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Donkin-Koppinen filtration for general linear supergroup

R. La Scala and A.N.Zubkov

Abstract

In this article we consider a generalization of Donkon-Koppinen filtrations for co-ordinate superalgebras of general linear supergroups. More precisely, if G = GL(m|n)is a general linear supergroup of (super)degree (m|n), then its coordinate superalgebraK[G] is a natural G × G-supermodule. For any finitely generated ideal Γ ⊆ Λ × Λthe largest supersubmodule OΓ(K[G]), whose all composition factors are L(λ)⊗L(µ)with (λ, µ) ∈ Γ, has a decreasing filtration OΓ(K[G]) = V0 ⊇ V1 ⊇ . . . , such that⋂

t≥0Vt = 0 and Vt/Vt+1 ≃ V−(λt)

∗⊗H0−(λt). Here H0

−(λ) and V−(λ) are couniversaland universal G-supermodules of highest weight λ ∈ Λ respectively (see [5]). We applythis result to describe adjoint action invariants of G.

Introduction

Let G be a reductive algebraic group defined over an algebraically closed field K. Thegroup G × G acts on G by (g, (g1, g2)) 7→ g−1

1 gg2, g, g1, g2 ∈ G. This induces a structureof a rational G-bimodule on K[G]. Koppinen (and previously Donkin) proved that K[G]has an increasing G-bimodule filtration 0 ⊆ V1 ⊆ V2 ⊆ . . . , such that

⋃

t≥1 Vt = K[G] and

Vt/Vt−1 ≃ V (λt)∗ ⊗ H0(λt), λt ∈ X(T )+ [10]. Besides, if k > l, then either λk ≤ λl or λk

and λl are not comparable.In the present article we generalize this result for the general linear supergroup G =

GL(m|n). In contrast to the classical case the G-superbimodule K[G] has not increasingfiltrations as above. In fact, the set of highest weights Λ of simple G-supermodules hasnot minimal elements. Nevertheless, one can prove that for any finitely generated idealΓ ⊆ Λ × Λ the largest supersubmodule OΓ(K[G]), whose all composition factors areL(λ) ⊗ L(µ) with (λ, µ) ∈ Γ, has a decreasing filtration OΓ(K[G]) = V0 ⊇ V1 ⊇ . . . , suchthat

⋂

t≥0 Vt = 0 and Vt/Vt+1 ≃ V−(λt)∗ ⊗H0

−(λt). Here H0−(λ) = indG

B−Kλ is an inducedsupermodule of highest weight λ, V−(λ) is a Weyl supermodule of highest weight λ, B− isa Borel supersubgroup of G consisting of all lower triangular matrices and Kλ is a natural(even) one dimensional B−-supermodule of weight λ (see [5] for more definitions). In thelast section we use the above filtrations to describe adjoint action invariants of G.

1 Preliminary definitions and notations

Let G ba an affine supergroup. In other words, G is a (representable) functor fromthe category of commutative superalgebras SAlgK to the category of groups Gr, suchthat G(A) = HomSAlgK

(K[G], A), A ∈ SAlgK . The Hopf superalgebra K[G] is calledcoordinate superalgebra of G. The category of (left) G-supermodules with even morphisms

1

is denoted by G − mod. The category G − mod is equivalent to the category of (right)K[G]-supercomodules with even morphisms [1, 5]. If V ∈ G−mod, then its coaction mapis denoted by ρV . From now on we use Sweedler’s notations : ρV (v) =

∑v1 ⊗ f2, v, v1 ∈

V, f2 ∈ K[G].In what follows HomG(V,W ) is a (superspace) of all (not necessary even) morphisms

between G-supermodules V and W . In particular, HomG−mod(V,W ) = HomG(V,W )0.In the same way, ExtiG(V, ?) = RiHomG(V, ?), i ≥ 0. Any ExtiG(V,W ) has a superspacestructure with ExtiG(V,W )ǫ = RiHomG(V,W )ǫ, ǫ = 0, 1. Moreover, ExtiG(V c,W ) ≃ExtiG(V,W c) = ExtiG(V,W )c, where V c is conjugated to V [5].

Let K[cij |1 ≤ i, j ≤ m + n] be a commutative superalgebra freely generated by theelements cij , where |cij | = 0 iff 1 ≤ i, j ≤ m or m + 1 ≤ i, j ≤ m + n, otherwise |cij | = 1.Denote the generic matrix (cij)1≤i,j≤m+n by C. Set

C00 = (cij)1≤i,j≤m, C01 = (cij)1≤i≤m,m+1≤j≤m+n,

C10 = (cij)m+1≤i≤m+n,1≤j≤n, C11 = (cij)m+1≤i≤m+n,m+1≤j≤m+n

and d1 = det(C00), d2 = det(C11). The general linear supergroup GL(m|n) is an alge-braic supergroup, whose coordinate (Hopf) superalgebra K[GL(m|n)] is isomorphic toK[cij |1 ≤ i, j ≤ m+n]d1,d2

. The comultiplication and the counit of K[GL(m|n)] are givenby δGL(m|n)(cij) =

∑

1≤t≤m+n cit ⊗ ctj and by ǫGL(m|n)(cij) = δij respectively. For thedefinition of the antipode see [5, 12].

Let C be a K-abelian and locally artinian Grothendieck category (see [3, 4] for defi-nitions). Assume that all simple objects in C are indexed by the elements of a partiallyordered poset (Λ,≤). Let L(λ) be a simple object and I(λ) be its injective envelope,λ ∈ Λ. The costandard object ∇(λ) is defined as largest subobject of I(λ) whose composi-tion quotients are L(µ), µ ≤ λ. For example, if C is a highest weight category in the senseof [3], then the costandard objects coincide with the first members of good filtrations ofinjective envelopes. Denote by Cf the full subcategory of C consisting of all finite objects.If all ∇(λ) belong to Cf and Cf has a duality τ , preserving simple objects, then one candefine standard objects in C by ∆(λ) = τ(∇(λ)), λ ∈ Λ. Any costandard object ∇(λ) isuniquely defined by the following universal property. If W ∈ C such that soc(W ) = L(λ)and all other composition quotients of W are L(µ), µ < λ, then W is isomorphic to a sub-object of ∇(λ). Symmetrically, if W/radW ≃ L(λ) and all other composition quotients ofradW are L(µ), µ < λ, then W is isomorphic to a factor of ∆(λ).

2 Hochschild-Serre spectral sequences

Let G be an affine supergroup and N is a normal supersubgroup of G. The dur K-sheaf˜̃

G/N is an affine supergroup (see Theorem 6.2, [6]). Moreover, K[˜̃

G/N ] ≃ K[G]N . To

simplify our notations we denote˜̃

G/N just by G/N . If V ∈ G − mod, then V N has anatural structure of a G/N -supermodule. More precisely, by Proposition 3.1 from [5], V isembedded (as a supercomodule) into a direct sum of several copies of K[G] and K[G]c. Itimplies that V N is the largest supersubmodule of G-supermodule V whose coefficient spacecf(V ) lies in K[G]N . The canonical epimorphism G → G/N is denoted by π. Followingnotations of [5] we have the restriction functor π0 : G/N −mod → G−mod. The proof ofthe following lemma can be copied from Lemma 6.4, II, [7].

