d and T M have the properties of 3.9 and P (D) has the required form. r As the derived subgroup of GL is simply connected and Mat is n n normal, the result follows from 3.8. (b) We have W ∼= S ⋉(Z/(2))l. We can identify W with a subgroup l of W′ ∼= S by letting σ ∈ S act on λ ∈ X(T′) via (σ.λ) = 2l l i λσ−1(i),(σ.λ)i′ = λ(σ−1(i))′ for 1 ≤ i ≤ l and letting the i-th unit vector of (Z/(2))l act as the transposition (ii′). With this action, the canonical projection π : X(T′) → X(T) is W-equivariant. For 1 ≤ i ≤ l, let di = ei + ei′ and Mi = {i,i′}. Then π(di) = π(dj) for 1 ≤ i,j ≤ l,the character d = π(d ) is a basis of X (T) and 1 0 the M form a partition of {1,...,n}. Inside W ∼= S ⋉ (Z/(2))l, i l (Z/(2))l corresponds toS ×...×S . Thus, 3.9yields thatG has M1 Ml the properties of 3.1 and P (D) has the desired form. The derived r subgroup of G is Sp , hence simply connected, and by the remark 2l below 2.11 in [3], G is normal. The result now follows from 3.8. (c) For 1 ≤ i ≤ l, let di = ei + ei′, Mi = {i,i′} and let dl+1 = el+1, M = {l + 1}. Then the M form a partition of {1,...,l}, the l+1 i character d = π(d ) is a basis of X (T) and π(d ) = 2π(d) for l+1 0 i 1 ≤ i ≤ l. The other properties of 3.9 can be checked as in the proof of (b), so that G has the properties of 3.1 and P (D) has the r desired form. The result now follows from 3.5. (d) The M form a partition of {1,...,n}. By definition of G, we have i W = S ×...×S ⊆ W′ = S and the d form a basis of X (T). M1 Ml n i 0 Thus, 3.9 shows that G has the properties of 3.1, so that the result follows from 3.8. (cid:3) We give an example showing that our approach does not work for the connected component G of the even orthogonal similitude group GO . Let l = 4 and choose p,r such that 4|pr − 1. Let T,T′,d 2l be as in 3.11.(b). Let ϕ : X(T′) → Z, t 7→ li=1min{ti,ti′}. The same proof shows that with the exception of (iv), all statements in the proof of 3.9 hold for ϕ and d. Let λ ,λ˜ ∈PX(T′) be the charac- 0 ters given by λ = (pr−1, pr−1, pr−1, pr−1, pr−1, pr−1, pr−1, pr−1) and λ˜ = 0 2 2 2 2 4 4 4 4 (0,0,0,0,1,1,−1,1). Then ϕ(λ ) = pr −1 > 0, ϕ(λ −prd) = −1 < 0 0 0 and λ | ∈ X (T), so that λ | ∈ P (D). In W ∼= S ⋉(Z/(2))l−1, S 0 T r 0 T r l l acts on X(T′) as in the proof of 3.11.(b) while (Z/(2))l−1 acts as the subgroup {x ∈ (Z/(2))l|x 6= 0 for an even number of i} of (Z/(2))l. i Thus, λ is invariant under S and we have ϕ(w.λ | + prλ˜| ) > 0 0 l 0 T T and hence w.λ | + prλ˜| ∈ X(D) for all w ∈ (Z/(2))l−1. Hence the 0 T T arguments in the proof of 3.5 are not applicable in this case. As the statement w.λ | + prλ˜| ∈ X(D) does not depend on ϕ and there 0 T T 10 CHRISTIANDRENKHAHN is no other choice for d in this case, we cannot find other ϕ or d in this example to salvage the proof of 3.5. However, since we have not computed the weights of L (λ | +prλ˜| ), we do not know whether all r 0 T T weights of this module are in fact polynomial. c 4. Equivalences between blocks of polynomial representations We adopt the notation of the previous section. Suppose that G has simply connected derived subgroup and that there is 0 6= b ∈ X (T)∩P(D) such that hbi = X (T). Choose representatives (ω ) 0 0 α α∈∆ ofthefundamental dominant weights ofthederived subgroupsuch that for all α ∈ ∆, ω ∈ P(D) and ω −b ∈/ P(D). Then the ω together α α α with b form a basis of X(T). Let ϕ : X(T) → Z be the projection on the coordinate corresponding to b and define P (D) = {λ ∈ X (T) ∩ r r P(D)|0 ≤ ϕ(λ) ≤ pr − 1}. Then each λ ∈ X(T) can be written uniquely as λ = λ +prλ˜ with λ ∈ P (D),λ˜ ∈ X(T). By [7, II.1.18], 0 0 r ∼ we have X(G) = X (T) via restriction, so that we can regard b as a 0 character ofGandhaveashift functor[b] : modG T → modG T,V 7→ r r V[b] = V ⊗ k , wherek istheone-dimensional G-moduleinduced byb. k b b This shift functor is an autoequivalence of modG T with inverse [−b], r hence it induces equivalences between blocks of modG T. Since −b is r not a polynomial weight, [−b] does not restrict to an autoequivalence of the category of polynomial G T-modules. However, we will see in r this section that the functor [b] sometimes still induces an equivalence between blocks of this category. Let W ∼= W⋉pZR be the affine Weyl p group of G, let R+ be a set of positive roots and ρ = 1α be α∈R+ 2 the half-sum of positive roots. We denote by w • λ = w(λ + ρ) − ρ the dot-action of w ∈ W on λ ∈ X(T). For a ∈ Z, if aP= qp+ r for p q,r ∈ Z such that 0 ≤ r ≤ p−1, we write r = a mod p. Lemma 4.1. Let the derived subgroup of G be simply connected. Let λ ∈ P (D) + prP(D) and a = max {ϕ(w.λ + w.ρ − ρ) mod p}. r w∈W Then for 1 ≤ i ≤ p−a−1, the shift [ib] defines a bijection W •λ∩(P (D)+prP(D)) → W •(λ+ib)∩(P (D)+prP(D)). p r p r Proof. Let (w,plµ) ∈ W . We show that (w,plµ)•λ ∈ P (D)+prP(D) p r iff (w,plµ)•(λ+ib) ∈ P (D)+prP(D). We have (w,plµ)•λ = w.λ+ r w.ρ−ρ+plµ and (w,plµ)•(λ+ib) = (w,plµ)•λ+ibby definition of the dot-actionandsincew.b = b. Writingw.λ+w.ρ−ρ+plµ = ν +prν˜with 0 ν ∈ P (D) and ν˜ ∈ X(T), we have ϕ(ν ) mod p = ϕ(w.λ+w.ρ−ρ) 0 r 0 mod p ≤ a. It follows that (ϕ(ν ) mod p) + i ≤ a + p − a − 1 = 0 p − 1. Thus, ϕ(ν + ib) ≤ pr, so that ν + ib ∈ P (D). By unicity 0 0 r of decomposition, we get (w,plµ) • λ ∈ P (D) + prP(D) ⇐⇒ ν˜ ∈ r P(D) ⇐⇒ (w,plµ)•(λ+ib) ∈ P (D)+prP(D). (cid:3) r