Complexity, Periodicity and One-Parameter Subgroups

We use the variety of one-parameter subgroups to define a numerical invariant for a representation of an infinitesimal group scheme. For an indecomposable module M of complexity 1, this number is related to the period of M.


Introduction
This paper is concerned with representations of finite group schemes that are defined over an algebraically closed field k of positive characteristic p > 0. Given such a group scheme G and a finite-dimensional G-module M , Friedlander and Suslin showed in their groundbreaking paper [32] that the cohomology space H * (G, M ) is a finite module over the finitely generated commutative k-algebra H • (G, k) := n≥0 H 2n (G, k). This result has seen a number of applications which have provided deep insight into the representation theory of G.
By work of Alperin-Evens [1], Carlson [9] and Suslin-Friedlander-Bendel [55,56], the notions of complexity, periodicity and infinitesimal one-parameter subgroups are closely related to properties of the even cohomology ring H • (G, k). In this paper, we employ the algebro-geometric techniques expounded in [32] and [55,56] in order to obtain information on the period of periodic modules, and the structure of the stable Auslander-Reiten quivers of algebraic groups. Our methods are most effective when covering techniques related to G r T -modules can be brought to bear. By way of illustration, we summarize some of our results in the following: Theorem. Let G be a smooth, reductive group scheme with maximal torus T ⊆ G. Suppose that M is an indecomposable G r T -module of complexity cx GrT (M ) = 1. Then the following statements hold: (1) There exists a unipotent subgroup U M ⊆ G r of height h M and a root α of G such that (2) The restriction M | Gr is periodic with period 2p r−h M .
Here cx GrT (M ) refers to the polynomial rate of growth of a minimal projective resolution of M and Ω GrT is the Heller operator of the Frobenius category of G r T -modules.
Our article can roughly be divided into two parts. Sections 1-6 are mainly concerned with the category mod G of finite-dimensional modules of a finite group scheme G. The Frobenius category mod G r T of compatibly graded G r -modules is dealt with in the remaining two sections. In Section 1 we exploit the detailed information provided by the Friedlander-Suslin Theorem [32] in order to provide an upper bound for the complexity of a G-module in terms of the dimension of certain Ext-groups. Section 2 lays the foundation for the later developments by collecting basic results concerning the cohomology rings of the Frobenius kernels G a(r) of the additive group G a .
The period of a periodic module is known to be closely related to the degrees of homogeneous generators of the ring H • (G, k). The Friedlander-Suslin Theorem thus implies that, for any infinitesimal group G of height ht(G), the number 2p ht(G)−1 is a multiple of the period of any periodic G-module. In Section 3, we analyze this feature more closely, showing how the period may be bounded by employing infinitesimal one-parameter subgroups of G. The relevant notion is that of the projective height of a module, which, for an indecomposable G-module M of complexity 1, coincides with the height of a certain unipotent subgroup U M ⊆ G.
Sections 4 and 5 are concerned with the Auslander-Reiten theory of finite group schemes. Following a discussion of components of Euclidean tree class, we determine in Section 5 those ARcomponents of the Frobenius kernels SL(2) r , that contain a simple module. Applications concerning blocks and AR-components of Frobenius kernels of reductive groups are given in Section 6. In particular, we attach to every representation-finite block B ⊆ kG a unipotent "defect group" U B ⊆ G, whose height is linked to the structure of B.
By providing an explicit formula for the Nakayama functor of the Frobenius category of graded modules over certain Hopf algebras, Section 7 initiates our study of G r T -modules. In Section 8 we come to the second central topic of our paper, the Auslander-Reiten theory of the groups G r and G r T , defined by the r-th Frobenius kernel of a smooth group G, and a maximal torus T ⊆ G. Our first main result, Theorem 8.1.1, links the projective height of a G r T -module of complexity 1 to the behavior of powers of the Heller operator Ω GrT . In particular, the Frobenius category mod G r T of a reductive group G is shown to afford no Ω GrT -periodic modules, and the Ω Gr -periods of G r T -modules of complexity 1 are determined by their projective height (cf. the Theorem above). It is interesting to compare this fact with the periods of periodic modules over finite groups, which are given by the minimal ranks of maximal elementary abelian p-groups, see [7, (2.2)]. The aforementioned results provide insight into the structure of the components of the stable Auslander-Reiten quiver of mod G r T and the distribution of baby Verma modules: Theorem. Suppose that G is defined over the Galois field F p with p ≥ 7. Let T ⊆ G be a maximal torus.
(1) If Θ is a component of the stable Auslander-Reiten quiver of mod G r T , then (2) If Θ contains two baby Verma modules Z r (λ) ∼ = Z r (µ), then cx GrT ( Z r (λ)) = 1, and there exists a simple root α of G such that { Z r (λ + np r α) ; n ∈ Z} is the set of baby Verma modules belonging to Θ.
(3) A stable Auslander-Reiten component of mod G r contains at most one baby Verma module.
Recall that a group scheme G is referred to as representation-finite if and only if mod G has only finitely many isoclasses of indecomposable objects. An indecomposable G-module is said to be periodic if there exists n ≥ 1 such that Ω n G (M ) ∼ = M . The first part of the following result refines [56, (7.6.1)]. (2) The group G is diagonalizable if and only if H 2np r−1 (G, k) = (0) for some n ≥ 1.
(3) The group G is representation-finite if and only if dim k H 2np r−1 (G, k) ≤ 1 for some n ≥ 1. Proof.
Let n ∈ N be such that dim k H 2np r−1 (G, k) ≤ 1. Then Theorem 1.1 implies cx G (k) ≤ 1 and our assertion is a consequence of [ (2) If M is indecomposable of length ℓ(M ) = 2 and ℓ(Ω 2np r−1 G (M )) ≤ 2 for some n ≥ 1, then M is projective or periodic.
Proof. By general theory, we have

Varieties for G a(r) -Modules
In Section 3 we shall study questions concerning the periodicity of G-modules by considering their rank varieties of infinitesimal one-paramenter subgroups of G. These are defined via the groups G a(r) := Spec k (k[T ]/(T p r )) (r ≥ 1).