2

Lemma 2.1 The functor V → V N is left exact and right adjoint to π0.

Proposition 2.1 (Proposition 6.6, II, [7]) For M ∈ G/N−mod, V, U ∈ G−mod,dim U <∞, we have the following spectral sequences :

1)En,m2 = ExtnG/N (M,ExtmN (U, V )) ⇒ Extn+m

G (M ⊗ U, V ).

2)En,m2 = ExtnG/N (M,Hm(N,V )) ⇒ Extn+m

G (M,V ).

3)En,m2 = Hn(G/N,Hm(N,V )) ⇒ Hn+m(G,V ).

Proof. The statements 2) and 3) are deduced from 1). To check it one has to set U = Kfor the statement 2) and then M = K for the statement 3). By Proposition 2.1, [5],HomN (U, V ) ≃ (U∗ ⊗ V )N and by Lemma 1.1

HomG/N (M, (U∗ ⊗ V )N ) ≃ HomG(M,U∗ ⊗ V ) ≃ HomG(M ⊗ U, V ).

In particular, the functor V → HomN (U, V ) is left exact and takes injective G-supermodulesto injective G/N -supermodules. Theorem 7, III, [8] completes the proof.

3 Adjoint and coadjoint actions

Let G be an affine supergroup. We have a right action of G × G on G by (g, (g1, g2)) 7→g−11 gg2, g, g1, g2 ∈ G(A), A ∈ SAlgK . Its dual morphism ρ : K[G] → K[G] ⊗ K[G]⊗2 is

defined asρ : f 7→

∑

(−1)|f1||f2|f2 ⊗ sG(f1) ⊗ f3, f ∈ K[G].

Compose with the diagonal inclusion G → G×G we obtain a right adjoint action of G onitself. Its dual morphism coincides with

νl : f 7→∑

(−1)|f1||f2|f2 ⊗ sG(f1)f3,

(see [6]).Let V be a L-supermodule and W be a H-supermodule, where L and H are affine

supergroups. The superspace V ⊗ W has natural structure of a L × H-supermodule by

ρV ⊗W (v ⊗ w) =∑

(−1)|w1||f2|v1 ⊗ w1 ⊗ f2 ⊗ h2,

where ρV (v) =∑

v1 ⊗ f2, ρW (w) =∑

w1 ⊗ h2. Let σ be a Hopf superalgebra anti-isomorphism of K[L] and τ be a Hopf superalgebra anti-isomorphism of K[H]. It is clearthat σ ⊗ τ is an anti-isomorphism of Hopf superalgebra K[L × H]. We have dualitiesV → V <σ>,W → W <τ> and U → U<σ⊗τ> of the categories L − smod,H − smod andL × H-smod correspondingly (see [5] for more definitions).

Lemma 3.1 In the above notations, if V and W are finite dimensional, then V <σ> ⊗W <τ> and (V ⊗W )<σ⊗τ> are canonically isomorphic as L×H-supermodules. In addition,if V ′ is a L-supermodule and W ′ is a H-supermodule, then (V ⊗ W ) ⊗ (V ′ ⊗ W ′) and(V ⊗ V ′)⊗ (W ⊗W ′) are canonically isomorphic as L×H-supermodules, where (V ⊗W )and (V ′ ⊗ W ′) are considered as L × H-supermodules, but (V ⊗ V ′) and (W ⊗ W ′) areconsidered as supermodules with respect to the diagonal action of L and H respectively.

3

Proof. Let v1, . . . , vs be a Z2-homogeneous basis of V and w1, . . . , wt be a Z2-homogeneousbasis of W . Denote by v∗1 . . . , v∗s and w∗

1 . . . , w∗t the dual homogeneous bases of V ∗ and W ∗

respectively. The required isomorphism is defined by v∗i ⊗w∗j 7→ (−1)|vi||wj|(vi ⊗wj)

∗, 1 ≤

i ≤ s, 1 ≤ j ≤ t. The second isomorphism is given by (v ⊗w)⊗ (v′ ⊗w′) 7→ (−1)|w||v′|(v ⊗v′) ⊗ (w ⊗ w′). We leave the routine checking to the reader.

Set G = GL(m|n). In what follows the anti-isomorphism of K[G], defined by cij 7→(−1)|i|(|j|+1)cji, 1 ≤ i, j ≤ m+n, is denoted by τ . The Borel supersubgroups of G consistingof lower (respectively, upper) triangular matrices is denoted by B− (respectively, by B+).The supergroup T = B−

⋂B+ is a maximal torus in G, called standard. The categroy

T -mod is semisimple. Any simple T -supermodule V is one dimensional and uniquelydefined by its character λ ∈ X(T ) = Zm+n, and by its parity ǫ = 0, 1. We denoteit by Kǫ

λ. The induced supermodules H0(G/B−,Kǫλ) and H0(G/B+,Kǫ

λ) are denoted byH0

−(λǫ) and H0+(λǫ) correspondingly. The Weyl supermodules H0

−(λǫ)<τ> and H0+(λǫ)<τ>

are denoted by V−(λǫ) and V+(λǫ) (all details can be found in [5]). All arguments of[5] can be symmetrically repeated for B+. In particular, the categroy G − mod is ahighest weight category with respect to the inverse dominant order on X(T ) (see Remark5.3, [5]). Precisely, the corresponding costandard and standard objects are H0

+(λǫ) andV+(λǫ) respectively, where ǫ = 0, 1, and λ ∈ X(T )−. Remind that H+(λ) = H0

+(λ0) =H0

+(λ1)c, V+(λ) = V+(λ0) = V+(λ1)c and

X(T )− = {λ = (λ1, . . . , λm+n) ∈ Zm+n|λ1 ≤ . . . ≤ λm, λm+1 ≤ . . . ≤ λm+n}.

The parity ǫ coincides with the parity of the one dimensional supersubspace H0+(λǫ)λ (re-

spectively, with the parity of the one dimensional supersubspace V+(λǫ)λ). The followinglemma is now obvious.

Lemma 3.2 For any λ ∈ X(T )+ and ǫ = 0, 1, we have natural isomorphisms H0−(λǫ)∗ ≃

V+(−λǫ) and V−(λǫ)∗ ≃ H0+(−λǫ).

It is clear that

V+(λǫ) ⊗ V−(µǫ′) ≃ V+(λπ) ⊗ V−(µ) ≃ V+(λ) ⊗ V−(µπ)

andH0

+(λǫ) ⊗ H0−(µǫ′) ≃ H0

+(λπ) ⊗ H0−(µ) ≃ H0

+(λ) ⊗ H0−(µπ),

where π = ǫ + ǫ′ (mod 2).

Proposition 3.1 For any λ1, µ1 ∈ X(T )−, λ2, µ2 ∈ X(T )+, ǫ, ǫ′ = 0, 1, a superspaceExtiG×G(V+(λǫ

1) ⊗ V−(λ2),H0+(µǫ′

1 ) ⊗ H0−(µ2)) is not equal to zero iff i = 0 and λ1 =

µ1, λ2 = µ2. In the last case HomG×G(V+(λǫ1) ⊗ V−(λ2),H

0+(µǫ′

1 ) ⊗ H0−(µ2)) is an one

dimensional superspace of parity ǫ + ǫ′ (mod 2).

Proof. We have the exact sequence of algebraic supergroups

1 → G → G × G → G → 1,

where the epimorphism G × G → G is dual to the monomorphism K[G] → K[G] ⊗ K[G]defined as f 7→ 1⊗f . The kernel of this epimorphism coincides with G×1 ≃ G. CombiningProposition 3.2, [5], with Lemma 3.1 we have

ExtiG×G(V+(λǫ1) ⊗ V−(λ2),H

0+(µǫ′

1 ) ⊗ H0−(µ2)) ≃

4

H i(G × G,V+(λǫ1)

∗ ⊗ V−(λ2)∗ ⊗ H0

+(µǫ′

1 ) ⊗ H0−(µ2)) ≃

H i(G × G,V+(λǫ1)

∗ ⊗ H0+(µǫ′

1 ) ⊗ V−(λ2)∗ ⊗ H0

−(µ2)).

On the other hand,

H i(G × 1, V+(λǫ1)

∗ ⊗ H0+(µǫ′

1 ) ⊗ V−(λ2)∗ ⊗ H0

−(µ2)) ≃

ExtiG(V+(λǫ1),H

0+(µǫ′

1 )) ⊗ (V−(λ2)∗ ⊗ H0

−(µ2)) = 0,

provided i > 0 (see [5], Theorem 5.5). Proposition 2.1 and the standard spectral sequencearguments infer that

ExtiG×G(V+(λǫ1) ⊗ V−(λ2),H

0+(µǫ′

1 ) ⊗ H0−(µ2)) ≃

HomG(V+(λǫ1),H

0+(µǫ′

1 )) ⊗ ExtiG(V−(λ2),H0−(µ2)),

where in the second tensor multiplier G is identified with 1×G. It remains to refer to theuniversal properties of standard/costandard objects.

Let G1, G2 be affine supergroups and let H1 ≤ G1,H2 ≤ G2 be their (closed) super-subgroups.

Lemma 3.3 (see Lemma 3.8, [7], part I) If Vi is a Hi-supermodule, i = 1, 2, then there isa canonical isomorphism indG1×G2

H1×H2V1⊗V2 ≃ indG1

H1V1⊗ indG2

H2V2 of G1×G2-supermodules.

Proof. The isomorphism is induced by the map V1 ⊗V2 ⊗K[G1]⊗K[G2] → V1 ⊗K[G1]⊗V2⊗K[G2] defined as v1⊗v2⊗f1⊗f2 7→ (−1)|v2||f1|v1⊗f1⊗v2⊗f2. In fact, by Proposition3.3, [5], and Lemma 3.1

indG1×G2

H1×H2V1 ⊗ V2 ≃ (V1 ⊗ V2 ⊗ (K[G1] ⊗ K[G2])l)

H1×H2 ≃

((V1 ⊗ K[G1]l)H1 ⊗ (V2 ⊗ K[G2]l)

H2 ≃ indG1

H1V1 ⊗ indG2

H2V2.

Here, for any affine supergroup L, K[L] = K[L]l is a L-supermodule via

f 7→∑

(−1)|f1||f2|f2 ⊗ sL(f1), δL(f) =∑

f1 ⊗ f2.

The character group X(T×T ) is identified with X(T )×X(T ). It is partially ordered by(λ, λ′) ≤ (µ, µ′) iff λ ≥ λ′, µ ≤ µ′. This ordering corresponds to the Borel supersubgroupB+ ×B− of G×G. The epimorphism K[B+ ×B−] → K[T ×T ] is split and K[T ×T ] canbe canonically identified with a Hopf supersubalgebra of K[B+ × B−] generated by thegroup-like elements c±1

ii ⊗ c±1jj , 1 ≤ i, j ≤ m + n. Denote the kernel of this epimorphism

by J . Let V be a B+ × B−-supermodule. Take a weight (λ, λ′) ∈ X(T × T ) and aZ2-homogeneous vector v ∈ V(λ,λ′). As in Proposition 5.3, [5], we have

ρV (v) = v ⊗ c(λ,λ′) + y, y ∈ (∑

(µ,µ′)<(λ,λ′)

V(µ,µ′)) ⊗ J,

where c(λ,λ′) =∏

1≤i≤m+n cλi

ii ⊗∏

1≤i≤m+n cλ′

i

ii . In particular, any simple B+ × B−-supermodule is one dimensional and isomorphic to Kǫ

(λ,λ′) ≃ Kǫλ ⊗ Kλ′ . Besides, if a

B+ × B−-supermodule is generated by a (Z2-homogeneous) vector v of weight (λ, λ′),

5

then V(µ,µ′) 6= 0 implies (µ, µ′) ≤ (λ, λ′) and dimV(λ,λ′) = 1. Using Proposition 5.4, [5],one can easily check that the morphism of superalgebras

K[G × G](π+−⊗π−+)δG×G

−→ K[B+ × B−] ⊗ K[B− × B+],

is an inclusion, where π+− : K[G] → K[B+ × B−] and π−+ : K[G] → K[B− × B+] arecanoical epimorphisms. Now, everything is prepared to prove the following lemmas.

Lemma 3.4 The supermodule H0+(λǫ) ⊗ H0

−(λ′) runs over all costandard objects in thecategory of G×G-supermodules, whenever (λ, λ′) runs over X(T )−×X(T )+ and ǫ = 0, 1.Besides, the simple socle of H0

+(λǫ) ⊗ H0−(λ′) is isomorphic to L(−λǫ) ⊗ L(λ′).

Proof. Word-by-word repetition of the proofs of Propositions 5.5 and 5.6, [5].

Lemma 3.5 The supermodule V+(λǫ) ⊗ V−(λ′) runs over all standard objects in the cat-egory of G × G-supermodules, whenever (λ, λ′) runs over X(T )− × X(T )+ and ǫ = 0, 1.

Proof. It obviously follows by Lemma 3.1.As in [5] a G×G-supermodule V is called restricted iff for any (λ, λ′) ∈ X(T )−×X(T )+,

we have dimHomG×G(V+(λ) ⊗ V−(λ′), V ) < ∞ and the set

V̂ = {(λ, λ′) ∈ X(T )− × X(T )+|HomG×G(V+(λ) ⊗ V−(λ′), V ) 6= 0}

does not contain infinite decreasing chains.