We denote the canonical generator of the coordinate ring by t. The algebra of measures kG a(r) of G a(r) is isomorphic to k[U 0 , . . . , U r−1 ]/(U p 0 , . . . , U p r−1 ), with U i + (U p 0 , . . . , U p r−1 ) corresponding to the linear form u i on k[G a(r) ] that sends t j onto δ p i ,j . Throughout, we assume that p ≥ 3. We write H * (k[u i ], k) = k[x i+1 ]⊗ k Λ(y i ) with deg(x i+1 ) = 2 and deg(y i ) = 1. The Künneth formula then provides an isomorphism of graded k-algebras, where Λ(y 0 , . . . , y r−1 ) denotes the exterior algebra in the variables y 0 , . . . , y r−1 . In this identification, The above notation derives from the grading associated to the action of a torus T on G a(r) . If T operates via a character α : T −→ k × , i.e., then, thanks to [45, (I.4.27)] (see also [12, (4.1)]), the induced action of T on H * (G a(r) , k) can be computed as follows: Lemma 2.1. The following statements hold: Given s ≤ r, we consider the standard embedding G a(s) ֒→ G a(r) , whose comorphism is the projec- The resulting embedding of algebras of measures is thus given by Let F : G a(r) −→ G a(r−1) ; x → x p be the Frobenius homomorphism. Setting u −1 := 0, we see that the corresponding homomorphism of Hopf algebras is given by We recall the notion of a p-point, introduced by Friedlander-Pevtsova [31]. Let A p = k[X]/(X p ) be the truncated polynomial ring with canonical generator u := X + (X p ). An algebra homomorphism α : α is left flat, and (b) there exists an abelian unipotent subgroup U ⊆ G such that im α ⊆ kU. If α : A p −→ kG is an algebra homomorphism, then α * : mod G −→ mod A p denotes the associated pull-back functor. Two p-points α and β are equivalent if for every M ∈ mod G the module α * (M ) is projective precisely when β * (M ) is projective. We denote by P (G) the space of equivalence classes of p-points.
The cohomological interpretation of p-points is based on the induced algebra homomorphisms is not projective}. According to [31, (3.10),(4.11)], the sets P (G) M are the closed sets of a noetherian topology on P (G) and the map Ψ G : P (G) −→ Proj(V G (k)) ; [α] → ker α • is a homeomorphism with P (G) M = Ψ G (Proj(V G (M )) for every M ∈ mod G. Moreover, Ψ G is natural with respect to flat maps H −→ G of finite group schemes.
Given f ∈ H • (G a(r) , k), we let Z(f ) be the zero locus of f , that is, the set of the maximal ideals of H • (G a(r) , k) containing f .
Since P (G a(r−1) ) N = ∅ (cf. [31, (4.11)]), we conclude that a r−1 = 0. As noted in [55, (1.13(2))], the iterated Frobenius homomorphism F r−1 : G a(r) −→ G a(1) induces a map Since F r−1 (α(u)) = a r−1 u 0 , the map F r−1 • α is an isomorphism of k-algebras. Consequently, Since ker α • ∈ Proj(V G a(r) (N )), it follows that the radical ideal is contained in ker α • . By the above, the image of x r in the coordinate ring k[V G a(r) (N )] is not zero, and Hilbert's Nullstellensatz provides a maximal ideal M ¢ H • (G a(r) , k) such that M ⊇ I N and x r ∈ M. Consequently, as desired.
Let G be an algebraic k-group. In [55, §1] the authors introduce the affine algebraic scheme V r (G) of infinitesimal one-parameter subgroups. By definition, is the homomorphism scheme, cf. [58, p.18]. Owing to [55, (1.14)], there exists a homomorphism ] of commutative k-algebras which multiplies degrees by p r 2 . Moreover, the map Ψ r G is natural in G. Let s ≤ r. We conclude this section with a basic observation concerning the map . In view of [55, (1.10)] (and its proof), the coordinate ring is reduced with Z-grading given by deg(T i ) = p i (see also [55, (1.12)]).
Proof. This is a direct consequence of the proof of [56, (6.5)].

Projective Height and Periodicity
Let k be an algebraically closed field of characteristic char(k) = p ≥ 3. Throughout this section, we let G be an infinitesimal k-group of height ht(G) = r. To each non-projective G-module M ∈ mod G we associate its projective height ph(M ). This numerical invariant, which will later be seen to be constant on the components of the stable Auslander-Reiten quiver of G, provides information on the period of periodic modules.
We let E(G) be the set of elementary abelian subgroups of G. Let M be a G-module of projective height ph(M ) = t > 0. Then there exists a subgroup U ⊆ G with U ∼ = G a(t−1) and M | U being projective. Consequently, M is a free module of the p t−1 -dimensional algebra kU, so that p t−1 | dim k M . Given a commutative k-algebra A, we denote by A red the associated reduced algebra.  (1) If M | U is not projective, then there exists ζ ∈ H 2p r−ph U (M ) (G, k) red such that Proof. Let 1 ≤ t ≤ s. Owing to [55, (1.14)] we have a commutative diagram of homomorphisms of graded, commutative k-algebras, where the horizontal arrows multiply degrees by p r 2 . Thanks to [55, (1.5)], the map π is surjective, and [55, (1.12)] shows that it respects degrees. Since U t ∼ = G a(t) we may consider the element T r−t ∈ k[V r (U t )] p r−t . As π is surjective, we can find v r−t ∈ k[V r (G)] p r−t with π(v r−t ) = T r−t . According to [56, (5.2)], we have v p r r−t ∈ im Ψ r G , so that there exists w ∈ H 2p r−t (G, k) with Ψ r G (w) = v p r r−t . In light of (2.3), we thus obtain Ψ r Ut (res(w)) = π(Ψ r G (w)) = π(v p r r−t ) = T p r r−t = Ψ r Ut (x p r−t t ).
Thanks to [56, (5.2)], we conclude that the residue class ζ t :=w ∈ H 2p r−t (G, k) red satisfies If we identify V Ut (k) with its image under the morphism res * : V Ut (k) −→ V G (k), whose comorphism is the restriction map res red : Let t := ph U (M ). In view of [56, (7.1) which, by choice of t, contradicts (2.2). This concludes the proof of (1). For the proof of (2), we set t := s, so that U t = U ∼ = G a(s) . Consider the homomorphism √ ker Φ M be the coordinate ring of the support variety V G (M ) and denote by the canonical projection map. According to our convention, the inclusion V U (k) ⊆ V G (M ) means that the image of the morphism res * : Thus, the map res * factors through the inclusion ι : By ( * ), we can find ζ := ζ s ∈ H 2p r−s (G, k) red such that res red (ζ) = 0. Consequently, ι * (ζ) = 0, and Hilbert's Nullstellensatz provides a maximal ideal M ⊇ √ ker Φ M which does not contain ζ. This implies We record a consequence concerning modules of complexity 1, which generalizes [19, (2.5)]. The proof employs a method of Carlson (cf. [10]), which is based on the following construction: By general theory, a cohomology class ζ ∈ H 2n (G, k)\{0} corresponds to an elementζ ∈ Hom G (Ω 2n G (k), k)\ {0}. We let L ζ := kerζ be the Carlson module of ζ. Proof. Let t := ph(M ) = ph U (M ) for some U ∈ E(G). Owing to (3.1(1)), we can find an element [56, (7.7)]). Thanks to [56, p.755] we have V G (L ζ ) = Z(ζ), so that an application of [56, (7.2)] gives Consequently, the module L ζ ⊗ k M is projective and the exact sequence Examples. (1) Consider the infinitesimal group G = SL(2) 1 T r , where r ≥ 2. Then ht(G) = r while any unipotent subgroup U of G has height ≤ 1. Consequently, every non-projective G-module has projective height 1, so that (3.2) yields 2p r−1 as an estimate for the period per(M ) of a periodic module M . According to [28, (5.6)] and [23, (4.5)] "most" periodic modules (namely those, whose rank varieties are not T -stable) have period 2, while those with a T -stable rank variety satisfy per(M ) = 2p r−1 .