Lemma 3.6 A restricted G×G-supermodule V has a costandard (or good) filtration withquotients H0

+(λǫ) ⊗ H0−(λ′) iff for all (λ, λ′) ∈ X(T )− × X(T )+ one of the following

equivalent conditions hold :

1. Ext1G×G(V+(λ) ⊗ V−(λ′), V ) = 0;

2. ExtiG×G(V+(λ) ⊗ V−(λ′), V ) = 0 for any i ≥ 1.

If V has a good filtration, then the multiplicity of a factor H0+(λǫ) ⊗ H0

−(λ′) is equal todim HomG×G(V+(λǫ) ⊗ V−(λ′), V )0 (see Remark 5.5 in [5]).

Corollary 3.1 Any injective G×G-supermodule has a good filtration with quotients H0+(λǫ)⊗

H0−(λ′). In particular, G×G-mod is a highest weight category with poset (X(T )−×{0, 1})×

X(T )+ ordered as above.

Remark 3.1 Let δ = (δ1, . . . , δt) be an element of the set {+,−}t. Set Gt = G × . . . × G︸ ︷︷ ︸

t

.

Mimic the above arguments one can prove that Gt-mod is a highest weight category withposet (X(T )δ1×{0, 1})×. . .×X(T )δt , whose costandard and standard objects are H0

δ1(λǫ

1)⊗. . . ⊗ H0

δt(λt) and Vδ1(λ

ǫ1) ⊗ . . . ⊗ Vδt

(λt) respectively.

Proposition 3.2 (see [7], Part II, Proposition 4.20) The superalgebra K[G], consideredas a G × G-supermodule via ρ, satisfies the following conditions:

1. ExtiG×G(V+(λ) ⊗ V−(λ′),K[G]) = 0 for all i ≥ 1, (λ, λ′) ∈ X(T )− × X(T )+;

6

2. HomG×G(V+(λǫ) ⊗ V−(λ′),K[G])0 6= 0 iff ǫ = 0 and λ = −λ′.

Besides, dim HomG×G(V+(λ) ⊗ V−(−λ),K[G])0 = 1.

Proof. By Proposition 2.1 we have a spectral sequence

Hn(1×G,Hm(G×1,H0−(−λǫ)⊗H0

+(−λ′)⊗K[G]) ⇒ Hm+n(G×G,H0−(−λǫ)⊗H0

+(−λ′)⊗K[G]).

On the left side K[G] is isomorphic to K[G]l as a G = G × 1-supermodule. As K[G]l ≃K[G] is injective (see [5]), this sequence degenerates and yields isomorphism

Hn(1 × G, (H0−(−λǫ) ⊗ K[G]l)

G ⊗ H0+(−λ′)) ≃ ExtnG×G(V+(λǫ) ⊗ V−(λ′),K[G]).

Finally, (H0−(−λǫ) ⊗ K[G]l)

G ≃ IndGGH0

−(−λǫ) = H0−(−λǫ) and the space on left is iso-

morphic toHn(G,H0

−(−λǫ) ⊗ H0+(−λ′)) ≃ ExtnG(V+(λǫ),H0

+(−λ′)).

Thus ExtnG×G(V+(λǫ) ⊗ V−(λ′),K[G]) 6= 0 iff n = 0, λ = −λ′. Besides, in the last caseHomG×G(V+(λǫ) ⊗ V−(−λ),K[G]) is one dimensional and even iff ǫ = 0.

4 Few results about filtrations

Let C be a highest weight categroy from Section 1, with a duality τ . We call τ a Chevalleyduality. In what follows we assume that all costandard objects are finite and Schurian,that is EndC(∇(λ)) = K for any λ ∈ Λ. For a weight λ ∈ Λ we denote by (λ] the (possibleinfinite) interval {µ ∈ Λ|µ ≤ λ}. The open interval {µ|µ < λ} is denoted by (λ). LetΓ ⊆ Λ and M ∈ C. We say that M belongs to Γ iff all composition factors of M areL(λ) with λ ∈ Γ. Any N ∈ C contains the largest subobject that belongs to Γ. Wedenote it by OΓ(N). Symmetrically, N contains a unique minimal subobject OΓ(N) suchthat N/OΓ(N) belongs to Γ. For example, ∇(λ) = O(λ](I(λ)), λ ∈ Λ. Finally, denote by[M : L(λ)] the supremum of multiplicities of a simple object L(λ) in composition series ofall finite subobjects of M .

The full subcategory consisting of all objects M with OΓ(M) = M is denoted by C[Γ].It is obvious that OΓ is a left exact functor from C to C[Γ] and it commutes with directsums. The functor OΓ : C → C also commutes with direct sums, but it is not right exactin general. In fact, for any exact sequence

0 → X → Y → Z → 0

we see that OΓ(Y ) → OΓ(Z) is an epimorphism and OΓ(X) ⊆ X⋂

OΓ(Y ). Nevertheless,it is possible that OΓ(X) is a proper subobject of X

⋂OΓ(Y ).

A subset Γ ⊆ Λ is called ideal, if µ ≤ λ implies µ ∈ Γ, provided λ ∈ Γ. IfΓ =

⋃

1≤k≤s(λk] , we say that Γ is finitely generated (by the elements λ1, . . . , λs). Wesuppose that any finitely generated ideal Γ ⊆ Λ has a decreasing chain of finitely gen-erated subideals Γ = Γ0 ⊇ Γ1 ⊇ Γ2 ⊇ . . ., such that Γ \ Γk is a finite set for all k ≥ 0and

⋂

k≥0 Γk = ∅. In particular, if Γ ⊆ Λ is a finitely generated ideal, then Γ is at mostcountable. From now on Γ is a finitely generated ideal, unless otherwise stated.

7

Example 4.1 The categroy G − mod satisfies all the above conditions. In fact, any λ ∈X(T )+ has finitely many predecessors µ < λ such that there is not π ∈ X(T )+ betweenλ and µ. More precisely, if µ is a such predecessor and

∑

1≤i≤m µi <∑

1≤i≤m λi, thenµ < λ′ < λ, where λ′ = (λ1, . . . , λm−1, λm − 1|λm+1 + 1, λm+2, . . . , λm+n). It implies thateither µ = λ′ or µ+ ≤ λ+, µ− ≤ λ−. It remains to notice that for any (ordered) partitionπ there are only finitely many partitions of the same length, less or equal π. Repeatingthese arguments as many times as we need, one can prove that Gt-mod satisfies all theabove conditions for any root data (X(T )δ1 × {0, 1}) × . . . × X(T )δt .

The subcategory C[Γ] is a highest weight category with costandard objects ∇(λ) andfinite injective envelopes IΓ(λ) = OΓ(I(λ)), λ ∈ Γ [3]. By Theorem 3.9 from [3] we haveExtiC(M,N) = ExtiC[Γ](M,N), for any M,N ∈ C[Γ].