(2) Let U be a unipotent infinitesimal group of complexity cx U (k) = 1. Owing to the main theorem of [26] such groups can have arbitrarily large height. The corresponding algebras kU are truncated polynomial rings k[X]/(X p n ), so that every indecomposable U-module M is periodic with per(M ) = 2. If U has height r ≥ 2, then only for those non-projective indecomposable modules M with dim k M = ℓ dim k kU r−1 does Corollary 3.2 provide the correct formula for per(M ).
Remark. We shall see in Section 8 below that 2p r−ph(M ) coincides with the period for graded modules of Frobenius kernels of reductive algebraic algebraic groups (see Theorem 8.  Proof. Since cx G (M ) = 1, an application of [56, (6.8) According to [56, (6.1)], the canonical action of k × on G a(r) endows the variety V r (G) M with the structure of a conical variety: In view of α.u r−1 = α p r−1 u r−1 , the group k × acts simply on V r (G) M \ {0}. Since M is indecomposable, Carlson's Theorem (see [56, (7.7)]) ensures that the variety As a result, U M = im ϕ is a subgroup of G. Being isomorphic to the factor group G a(r) / ker ϕ, the group U M is elementary abelian. Let s := ph U M (M ). We propose to show that s = ph(M ) = ht(U M ). Let U be an elementary abelian subgroup of G such that M | U is not projective. Owing to [56, (6.6) By applying this observation to the Frobenius kernels of U M , we conclude that s = ht(U M ). Thus, if U is elementary abelian and r ′ ∈ N is a natural number such that the restriction We now specialize to the case, where G = G r is a Frobenius kernel of a smooth group scheme G. Then G acts on G r via the adjoint representation, and we obtain an action of G on V r (G). If M is a G r -module and g ∈ G, then M (g) denotes the G r -module with underlying k-space M and action One readily verrifies that We refer to a G r -module as Proof. This follows directly from Theorem 3.3 and the definition of the G-action on V r (G) M .
We fix a maximal torus T ⊆ G and denote by X(T ) its character group. Since T acts diagonally on the Lie algebra g := Lie(G), there exists a finite subset Ψ ⊆ X(T ) \ {0} such that The maximal torus T acts canonically on H • (G r , k) red and H • (U, k) red . In light of (2.1), the weights of this action on the latter space are of the form −nα for n ≥ 0. We thus have the following refinement of Proposition 3.1: Proposition 3.5. Let G be a smooth group scheme, U ⊆ G r be a T -invariant elementary abelian subgroup, M be a G r -module. Then there exists α ∈ Ψ ∪ {0} with the following properties: Hence T also acts on Lie(U) ⊆ g via α, so that α ∈ Ψ ∪ {0}. We return to the proof of Proposition 3.1. For t ≤ r we found elements We may now adopt the arguments of the proof of (3.1) verbatim to obtain our result.
Now assume G to be reductive, with Borel subgroup B = U T and sets Ψ + and Σ of positive roots and simple roots, respectively. As usual, ρ denotes the half-sum of the positive roots. Given d ≥ 0, we recall that where U α ⊆ G is the root subgroup of α. Moreover, for a subset Φ ⊆ Ψ, we put Let Y (T ) be the set of co-characters of T and denote by , : Here α ∨ ∈ Y (T ) denotes the root dual to α. Owing to [43, (2.7)], the set Ψ s λ is a subsystem of Ψ whenever the prime number p is good for G.
Given λ ∈ X(T ) with λ + ρ, α ∨ = 0 for some α ∈ Ψ, we define the depth of λ via dp(λ) := min{s ∈ N 0 ; Ψ s λ = Ψ}, and put dp(λ) = ∞ otherwise, see [25]. The following result links the depth of a weight to the projective height ph Σ (Z r (λ)) of the baby Verma module Suppose that G is a smooth group scheme that is defined over the Galois field F p . Let M be a G-module. Given d ∈ N, we let M [d] be the G-module with underlying k-space M and action defined via pull-back along the iterated Frobenius endomorphism F d : G −→ G. Proposition 3.6. Suppose that G is semi-simple, simply connected, defined over F p and that p is good for G. Let λ ∈ X(T ) be a weight such that dp(λ) ≤ r. Then the following statements hold: Proof. Let dp(λ) = d + 1 with d ≥ 0. Owing to [25, (6.2)], there exists a weight µ of depth 1 and an isomorphism According to [25, (5.2)], the inclusion where the vertical arrows are the canonical inclusions induced by the restriction maps. Thanks to [25, (6.2(b))], the lower horizontal map is an isomorphism sending V Gr so that µ+ρ, α ∨ ∈ pZ, whence α ∈ Σ\Ψ 1 µ . In view of the identification discussed at the beginning of this section, the inclusion ). The diagram above then implies Let L be the Levi subgroup of G associated to the simple root α and let St L d be the d-th Steinberg module of L. Thanks to [49, (4 The rank varieties of the former module were essentially computed in [56, (7.9)] (The quoted result deals with Frobenius kernels of SL (2)). This result implies in particular that As a result, the module Z r (λ)| (Uα) d+1 is not projective, so that ph α (Z r (λ)) ≤ d + 1.

Euclidean Components of finite group Schemes
This section is concerned with the Auslander-Reiten theory of a finite group scheme G. Given a self-injective algebra Λ, we denote by Γ s (Λ) the stable Auslander-Reiten quiver of Λ. By definition, the directed graph Γ s (Λ) has as vertices the isomorphism classes of the non-projective indecomposable Λ-modules and its arrows are defined via the so-called irreducible morphisms. We refer the interested reader to [4, Chap. VII] for further details. The AR-quiver is fitted with an automorphism τ Λ , the so-called Auslander-Reiten translation. Since Λ is self-injective, τ Λ coincides with the composite Ω 2 Λ • ν Λ of the square of the Heller translate Ω Λ and the Nakayama functor ν Λ , cf.
The connected components of Γ s (Λ) are connected stable translation quivers. By work of Riedtmann [51,Struktursatz], the structure of such a quiver Θ is determined by a directed tree T Θ , and an admissible group Π ⊆ Aut k (Z[T Θ ]), giving rise to an isomorphism of stable translation quivers. The underlying undirected treeT Θ , the so-called tree class of Θ is uniquely determined by Θ. We refer the reader to [5, (4.15.6)] for further details. For group algebras of finite groups, the possible tree classes and admissible groups were first determined by Webb [59].
Following Ringel [52], an indecomposable Λ-module M is called quasi-simple, provided it lies at the end of a component of tree class A ∞ of the stable AR-quiver Γ s (Λ).
Throughout this section, G is assumed to be a finite algebraic group, defined over an algebraically closed field k of characteristic p > 0. By general theory (see [30, (1.5)]), the algebra kG affords a Nakayama automorphism ν = ν G of finite order ℓ. For each n ∈ N 0 , the automorphism ν n induces, via pull-back, an auto-equivalence of mod G and hence an automorphism on the stable AR-quiver Γ s (G) of kG. In view of [4, Chap.X], the Heller operator Ω G also gives rise to an automorphism of Γ s (G). Given an AR-component Θ ⊆ Γ s (G), we denote by Θ (n) the image of Θ under ν n and put . We say that a component Θ ⊆ Γ s (G) has Euclidean tree class if the graphT Θ is one of the Euclidean diagramsÃ 12 ,D n (n ≥ 4) orẼ n (6 ≤ n ≤ 8).
According to [17, (I.8.8)], the map In view of (b), the same reasoning now implies  [36,Theorem], the tree classT Θ is either a finite Dynkin diagram or A ∞ , contradicting our hypothesis on Θ.

Representations of SL(2) r
Throughout, we consider the infinitesimal group scheme SL(2) r , defined over the algebraically closed field k of characteristic p ≥ 3. Recall that the algebra Dist(SL (2) The blocks of Dist(SL(2) r ) are well understood (cf. [50]). In the following, we identify the character group X(T ) of the standard maximal torus T of diagonal matrices of SL(2) with Z, by letting n ∈ Z correspond to nρ ∈ X(T ). Accordingly, the modules L r (0), . . . , L r (p r −1) constitute a full set of representatives for the isomorphism classes of the simple SL (2) so that the corresponding block consists of those modules, whose composition factors are of the form L r (λ) with λ ∈ B i,s , sending L r−s (n) onto L r (np s +(p s −1)).
Proof. This follows directly from [ i,0 -modules of complexity 2 are given by the highest weights p r −p+i and p r −2−i.
(2) The composition factors of Ht r (λ) are of the form L r (µ), with Proof.
We turn to the Auslander-Reiten theory of SL(2) r and begin by determining the stable ARcomponents of Euclidean tree class. We recall that the standard almost split sequence is the only almost split sequence involving the principal indecomposable module P r (λ) (cf. [4, (V.5.5)]). Proof. In view of Lemma 5.1, we may assume that Θ ⊆ Γ s (B (r) i,0 ). According to [59, (2.4)], the component Θ is attached to a principal indecomposable module, so by passing to the isomorphic component Ω −1 SL(2)r (Θ) we may assume without loss of generality that Θ contains a simple module L r (λ).
Recall that a non-projective indecomposable SL(2) r -module M is referred to as quasi-simple, provided (a) the module M belongs to a component of tree class A ∞ , and (b) M has exactly one predecessor in Γ s (SL(2) r ).
We thus have an almost split sequence Since Soc(Ht r (λ)) is simple, we may apply [4, (V.3. 2)] to conclude that Y is simple. As cx SL (2)

Blocks and AR-Components
In this section we illustrate our results by considering blocks and Auslander-Reiten components of infinitesimal group schemes.  Proof. Let U ∈ E be an elementary abelian subgroup and consider M ∈ Θ. We write t := ph U (M ). According to [30, (1.5)], the Nakayama functor ν G is induced by the automorphism id G * ζ ℓ , which is the convolution of id G with the left modular function ζ ℓ : kG −→ k. As U s is unipotent, the restriction ζ ℓ | kUs of ζ ℓ coincides with with the co-unit, so that (id G * ζ ℓ )| kUs = id kUs . It follows that ν G (M )| Us = M | Us for all s.

Now let
be the almost split sequence terminating in M . If s ≤ t − 1, then M | Us is projective and E M | Us splits, so that each E i | Us is a direct summand of the projective module M | Us ⊕ τ G (M )| Us . Hence E i | Us is projective and ph U (E i ) ≥ t for 1 ≤ i ≤ n. Let M, N ∈ Θ. By the above, we have ph U (N ) ≥ ph U (M ), whenever N is a predecessor of M . In that case, there also exists an arrow τ G (M ) → N , so that ph Since Θ is connected, it follows that ph U (M ) = ph U (N ) for all M, N ∈ Θ. This readily implies our assertion.  The group U B may be thought of as a "defect group" of the block B. Recall that a finite-dimensional k-algebra Λ is called a Nakayama algebra if all of its indecomposable projective left modules and indecomposable injective left modules are uniserial (i.e., they only possess one composition series).
The following result provides a first link between properties of U B and B: Proposition 6.2.2. Let S be a simple G-module such that cx G (S) = 1 and ph(S) = r. Then the block B ⊆ kG containing S is a Nakayama algebra with simple modules {ν i G (S) ; i ∈ N 0 }. Proof. Since S has complexity 1 with ph(S) = r, Corollary 3.2 implies Ω 2 G (S) ∼ = S. The arguments employed in the proof of [19, (3.2)] now yield the assertion.
6.3. Frobenius kernels of reductive groups. In this section we consider a smooth reductive group G of characteristic p ≥ 3. Our objective is to apply the foregoing results in order to correct the proof of [21, (4.1)]. Given r ≥ 1, we are interested in the algebra Dist(G r ) = kG r of distributions of G r .