Let R be a class of objects from C. An increasing filtration

0 = M0 ⊆ M1 ⊆ M2 ⊆ . . .

of an object M such that Mi/Mi−1 ∈ R, i ≥ 1, and⋃

i≥1 Mi = M , is called increasingR-filtration. If M has decreasing filtration

M = M0 ⊇ M1 ⊇ M2 ⊇ . . .

such that Mi/Mi+1 ∈ R, i ≥ 0, and⋂

i≥0 Mi = 0, then we call it decreasing R-filtration.For example, any injective envelope I(λ) has an increasing ∇-filtration, where ∇ ={∇(λ)|λ ∈ Λ}. To complete our notations, let us denote the class {∆(λ)|λ ∈ Λ} by∆. Remind that L(λ) = ∆(λ)/rad∆(λ) and all other composition factors L(µ) of ∆(λ)satisfy µ < λ.

Lemma 4.1 If M belongs to Γ, then ExtiC(M,∇(λ)) 6= 0, for i > 0, infers that there is acomposition factor L(µ) of M such that µ > λ.

Proof. Without loss of generality one can suppose that λ ∈ Γ. Consider the short exactsequence

0 → ∇(λ) → IΓ(λ) → Q → 0,

where Q has a ∇-filtration with quotients ∇(µ), µ > λ. The fragment of long exactsequence

. . . → Exti−1C (M,Q) → ExtiC(M,∇(λ)) → 0

shows that Exti−1C (M,Q) 6= 0. The induction on i implies that HomC(M,∇(µ)) 6= 0 for

some µ > λ.

Lemma 4.2 The category C[Γ] has enough projectives.

Proof. Denote τ(IΓ(λ)) by PΓ(λ), λ ∈ Γ. It is an easy exercise to prove that PΓ(λ) is aprojective cover of L(λ). We leave it for the reader.

It is obvious that any PΓ(λ) has the ∆-filtration which is Chevalley dual to the cor-responding ∇-filtration of IΓ(λ), λ ∈ Γ. The following lemma is the symmetric variant ofLemma 4.1.

Lemma 4.3 Let M ∈ C[Γ]. If ExtiC(∆(λ),M) 6= 0, for i > 0, then there is compositionfactor L(µ) of M such that µ > λ.

8

Corollary 4.1 (compare with Theorem 3.11 from [3]) For any λ, µ ∈ Λ and i > 0, wehave ExtiC(∆(λ),∇(µ)) = 0.

Corollary 4.2 Let M ∈ C. If Ext1C(∆(λ),M) = 0 (respectively, Ext1C(M,∇(λ)) = 0) forall λ ∈ Λ, then the same still holds for OΓ(M) (respectively, for M/OΓ(M)).

Proof. If Ext1C(∆(λ), OΓ(M)) 6= 0, then λ ∈ Γ. It remains to consider the followingfragment of long exact sequence

HomC(∆(λ),M/OΓ(M)) → Ext1C(∆(λ), OΓ(M)) → 0

and notice that HomC(∆(λ),M/OΓ(M)) = 0.An object M ∈ C[Γ] is called Γ-restricted iff [M : L(λ)] is finite for any λ ∈ Γ. If

Γ =⋃

1≤j≤k(πj ] and λ ∈ Γ, we denote by (λ,Γ] (respectively, by [λ,Γ]) the finite set⋃

1≤j≤k(λ, πj ] (respectively, the finite set⋃

1≤j≤k[λ, πj ]). The following theorem general-izes Corollary 4.1.

Theorem 4.1 Assume that a Γ-restricted object M has an increasing (decreasing) ∆-filtration. Then ExtiC(M,∇(λ)) = 0 for all λ ∈ Λ and i ≥ 1. Moreover, if we assume thatM has an increasing (decreasing) ∇-filtration, then ExtiC(∆(λ),M) = 0 for all λ ∈ Λ andi ≥ 1.

Proof. We consider the case of decreasing ∆-filtration only, all other cases are similar.Without loss of generality, one can assume that λ ∈ Γ. As the set A = (λ,Γ] is finite,there is a finite subobject N ⊆ M such that [N : L(ν)] = [M : L(ν)] for any ν ∈ A.For sufficiently large k it holds that N

⋂Mk = 0, where Mk is the k-th member of the

corresponding decreasing ∆-filtration of M . In particular, [Mk : L(ν)] = 0 for ν ∈ A. Wehave the fragment of long exact sequence

. . . → ExtiC(M/Mk,∇(λ)) → ExtiC(M,∇(λ)) → ExtiC(Mk,∇(λ)) → . . .

It remains to notice that ExtiC(M/Mk,∇(λ)) = 0 by Corollary 4.1 and ExtiC(Mk,∇(λ)) =0 by Lemma 4.1.

Lemma 4.4 Let M be a Γ-restricted object such that Ext1C(∆(λ),M) = 0 (respectively,Ext1C(M,∇(λ)) = 0) for all λ ∈ Λ. Then for all λ ∈ Λ and i > 1 we have ExtiC(∆(λ),M) =0 (respectively, ExtiC(M,∇(λ)) = 0).

Proof. As in Theorem 4.1 any increasing chain π < π1 < π2 < . . . in Γ has cardinality atmost |[π,Γ]|. Again, one can assume that λ ∈ Γ. We work in the category C[Γ]. The shortexact sequence

0 → Q → PΓ(λ) → ∆(λ) → 0

induces. . . → Exti−1

C (Q,M) → ExtiC(∆(λ),M) → 0.

Since Q has a ∆-filtration with factors ∆(µ), where µ > λ, one can argue by induction oni and the partial order. The proof of the second statement is similar.

Theorem 4.2 Let M be a Γ-restricted object and Ext1C(∆(λ),M) = 0 for all λ ∈ Λ.Then M has a decreasing ∇-filtration. The symmetrical statement is also true, that is ifExt1C(M,∇(λ)) = 0 for all λ ∈ Λ, then M has an increasing ∆-filtration.

9

Proof. Suppose that Ext1C(∆(λ),M) = 0 for all λ ∈ Λ. By our assumption there is adecreasing chain of finitely generated ideals

Γ = Γ0 ⊇ Γ1 ⊇ . . .

such that Γ \ Γk is finite for any k ≥ 0 and⋂

k≥0 Γk = ∅. For the sake of shortness wedenote OΓk

(M) just by Mk. We have decreasing chain of subobjects

M = M0 ⊇ M1 ⊇ M2 ⊇ . . . ,

where any quotient M/Mk is finite and⋂

1≤k Mk = 0. In fact, the socle of any M/Mk

belongs to Γ \ Γk and therefore, it is finite. Thus M/Mk can be embedded into a finitesum of finite indecomposable injectives from C[Γ]. Consider the following fragment of longexact sequence

HomC(∆(µ),M/Mk) → Ext1C(∆(µ),Mk) → 0 →

→ Ext1C(∆(µ),M/Mk) → Ext2C(∆(µ),Mk).