The following Lemma, which follows directly from the arguments of [20, (7.2)], reduces a number of issues to the special case G = SL(2): Lemma 6.3.1. Let B ⊆ Dist(G r ) be a block. If B has a simple module of complexity 2, then B is Morita equvalent to a block of Dist(SL(2) r ).
The proof of the following result corrects the false reference to SL(2)-theory on page 113 of [21]. Theorem 4.1 of [21] is correct as stated. Proof. This follows from a consecutive application of Lemma 6.3.1 and Proposition 5.5.
A finite-dimensional k-algebra Λ is called tame if it is not of finite representation type and if for every d > 0 the d-dimensional indecomposable Λ-modules can by parametrized by a one-dimensional variety. The reader is referred to [17, (I.4)] for the precise definition.
The structure of the representation-finite and tame blocks of the Frobenius kernels of smooth groups is well understood, see [22,Theorem]. The following result shows that, for smooth reductive groups, such blocks may be characterized via the complexities of their simple modules. Theorem 6.3.3. Suppose that G is reductive, and let B ⊆ Dist(G r ) be a block. Then the following statements are equivalent: (1) B is tame.
(2) Every simple B-module has complexity 2. Proof. (1) ⇒ (2) Passing to the connected component of G if necessary, we may assume that G is connected. Let S be a simple B-module. Since B is tame, [24, (3.2)] implies cx Gr (S) ≤ 2. Since B is not representation finite, the simple B-module S is not projective, so that cx Gr (S) ≥ 1. Suppose that cx Gr (S) = 1. Since S is G-stable, Corollary 3.4 shows that U B ¢ G r is a unipotent, normal subgroup of G r . Passage to the first Frobenius kernels yields the existence of a non-zero unipotent p-ideal u¢ g. Since G is reductive, [38, (11.8)] rules out the existence of such ideals, a contradiction.
(2) ⇒ (3) By Lemma 6.3.1, the block B is Morita equivalent to a block B ′ ⊆ Dist(SL(2) r ), all whose simple modules have complexity 2. A consecutive application of Lemma 5.2 and Lemma 5.3 implies r = 1. Consequently, B ′ is tame and so is B. Remark. Let G be a smooth algebraic group scheme. According to [22, (4.6)], the presence of a tame block B ⊆ Dist(G r ) implies that G is reductive.

The Nakayama Functor of mod gr Λ
In preparation for our analysis of G r T -modules we study in this section the category of graded modules of an associative algebra. Let k be a field, Λ = i∈Z n Λ i be a finite-dimensional, Z n -graded k-algebra. We denote by mod gr Λ the category of finitedimensional Z n -graded Λ-modules and degree zero homomorphisms. Given i ∈ Z n , the i-th shift functor the Nakayama functor of mod gr Λ. The purpose of this section is to determine this functor for certain Hopf algebras Λ. We begin with a few general observations. 7.1. Graded Frobenius Algebras. Suppose that Λ is a Frobenius algebra with Frobenius homomorphism π : Λ −→ k and associated non-degenerate bilinear form ( , ) π : Λ × Λ −→ k ; (x, y) → π(xy).
Our first result extends the classical formula for the Nakayama functor of Frobenius algebras to the graded case.
(2) There are natural isomorphisms for all M ∈ mod gr Λ.
(1) Since π has degree d Λ , we have π(Λ i ) = (0) for i = −d Λ , whence Let a ∈ Λ i and write µ(a) = j∈Z n x j . Assuming j = i, we consider b ∈ Λ ℓ . Then we have As a result, x j = 0 whenever j = i, so that µ(a) = x i ∈ Λ i .
(2) The block B ⊆ Λ is a homogeneous subspace.
(3) Suppose that Λ is a Frobenius algebra with homogeneous Frobenius homomorphism of degree d Λ . Then B is a Frobenius algebra, and every homogeneous Frobenius homomorphism of B has degree d Λ .
Proof. (1) General theory provides a torus T , which acts on Λ via automorphisms such that the given grading coincides with the weight space decomposition Λ = λ∈X(T ) Λ λ of Λ relative to T . Let I ⊆ Λ be the set of central primitive idempotents of Λ. Then I is a finite, T -invariant set, so that the connected algebraic group T acts trivially on I. Hence I ⊆ Λ 0 , and our assertion follows from the fact that every central idempotent is a sum of elements of I.
(2) There exists a central primitive idempotent e such that B = Λe. Thanks to (1), we have e ∈ Λ 0 , whence as desired.
(3) Let π : Λ −→ k be a Frobenius homomorphism of degree d Λ . If I ⊆ B is a left ideal of B which is contained in ker π| B , then I is also a left ideal of Λ, so that I = (0). In view of (2), the block B is a homogeneous subspace. We conclude that π| B is a homogeneous Frobenius homomorphism of B of degree d Λ . Our assertion now follows from Lemma 7.1.1.
A k-algebra Λ is called connected if its Ext-quiver is connected. This is equivalent to Λ having exactly one block. The following result in conjunction with the observations above shows when graded Frobenius algebras afford homogeneous Frobenius homomorphisms.
Theorem 7.1.3. Let Λ = i∈Z n Λ i be a connected graded Frobenius algebra. Then there exists a homogeneous Frobenius homomorphism π : Λ −→ k.
Remark. Let Λ and Γ be two connected Z n -graded Frobenius algebras with homogeneous Frobenius homomorphisms of degrees d Λ and d Γ , respectively. If d Λ = d Γ , then Lemmas 7.1.1 and 7.1.2 imply that Λ ⊕ Γ is a graded Frobenius algebra, which does not afford a homogeneous Frobenius homomorphism. We shall see in the next section that such phenomena do not arise within the context of graded Hopf algebras.
Suppose that Λ is a Frobenius algebra. Then mod gr Λ is a Frobenius category, and [35, (3.5)] ensures that mod gr Λ has almost split sequences. In view of [35, §1], the Auslander-Reiten translation τ gr Λ is given by τ gr Λ = N • Ω 2 gr Λ , where Ω gr Λ denotes the Heller operator of the Frobenius category mod gr Λ. We denote by Γ s (gr Λ) the stable Auslander-Reiten quiver of mod gr Λ. Suppose that M ∈ Θ. Since M is indecomposable, there exists a block B ℓ ⊆ Λ such that M ∈ mod gr B ℓ . This readily implies that Θ ⊆ mod gr B ℓ , whence Θ ⊆ Γ s (gr B ℓ ). Our assertion now follows from Theorem 7.1.3.

Graded Hopf Algebras.