If Ext1C(∆(µ),Mk) 6= 0, then µ ∈ Γk. On the other hand, the socle of M/Mk does notbelong to Γk, that is HomC(∆(µ),M/Mk) = 0. In particular, Ext1C(∆(µ),Mk) = 0 for anyµ. By Lemma 4.4 Ext2C(∆(µ),Mk) = 0 for any µ also and therefore, Ext1C(∆(µ),M/Mk) =0. Since Mk−1/Mk = OΓk−1

(M/Mk), one can repeat the above arguments to obtainthat Ext1C(∆(µ),Mk−1/Mk) = 0 for all µ. Finally, any object Mk−1/Mk is finite and weconclude the proof by the standard arguments from [7, 9].

For the second statement it is enough to prove that all subobjects OΓk(M) are finite.In fact, OΓk(M) contains a finite subobject N such that [N : L(µ)] = [OΓk(M) : L(µ)] forall µ ∈ Γ \ Γk. In particular, OΓk(M)/N belongs to Γk, that is N = OΓk(M). The finalarguing is the same as above.

Corollary 4.3 Assume that M is Γ-restricted and we have an exact sequence

0 → N → M → S → 0.

Theorems 4.1 and 4.2 imply

1. If both M and N have decreasing ∇-filtrations, then S has a decreasing ∇-filtration.

2. If both M and S have increasing ∆-filtrations, then N has an increasing ∆-filtration.

3. If M has a decreasing (respectively, increasing) ∇ (respectively, ∆)-filtration thenany its direct summand has a decreasing (respectively, increasing) ∇ (respectively,∆)-filtration.

4. If M has a decreasing ∇-filtration, then an object ∇(λ) appears exactly (M : ∇(λ)) =dimHomC(∆(λ),M) times as a factor of it. Besides, (M : ∇(λ)) = (N : ∇(λ))+(S :∇(λ)), provided N has a decreasing ∇-filtration.

5. If M has an increasing ∆-filtration, then an object ∆(λ) appears exactly (M :∆(λ)) = dim HomC(M,∇(λ)) times as a factor of it. Moreover, (M : ∆(λ)) =(N : ∆(λ)) + (S : ∆(λ)), provided S has an increasing ∆-filtration.

10

5 Donkin-Koppinen filtration and coadjoint action invari-

ants

Lemma 5.1 The superalgebra K[G] is a Λ-restricted G × G-supermodule, where Λ =(X(T )− × {0, 1}) × X(T )+.

Proof. It is enough to prove that OΓ(K[G]) is Γ-restricted for any (finitely) generatedideal Γ. By Corollary 4.2, Theorem 4.2, Corollary 4.3 and Proposition 3.2 we have

[OΓ(K[G]) : L(−λǫ) ⊗ L(λ′)] = dim HomG×G(PΓ(−λǫ, λ′), OΓ(K[G])) ≤

∑

(µǫ,µ′)∈[(λǫ,λ′),Γ]

dim HomG×G(V+(µǫ) ⊗ V−(µ′), OΓ(K[G])) ≤|[(λǫ, λ′),Γ]|

2.

The following theorem is now obvious.

Theorem 5.1 For any Γ the G × G-supersubmodule OΓ(K[G]) has an decreasing (good)filtration OΓ(K[G]) = V0 ⊇ V1 ⊇ V2 ⊇ . . . , such that Vk/Vk+1 ≃ V−(λk)

∗ ⊗H0−(λk), k ≥ 0.

Moreover, for any pair of indexes k < l we have either λk > λl or these weights are notcomparable each to other.

Remark 5.1 By Proposition 3.2 a supermodule V−(λ)∗ ⊗ H0−(λ) appears as a factor of

some good filtration of OΓ(K[G]) iff (−λ, λ) ∈ Γ and (OΓ(K[G]) : V−(λ)∗ ⊗ H0−(λ)) = 1.

For the obvious reason the above filtration can be called Donkin-Koppinen filtration (see[10] and also notice in the end of this article).

Consider the coadjoint action νl of G on K[G]. Define supersubalgebra R = K[G]G ofcoadjoint (rational) invariants. By definition, R = {f ∈ K[G]|νl(f) = f ⊗1}. As νl is dualto the adjoint action of G on itself, these invariants can be also called adjoint. Moreover,an rational function f ∈ K[G] belongs to R iff for any A ∈ SAlgK , g1, g2 ∈ G(A) we havef(g−1

1 g2g1) = f(g1). In other words, invariants from R are absolute ones.The subalgebra of R, generated by polynomial invariants, is denoted by Rpol.Let H be an algebraic supergroup and V ∈ H − mod,dim V < ∞. Fix a Z2-

homogeneous basis of V , say v1, . . . , vp, vp+1, . . . , vp+q, where |vi| = 0 iff 1 ≤ i ≤ p,otherwise |vi| = 1. Set ρV (vi) =

∑

1≤j≤p+q vj ⊗ fji, 1 ≤ i ≤ p + q. Denote by Tr(ρ) (or byTr(V ), see [11]) the supertrace

∑

1≤i≤p fii −∑

p+1≤i≤p+q fii. It is well known that Tr(ρ)dos not depend on the choice of Z2-homogeneous basis of V [11, 12, 13].

Lemma 5.2 The supertrace Tr(ρ) belongs to the superalgebra of (co)adjoint invariantsK[H]H .

Proof. Denote by S the Hopf supersubalgebra of K[H], generated by the elements fij. Wehave the natural Hopf superalgebra epimorphism K[GL(p|q)] → S, induced by the mapcij 7→ fij. It remains to refer to [12].

Consider K[G] as a T -supermodule. Notice that this action is a restriction of thecoaction ρ on the (super)subgroup 1 × T ⊆ T × T . Remind that a G-supermodule V issemisimple as a T -supermodule and V =

⊕

λ∈X(T ) Vλ, where Vλ is a direct sum of onedimensional T -supermodules of weight λ. In particular, any element f ∈ K[G] can be

11

represented as a sum∑

λ∈X(T ) fλ, where fλ ∈ K[G]λ. A non-zero summand fλ of f iscalled leading if λ is maximal among all µ with fµ 6= 0. The corresponding weight λ isalso called leading.

Lemma 5.3 Let V be a G-supermodule. Choose a Z2-homogeneous basis v1, . . . , vt of Vsuch that vi ∈ Vλ(i) (it is possible that λ(i) = λ(j) for i 6= j). Set ρV (vi) =

∑

1≤j≤t vj ⊗cji, 1 ≤ i ≤ t. Then cji ∈ K[G]λ(i), 1 ≤ i, j ≤ t.

Proof. The map ρV : V → Vtriv ⊗ K[G]|T is a morphism of T -supermodules, where Vtriv

is considered as a trivial T -supermodule.

Corollary 5.1 If some λ = λ(i) is maximal (largest), then Tr(V )λ is a leading (respec-tively, a unique leading) summand of Tr(V ).

Let E be a standard G-supermodule with Z2-homogeneous basis e1, . . . , em+n, where|ei| = 0 iff 1 ≤ i ≤ m, otherwise |ei| = 1, and such that ρE(ei) =

∑

1≤j≤m+n ej ⊗ cji, 1 ≤i ≤ m + n. Denote by Ber(E) the one dimensional G-supermodule corresponding to thegroup-like element (berezinian) Ber((cij)) = det(C00 −C01C

−111 C10) det(C11)

−1. It is clearthat Ber(E) ∈ K[G]θ, where θ = (1m|(−1)n) = (1, . . . , 1

︸ ︷︷ ︸

m

| −1, . . . ,−1︸ ︷︷ ︸

n

).

Let I(r) be a set of all maps I : r → m + n. One can consider any I ∈ I(r) as a multi-index (i1, . . . , ir), where ik = I(k), 1 ≤ k ≤ r. For I, J ∈ I(r) we set xIJ = (−1)s(I,J)cIJ ,where cIJ =

∏

1≤k≤r cikjk, s(I, J) =

∑

t |it|(∑

s<t |is| + |js|) and |i| = |ei|. The superspaceE⊗r has a basis eI = ei1 ⊗ . . . ⊗ eir , I ∈ I(r). It is known that E⊗r has the naturalstructure of G-supermodule, as well as exterior and symmetric powers Λr(E) and Sr(E)respectively [15].

Lemma 5.4 The structure of G-supermodule on E⊗r is given by

ρE⊗r(eI) =∑

J∈I(r)

eJ ⊗ xJI .

Proof. Straightforward calculations.Denote by LI(r) the subsets of I(r), consisting of all multi-indexes I such that for

some k ≤ r we have i1 < . . . < ik ≤ m < jk+1 ≤ . . . ≤ jr. Analogously, denote by SI(r)the subsets of I(r), consisting of all multi-indexes I such that for some k ≤ r we havei1 ≤ . . . ≤ ik ≤ m < jk+1 < . . . < jr. Let π1 : E⊗r → Λr(E) and π2 : E⊗r → Sr(E) arecanonical epimorphisms. Then π1(eI) (π2(eI)) form a basis of Λr(E) (respectively, basisof Sr(E)), when I runs over LI(r) (respectively, over SI(r)). Remind that the symmetricgroup Sr acts on E⊗r as eIσ = (−1)s(I,σ)eIσ, where

s(I, σ) = |{(k, l)|1 ≤ k < l ≤ r, σ(k) > σ(l), |iσ(k)| = |iσ(l)| = 1}|,

see [16]. If we replace E by Ec but preserve our notations, then this action turns intoeIσ = (−1)s

′(I,σ)eIσ, where

s′(I, σ) = |{(k, l)|1 ≤ k < l ≤ r, σ(k) > σ(l), |iσ(k)| = |iσ(l)| = 0}|.

As in [16] one can define two actions of Sr on the set of monomials xIJ . Precisely,xIJ⋆σ = (−1)s(J,σ)xIJσ (xI⋆σ,J = (−1)s(I,σ)xIσ,J) and xIJ◦σ = (−1)s

′(J,σ)xIJσ (xI◦σ,J =(−1)s

′(I,σ)xIσ,J). It can be easily checked that xI⋆σ,J⋆σ = xIJ and xI◦σ,J◦σ = xIJ .

12

Lemma 5.5 For any r ≥ 0 we have

Tr(Λr(E)) =∑

I∈LI(r)

(−1)|I|∑

σ∈Stab(I)\Sr

xI◦σ,I ,

andTr(Sr(E)) =

∑

I∈SI(r)

(−1)|I|∑

σ∈Stab(I)\Sr

xI⋆σ,I ,

where |I| =∑

1≤k≤m+n |ik| = |eI |.

Proof. We consider only the first equality, the second one is symmetrical. It is clear thatfor any I ∈ LI(r) the vector π1(eI) appears in ρ(π1(eI)) as follows :

∑

σ∈Stab(I)\Sr

π1(eIσ) ⊗ xIσ,I .

It remains to notice that π1(eIσ) = (−1)s′(I,σ)π1(eI).

Corollary 5.2 An invariant Cr = Tr(Λr(E)) has the unique leading summand of weight(1r, 0m−r|0n) iff r ≤ m, otherwise this summand has weight (1m|r − m, 0n−1).

Consider the dual G-supermodule E∗. By definition, ρ(e∗i ) =∑

1≤j≤m+n e∗j ⊗c∗ji, where

e∗i (ek) = δik, c∗ji = (−1)|j|(|i|+|j|)sG(cij), 1 ≤ i, k ≤ m + n [5]. Denote Tr(Sr(E∗)) by Dr.

Corollary 5.3 An invariant Dr has the unique leading summand of weight (0m|0n−r, (−1)r)iff r ≤ n, otherwise this summand has weight (0m−1, n − r|(−1)n).

Proof. Any e∗i has weight (0, . . . , 0, −1︸︷︷︸

i−th place

, 0, . . . , 0). Since the lowest weight, appearing

in Sr(E), is (0m|0n−r, 1r) in the case r ≤ n, otherwise it is (0m−1, r − n|1n), we are doneby Corollary 5.1.

Let f be an invariant from R. Denote by M the G×G-supersubmodule generated byf . Since M is finite dimensional, the ideal

Γ =⋃

(µǫ,µ′),[M :L(−µǫ)⊗L(µ′)] 6=0

((µǫ, µ′)]

is finitely generated. Fix a Donkin-Koppinen filtration of OΓ(K[G]), say OΓ(K[G]) =V0 ⊇ V1 ⊇ V2 ⊇ . . . , as in Theorem 5.1.

Lemma 5.6 For any t ≥ 0 the superspace (Vt/Vt+1)G is even and one dimensional. More-

over, a non-zero basic vector from (Vt/Vt+1)G has the unique leading summand of weight

λt.

Proof. A non-zero basic vector g of (Vt/Vt+1)G ≃ HomG(V−(λt),H

0−(λt)) can be repre-

sented as a sum∑

µ≤λt

∑

1≤i≤kµφi,µ⊗vi,µ, where φi,µ ∈ V−(λt)

∗ and vi,µ runs over a basis

of H0−(λt)µ. Besides, gµ =

∑

1≤i≤kµφi,µ ⊗ vi,µ with respect to the right T -action, kλt

= 1

and φ1,λt(V−(λt)) 6= 0.

13

Corollary 5.4 The superalgebra R is pure even.