Suppose that H is a finite-dimensional Hopf algebra. We say that H is graded if H = i∈Z n H i is a graded k-algebra such that the comultiplication ∆ : H −→ H⊗ k H is homogeneous of degree 0. In that case, the counit ε : H −→ k and the antipode η : H −→ H are also maps of degree 0. The subspace  (1) There exists i ∈ Z n such that ℓ H ⊆ H i . (2) H is a Frobenius algebra with a homogeneous Frobenius homomorphism π : H −→ k.
Since deg ε = 0, we obtain hx j = 0 for d = 0, and hx j = ε(h)x j for d = 0. Consequently, x j ∈ ℓ H for every j ∈ Z n . Since dim k ℓ H = 1, it follows that ℓ H ⊆ H i for some i ∈ Z n . (2) Note that the dual Hopf algebra H * is graded. Let π ∈ H * be a non-zero left integral. A result due to Larson and Sweedler [47] ensures that H is a Frobenius algebra with Frobenius homomorphism π : H −→ k. In view of (1), the linear map π is homogeneous.
Example. Suppose that char(k) = p > 0, and let (g, [p]) be a finite-dimensional restricted Lie algebra with restricted enveloping algebra U 0 (g). Assume that g = i∈Z n g i is restricted graded, that is, g i ⊆ g ip , so that U 0 (g) is also Z n -graded. Given a homogeneous basis {e 1 , . . . , e m } of g, we write e a := e a 1 1 · · · e am m for every a ∈ N m 0 and put τ := (p−1, . . . , p−1) as well as a ≤ τ :⇔ a i ≤ p−1 for 1 ≤ i ≤ m. Then the set {e a ; 0 ≤ a ≤ τ } is a homogeneous basis of U 0 (g), and π : U 0 (g) −→ k ; 0≤a≤τ α a e a → α τ is a homogeneous Frobenius homomorphism of degree d g := −(p−1) i∈Z n (dim k g i )i. Moreover, by [45, (I.9.7)], the unique automorphism µ : U 0 (g) −→ U 0 (g), given by µ(x) = x + tr(ad x)1 for all x ∈ g, is the Nakayama automorphism corresponding to π. By Lemma 7.1.1, we have for every M ∈ mod gr U 0 (g). Let P (S) be the projective cover of the graded simple module S. General theory implies that which retrieves [44, (1.9)].
We are going to apply the foregoing result in the context of Frobenius kernels. Suppose that char(k) = p > 0, and let G be a reduced algebraic k-group scheme with maximal torus T . We denote the adjoint representation of G on Lie(G) by Ad. For r > 0, the algebra kG r obtains a grading via the adjoint action of T on kG r : We shall identify characters of G, T and G r with the corresponding elements of the coordinate rings or the duals of the algebras of measures. Let λ G ∈ X(G) be the character given by λ G (g) = det(Ad(g)) for all g ∈ G. If g = α∈Ψ∪{0} g α is the root space decomposition of g := Lie(G) relative to T , then det(Ad(t)) = α∈Ψ α(t) dim k gα for every t ∈ T , so that Given r > 0, we denote by mod G r T the category of finite-dimensional modules of the group scheme G r T ⊆ G. In view of [23, (2.1)], which also hols for r > 1, this category is a direct sum of blocks of the category mod(G r ⋊ T ) of finite-dimensional (G r ⋊ T )-modules. The latter category is just the category of X(T )-graded kG r -modules. It now follows from work by Gordon-Green [35, (3.5)], [33] that the Frobenius category mod G r T affords almost split sequences. By the same token, the Auslander-Reiten translation τ GrT of mod G r T is given by GrT . The following result provides a formula for the Nakayama functor of mod G r T . Proposition 7.2.2. Let G be a reduced algebraic k-group scheme with maximal torus T ⊆ G.
(2) Let ζ ℓ be the left modular function of kG r . By definition, ζ ℓ : kG r −→ k is given by xh = ζ ℓ (h)x for every h ∈ kG r and x ∈ ℓ kGr . Owing to [30, (1.5)], the automorphism id kGr * ζ ℓ is the Nakayama automorphism associated to ( , ) π . It now follows from [45, (I.9.7)] that ζ ℓ (g) = λ G (g) for all g ∈ G r . Let M be a graded kG r -module. In view of Lemma 7.1.1, the Nakayama functor N of mod gr kG r satisfies For λ ∈ X(T ), we denote by k λ the one-dimensional (G r ⋊T )-module on which T and G r act via λ and 1, respectively. Then we have For a character γ ∈ X(G), we define charactersγ,γ ∈ X(G r ⋊T ) viâ γ(g, t) = γ(g)γ(t) andγ(g, t) = γ(g), respectively. Given M ∈ mod G r ⋊T , we now obtain As mod G r T is a sum of blocks of mod G r ⋊T , it follows that for every M ∈ mod G r T .
For future reference, we record the following result: Suppose that G is a reduced group scheme with maximal torus T ⊆ G and Lie algebra g = α∈Ψ∪{0} g α .
(1) By assumption, we have λ G | T ≡ 1, so that T ⊆ ker λ G . Let Z G (T ) be the Cartan subgroup of G defined by T . According to [54, (7.2.10)], the group Z G (T ) is connected and nilpotent, and [54, (6.8)] implies that Z G (T ) ⊆ ker λ G . As all Cartan subgroups of G are conjugate, their union U is also contained in ker λ G . By virtue of [54, (7.3.3)], the set U lies dense in G, so that λ G ≡ 1. Our assertion now follows from Proposition 7.2.2.
8. Complexity and the Auslander-Reiten Quiver Γ s (G r T ) As before, we fix a smooth group scheme G as well as a maximal torus T ⊆ G. The set of roots of G relative to T will be denoted Ψ. Recall that the category mod G r T of finite-dimensional G r Tmodules is a Frobenius category, whose Heller operator is denoted Ω GrT . By work of Gordon and Green [34], the forgetful functor F : mod G r T −→ mod G r preserves and reflects projectives and indecomposables, respectively. Moreover, we have Ω n Gr • F = F • Ω n GrT for all n ∈ Z, and the fiber F −1 (F(M )), defined by an indecomposable G r T -module M , consists of the shifts {M ⊗ k k p r λ ; λ ∈ X(T )}, with different shifts giving non-isomorphic modules (see [23,25] for more details). Barring possible ambiguities, we will often suppress the functor F.

8.1.
Modules of complexity 1. Modules of complexity 1 play an important rôle in the determination of the Auslander-Reiten quiver of a finite group scheme. Our study of the stable AR-quiver of mod G r T also necessitates some knowledge concerning such modules. Since mod G r T has projective covers, we have the concept of a minimal projective resolution, so that we can speak of the complexity cx GrT (M ) of a G r T -module M . Note that cx Gr (F(M )) = cx GrT (M ) for every M ∈ mod G r T .
Recall that the conjugation action of T on G r induces an operation of T on the variety V r (G). Standard arguments then show that the rank varieties of G r T -modules are T -invariant subvarieties of V r (G). In analogy with §3, the cohomology class ζ can be interpreted as an element of the weight space Hom Gr (Ω 2p r−s GrT (k), k) −p r α , or equivalently, as a non-zero G r T -linear mapζ : Ω 2p r−s GrT (k)⊗ k k −p r α −→ k, see [45, (I.6.9(5))]. There results an exact sequence Remark. For r = 1 our result specializes to [23, (2.4(2))].
Suppose that G is reductive. For every root α ∈ Ψ, there exists a subgroup U α ⊆ G on which T acts via α. The group U α is isomorphic to the additive group G a and it is customarily referred to as the root subgroup of α. An indecomposable G r T -module M is called periodic, provided there is an isomorphism Ω m GrT (M ) ∼ = M for some m ∈ N.  so that [34, (4.1)] implies α = ℓγ. Let W be the Weyl group of the root system Ψ. General theory (cf. [40, (1.5)]) provides an element w ∈ W such that α i := w(α) is simple. If Ψ is not a union of copies of A 1 , then there exists a simple root α j such that α i (α ∨ j ) = −1. Thus, −1 = w(γ)(α ∨ j )ℓ, proving that ℓ = 1. Consequently, s = 2p r−ph(M ) , as desired.
It remains to consider the case, where the connected components of the root system all have type A 1 . In that case, we have 2 = α(α ∨ ) = γ(α ∨ )ℓ, so that ℓ ∈ {1, 2}.

GrT
(M ) are indecomposable, they belong to the same block of mod G r T , see [45, (7.1)]. Consequently, all their composition factors belong to the same linkage class. Let W p be the affine Weyl group and denote by the dot action of W p on X(T ), cf. [45,(II.6.1)]. Suppose that L r (µ) is a composition factor of M . According to [45,(II.9.6)], the module L r (µ + p r γ) ∼ = L r (µ) ⊗ k p r γ is a composition factor of N ∼ = M ⊗ k k p r γ . The linkage principle [45,(II.9.19)] implies that µ + p r γ ∈ W p .µ. Since w.λ ≡ λ mod(ZΨ) for all w ∈ W p and λ ∈ X(T ), we conclude that p r γ = p r 2 α ∈ ZΨ. As p ≥ 3, we have reached a contradiction.
As a result, ℓ = 1, so that we have s = 2p r−ph(M ) in this case as well.
We turn to the question, which periods can actually occur. Our approach necessitates the following realizability criterion. Proof. Thanks to [56, (6.8)], it suffices to establish the corresponding result for support varieties.
Since T r acts trivially on H • (G r , k) (cf. [45, (I.6.7)]), the T -action on H • (G r , k) gives rise to the following decomposition If V ⊆ V Gr (k) is a conical, T -invariant closed subvariety, then there exists a homogeneous T - We let ζ 1 , . . . , ζ r be homogeneous generators of I V , that is, ζ i ∈ H 2n i (G r , k) p r λ i for suitable n i ≥ 0 and λ i ∈ X(T ). As noted earlier, each ζ i corresponds to a mapζ i : see [56, (7.5)]. The result now follows from the tensor product theorem [56, (7.2)].
Proof. (1) Let α ∈ Ψ be a root, U α ⊆ G be the corresponding root subgroup. We consider the subgroup U := (U α ) r−s ⊆ G r and note that U ∼ = G a(r−s) is a T -invariant elementary abelian subgroup of height ht(U) = r −s. Let ϕ : G a(r) −→ U be the map such that im ϕ = U. Since T acts on U via the character α, the one-dimensional closed subvariety According to Theorem 8.1.3, the module M | Gr is periodic, with period 2p r−ht(U) = 2p s .
Remark. The example of the groups SL(2) 1 T r with r ≥ 3 shows that for infinitesimal groups that are not Frobenius kernels of smooth groups, the ranks of tubes may be more restricted, see [28, (5.6)]. [36] show that the presence of so-called subadditive functions imposes constraints on the structure of the tree class of a connected stable representation quiver. This approach was first effectively employed by Webb [59] in his determination of the tree classes for AR-components of group algebras of finite groups. We shall establish an analogue for mod G r T , with a refinement for the case, where G is reductive.

Webb's Theorem. Results by Happel-Preiser-Ringel
Let G be a smooth algebraic group scheme. In the sequel, we let Γ s (G r T ) be the stable Auslander-Reiten quiver of the Frobenius category mod G r T . For a component Θ ⊆ Γ s (G r T ), we have for all M, N ∈ Θ.
Accordingly, we can attach a variety V r (G) Θ to the component Θ. By combining this fact with results by Happel-Preiser-Ringel [36] one obtains: Proposition 8.2.1 (cf. [23]). Let Θ ⊆ Γ s (G r T ) be a component. Then the tree classT Θ is a simply laced finite or infinite Dynkin diagram, a simply laced Euclidean diagram, orÃ 12 .
Proof. A consecutive application of Corollary 7.2.3 and Corollary 8.1.2 shows that mod G r T has no τ GrT -periodic modules. We may thus adopt the arguments of [23, (3.2)].
In case the underlying group is reductive, only three types of components can occur: Since H is an almost simple group of rank 1, it follows that the central subgroup H ∩ K ⊆ H is either trivial or isomorphic to µ (2) (cf. [54, (8.2.4)]). Hence, if H ∩ K = e k , then there exists a character λ ∈ X(H ∩ K) ∼ = Z/(2) such that H ∩ K acts on every vertex M ∈ Θ via λ. As H ∩ K is contained in the maximal torus T , we can find γ ∈ X(T ) with γ| H∩K = λ. In view of p being odd, we also have p r γ| H∩K = λ.  Consequently, the isomorphism kG r ∼ = kH r ⊗ k kK r induced by (c) is compatible with the T -action. Thus, the outer tensor product defines a functor Let B ⊆ kG r be the block containing the simple G r -module S, so that Θ ⊆ Γ s (BT ). Owing to [13,Section 10.E], the G r -module S is an outer tensor product S ∼ = S 1 ⊗ k S 2 with a simple projective K r -module S 2 . In view of [45,(II.9.6)], we may assume that S 1 ∈ mod H r T and S 2 ∈ mod K r T . It now follows from [34, (4.1) for a suitable γ ∈ X(T ). (Since the right-hand module lies in mod G r T , we actually have γ ∈ p r X(T ).) Letting B 1 ⊆ kH r be the block containing S 1 , we obtain inverse equivalences , so that the first functor induces an equivalence Thus, Θ is isomorphic to a component Θ 1 ⊆ Γ s (H r T H ), which, by (b), has a two-dimensional rank variety.