Let λ = (λ1, . . . , λm|λm+1, . . . , λm+n) ∈ X(T )+. As in [5] we denote (λ1, . . . , λm) and(λm+1, . . . , λm+n) by λ+ and λ− correspondingly. The invariant

fλ = Ber(E)λm+n

∏

1≤s≤n−1

(Ber(E)Ds)λm+n−s−λm+n−s+1C |λ−|+λm

m

∏

1≤t≤m−1

Cλt−λt+1

t ,

has the unique leading summand of weight λ.If k is sufficiently large, then M

⋂Vk+1 = 0. In particular, all weights of f are among

the weights of⊕

0≤t≤k Vt/Vt+1. Assume that f ∈ Vt \ Vt+1, t ≤ k. By Lemma 5.5 f has aleading summand of weight λt. By Corollary 4.2 and Remark 5.1 the invariant f∗ = fλt

also belongs to Vt \ Vt+1. It infers that the leading summand of fλtcoincides with the

above leading summand of f up to a non-zero scalar. We call f∗ = fλta companion of f .

Denote by A the free Laurent polynomial algebra K[x±11 , . . . , x±1

m , y±11 , . . . , y±1

n ]. Wehave epimorphism of (super)algebras φ : K[G] → A, defined by cij 7→ δijxi if i ≤ m,otherwise cij 7→ δijyi. The algebra A can be obviously identified with K[T ], that is Ahas an obvious T -supermodule structure. Moreover, for any λ ∈ X(T ) the epimorphismφ takes K[G]λ to Aλ.

Theorem 5.2 (Chevalley’s restriction theorem) The restriction of φ on R is a monomor-phism.

Proof. Let f ∈ R \ 0. It is sufficient to prove that the image of a leading term of f isnot zero. In particular, if the image of the leading summand of its companion is not zero,then we are done. Since A is an integral domain, it remains to notice that the images ofthe leading summands of C1, . . . , Cm,D1, . . . ,Dn−1 and Ber(E) are non-zero and use theabove product representation of f∗.

Denote the images of Cr and Dr in A by cr and dr respectively. It is clear that

cr(x1, . . . , xm, y1, . . . , yn) = cr(x|y) =∑

0≤i≤min{r,m}

(−1)r−iσi(x)pr−i(y),

where σi(x) and pj(y) are elementary and complete symmetric functions correspondingly.The subalgebra of A consisiting of polynomials f(x|y) = f(x1, . . . , xm, y1, . . . , yn), sym-

metric in the x and y separately and such that, ddt(f |x1=y1=t) = 0, is denoted by As. If

charK = 0, then they have already been considered in [13] and in [17] (as pseudosym-metric and supersymmetric polynomials respectively). Moreover, it was proved that thesubalgebra As is generated by the elements cr. Since φ(Rpol) ⊆ As (see [13]) it impliesthat φ(Rpol) = As and Rpol is generated by the elements Cr. If charK = p > 0 it is stillopen question : what are the generators of Rpol?

Let V = {V } be a collection of polynomial G-supermodules. It is called good iff forany λ ∈ X(T )+ there is V ∈ V such that λ is the highest weight of V . For example,the collection of all simple polynomial G-supermodules is good. As in [5] denote thelargest polynomial supersubmodule of H0

−(λ) by ∇(λ). Remind that ∇(λ) 6= 0 iff L(λ) ispolynomial iff λ ∈ X(T )++. The subset X(T )++ ⊆ X(T )+ is completely described in [2].The collection {∇(λ)}λ∈X(T )++ is also good.

14

Lemma 5.7 Denote by X(T )+≥0 the set {λ ∈ X(T )+|λm, λm+n ≥ 0} (it does not imply

λ ∈ X(T )++). Then for any µ ∈ X(T )+≥0 the set (µ]⋂

X(T )+≥0 is finite.

Proof. Use the arguments from Example 4.1 and induction on |µ+|.

Theorem 5.3 The algebra Rpol is generated (as a vector space) by Tr(V ), where V runsover a good collection of G-supermodules.

Proof. Consider f ∈ Rpol. As above, M is a G × G-supersubmodule generated by f andM ⊆ OΓ(K[G]) for a finitely generated ideal Γ. Besides, M ⊆ Vt, t ≤ k and M

⋂Vk+1 = 0

for a Donkin-Koppinen filtration {Vi}i≥0 of OΓ(K[G]). The ideal generated by (finitelymany) µ with fµ 6= 0 denote by Γ′. It is obvious that Γ′ is also generated by leadingweights of f . Since M |1×G contains a polynomial G-supersubmodule (generated by f), inthe quotient Vt/Vt+1 ≃ V−(λt)

∗ ⊗ H0−(λt) the right hand side factor H0

−(λt) has non-zeropolynomial part. In other words, λt ∈ X(T )++ and there is V ∈ V whose highest weightis λt. Again, Tr(V ) ∈ Vt \ Vt+1 and for a non-zero scalar a all weights of polynomialinvariant f − aTr(V ) belong to (Γ′

⋂X(T )+≥0) \ {λt}. Lemma 5.7 concludes the proof.

Let V be a G-supermodule with a basis as in Lemma 5.3. The (Laurent) polyno-mial χ(V ) =

∑

1≤i≤t(−1)|vi|φ(cii) is called formal supercharacter of V . It is clear that

χ(V ) =∑

λ∈X(T )(dim(Vλ)0 − dim(Vλ)1)xλ+yλ− , where xλ+ =

∏

1≤i≤m xλi

i and yλ− =∏

m+1≤i≤m+n yλi

i−m.

Example 5.1 (see [14]) Set m = n = 1. A simple polynomial GL(1|1)-supermodule is atmost two dimensional. More precisely, set X(T )++ =

⋃

r≥0 X(T )++r , where X(T )++

r ={λ = (λ1|λ2) ∈ X(T )++||λ| = r}. If p|r, then X(T )++

r = {(i, r − i)|0 ≤ i ≤ r} andL(i) = L((i|r − i) is (even) one dimensional. In particular, Tr(L(i)) = xi

1yr−i1 obviously

belongs to As. Otherwise, X(T )++r = {(i, r − i)|1 ≤ i ≤ r} and L(i) = L((i|r − i) is a two

dimensional supermodule. By definition, its highest vector is even and the second basicvector is odd of weight (i − 1|r − i + 1). Thus Tr(L(i)) = xi

1yr−i1 − xi−1

1 yr−i+11 ∈ As.

A homogeneous polynomial f(x|y) =∑

λ∈X(T ) aλxλ+yλ− is said to be p-balanced iff forany λ with aλ 6= 0 and any 1 ≤ i ≤ m < j ≤ m + n it satisfies p|(λi + λj).

Hypothesis 5.1 The algebra φ(Rpol) coincides with As. Moreover, it is generated by allcr and by all p-balanced polynomials symmetric in x and y separately.

Problem 5.1 What are the preimages of the p-balanced polynomials from As?

Acknowledgements

This work was supported by RFFI 07-01-00392 and by INDAM.

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