Since H has rank 1, we have H ∼ = SL(2), PSL (2). Since mod PSL(2) r−s T ′ is a sum of blocks of mod SL(2) r−s T , it suffices to address the case, where H = SL (2). We shall write T := T H . As noted in Section 5, the block B 1 is of the form B Thanks to [45,(II.10.4)], this functor and its inverse take SL(2) r−s T -modules to SL(2) r T -modules, so that Θ 1 is isomorphic to a component Θ 2 of Γ s (SL(2) r−s T ), whose modules have complexity 2.
If r − s = 1, then standard SL(2) 1 -theory (see [23, §3, Example]) implies Θ 2 ∼ = Z[A ∞ ∞ ]. Now assume that r − s ≥ 2 and let L r−s (λ) be the simple module belonging to Θ 2 which corresponds to S. According to [25, (3.3)], the forgetful functor F : mod SL(2) r−s T −→ mod SL(2) r−s takes our component Θ 2 to the component Ψ 2 := F(Θ 2 ) ⊆ Γ s (SL(2) r−s ) containing the simple module L r−s (λ). Since r − s ≥ 2, Lemma 5.3 shows that Ht r−s (λ) is indecomposable. From the standard almost split sequence (0) −→ Rad(P r−s (λ)) −→ Ht r−s (λ) ⊕ P r−s (λ) −→ P r−s (λ)/ Soc(P r−s (λ)) −→ (0) we see that P r−s (λ)/ Soc(P r−s (λ)) has exactly one predecessor. Consequently, the module L r−s (λ) ∼ = Ω SL(2) r−s (P r−s (λ)/ Soc(P r−s (λ))) enjoys the same property and Proposition 5.5 guarantees that Let ϕ ∈ V r−s (G) Θ 2 and consider the module M ϕ := k SL(2) r−s ⊗ k[u r−1 ] k. As argued in [23, (3.2)], Remark. It is not known whether components of tree class D ∞ actually occur. According to [25, (4.5)], components containing baby Verma modules have tree class A ∞ . 8.3. Components containing Verma modules. Throughout, G denotes a smooth reductive group scheme with maximal torus T ⊆ G and root system Ψ. By picking a Borel subgroup B ⊆ G containing T we obtain the sets Ψ + and Σ of positive and simple roots, respectively. Given λ ∈ X(T ), we denote by k λ the corresponding one-dimensional T -module. Since B = U T is a product of T and the unipotent radical U ⊆ B, this module is also a B-module. Given r ∈ N, the adjoint representation endows the induced Dist(G r )-module Z r (λ) := Dist(G r )⊗ Dist(Br) k λ with the structure of a G r T -module. We denote this module by Z r (λ) and refer to Z r (λ) and Z r (λ) as baby Verma modules defined by λ. Given λ ∈ X(T ), the modules Z r (λ) and Z r (λ) have simple tops L r (λ) and L r (λ), and all simple objects of mod G r and mod G r T arise in this fashion. Moreover, every simple G r T -module L r (λ) has a projective cover P r (λ), see [45, §II.3, §II.9, §II.11] for more details. In what follows, B − denotes the Borel subgroup opposite to B.
The main result of this section, Theorem 8.3.3, employs support varieties to study the Heller translates of the baby Verma modules Z r (λ). This is motivated by the Auslander-Reiten theory of the Frobenius categories mod G r T and mod G r , where non-projective Verma modules are quasi-simple (cf. [25, (4.4),(4.5)]). Our result implies that the connected components of the stable Auslander-Reiten quiver Γ s (G r ) contain at most one baby Verma module Z r (λ).
We let F(∆) ⊆ mod G r T be the subcategory of ∆-good modules. By definition, every object M ∈ F(∆) possesses a filtration, a so-called Z r -filtration, whose factors are baby Verma modules Z r (λ). The filtration multiplicities [M : Z r (λ)] do not depend on the choice of the filtration, and each projective indecomposable module P r (λ) belongs to F(∆), with its filtration multiplicities being linked to the Jordan-Hölder multiplicities by BGG reciprocity: see [45,§II.11]. Since [ Z r (λ) : L r (λ)] = 1, this readily implies the following: Then Ω m GrT ( Z r (λ)) ∈ F(∆) with filtration factors Z r (µ) for µ > λ.
We require the following subsidiary result concerning the subset wt(M ) ⊆ X(T ) of weights of a G r T -module M .
Proof. We let T act on Dist(G r ) via the adjoint representation and put A : Given γ ∈ X(T ), we consider the G r -module P γ := Dist(G r )⊗ Dist(Tr ) k γ .
Suppose that G is defined over F p and that p is good for G. Let λ, µ ∈ X(T ) be characters such that there exists m > 0 with GrT ( Z r (λ)) ∼ = Z r (µ). Then the following statements hold: (1) We have dp(λ) = dp(µ) = r, and there exists a simple root α ∈ Σ\Ψ r λ such that µ = λ+mp r α.
Since the baby Verma module Z r (λ) has a simple top, at least one summand has to coincide with Z r (λ).
The exact sequence ( * * ), however, yields dim k P r (λ) = dim k P r (γ), a contradiction. Q As an upshot of the above, our result holds for weights of depth dp(λ) = 1, and we now suppose that 2 ≤ d + 1 = dp(λ) ≤ r.
Given λ ∈ X(T ), we recall that P r (λ) denotes the projective cover of the simple G r -module L r (λ). If L r (λ) is not projective, we let Ht r (λ) = Rad(P r (λ))/ Soc(P r (λ)) be its heart. Recall that Γ s (G r ) denotes the stable Auslander-Reiten quiver of the self-injective algebra kG r = Dist(G r ). We record an immediate consequence of Theorem 8.3.3, which generalizes and corrects [23, (4.3)].
Corollary 8.3.4. Suppose that G is defined over F p and that p is good for G. Let λ ∈ X(T ) be a weight of depth dp(λ) ≤ r. If Z r (λ) and Z r (µ) belong to the same component of Γ s (G r ), then Z r (µ) ∼ = Z r (λ).
(3) There exists a simple root α ∈ Ψ such that { Z r (λ+np r α ; n ∈ Z} is the set of those baby Verma modules that belong to Θ(λ).