# On complete reducibility of tensor products of simple modules over simple algebraic groups

## Abstract

Let be a simply connected simple algebraic group over an algebraically closed field of characteristic The category of rational . is not semisimple. We consider the question of when the tensor product of two simple -modules -modules and is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel of in tensor products, we obtain a reduction to the case where the highest weights ) and are In this case, we also prove that -restricted. is completely reducible as a if and only if -module is completely reducible as a -module.

## 1. Introduction

Let be a simply connected simple algebraic group over an algebraically closed field of positive characteristic The simple . are parametrized by the set -modules of dominant weights of (with respect to a fixed maximal torus and Borel subgroup) and for we write , for the unique simple of highest weight -module One of the most powerful tools in examining the simple modules . is Steinberg’s tensor product theorem: Given there is a unique , decomposition -adic where , is a weight and -restricted Then the simple module . has a tensor product decomposition

where denotes the Frobenius twist of the simple module Furthermore, the simple . -module remains simple upon restriction to the first Frobenius kernel of by a result of C. W. Curtis; see Reference Cur60. This allows one to reduce many questions about simple to questions about simple -modules with -modules highest weight, or to questions about simple -restricted -modules.

Given weights with decomposition -adic and respectively, the tensor product , admits a decomposition

Thus, a lot of structural information about can be obtained by understanding the structure of and One of our main results, see Theorem .C below, is an illustration of this principle.

Our first main result is the following:

Additionally, we obtain a theorem relating complete reducibility of and -modules -modules:

Combining Theorems A and B, we obtain the following reduction theorem:

The question of complete reducibility of tensor products of has previously been considered by J. Brundan and A. Kleshchev in -modulesReference BK99 and Reference BK00, and by J.-P. Serre in Reference Ser97. Some of their results will be recalled in Sections 4 and 7 below.

We prove our results using some new techniques for weakly maximal vectors (that is, maximal vectors for the action of in tensor products of ) More precisely, we give criteria under which weakly maximal vectors of -modules.non- weights generate non-simple -restricted (see Propositions -submodules3.9 and 3.13) and we show how to construct explicitly a weakly maximal vector of weight given a weakly maximal vector of weight , (see Propositions 3.3 and 3.5). In the proofs of Theorems A and B, we will use these results to construct weakly maximal vectors that generate non-simple of -submodules thus showing that , is not completely reducible as a -module.

The paper is organized as follows: In Section 2, we summarize the basic definitions and recall some important results. Section 3 is concerned with the results on weakly maximal vectors in tensor products of that will be required to prove Theorems -modulesA and B. In section 4, we cite results about complete reducibility from the literature and derive some consequences. The results we are using are due to H. H. Andersen, J. Brundan, A. Kleshchev, J.-P. Serre and I. Suprunenko. Some of the results in Section 3 are only valid for groups of type different from and for primes that are not too small with respect to the root system. Therefore, the proofs of Theorems A and B are split up over several sections. In Section 5, we will consider the case where is of type different from and if is of type , or In Section .6, we give proofs of the theorems for of type when Finally, if . is of type , or and or is of type and then the simple of -modules weight admit a refined tensor product decomposition corresponding to the decomposition of the root system of -restricted into short roots and long roots. We make use of this in Section 8 in order to prove Theorems A and B in the remaining cases. Our treatment of groups of type in characteristic relies on a detailed study of tensor products of simple modules for the Levi subgroup of type These results are given in Section .7, along with a complete classification of the pairs of weights -restricted and such that is completely reducible for of type when In the final Section .9, we give the proof of Theorem C.

## 2. Preliminaries

In this section, we give the basic definitions and cite some important results from the literature.

### 2.1. Notation

Our notational conventions are essentially the same as in Reference Jan03, except that we write for the induced module and for the Weyl module of highest weight The following basic notations will be used throughout: .

We fix to be an algebraically closed field of characteristic and to be a simply connected simple algebraic group scheme over defined and split over the finite field , The assumption of . being simple and simply connected is for convenience and our main results generalize to connected reductive groups over Let . be a split maximal torus in and denote by the character group of Let . be the root system of with respect to with a fixed choice of base , Unless otherwise specified, we adopt the standard labeling of simple roots as given in .Reference Bou02. We write for the positive system defined by and Let . be the Weyl group of and let be a inner product on the real space -invariant normalized so that , for all short roots The coroot of . is defined by Let .

be the set of dominant weights, define

and set the set of , (dominant) weights. Let -restricted be the fundamental dominant weights with respect to that is , and let , There is a partial order on . defined by if and only if is a non-negative integer linear combination of positive roots. Denote by the highest root with respect to this partial order.

Denote by a Frobenius endomorphism and let be the first Frobenius kernel of For a rational . -module we denote by , the Frobenius twist of If . is finite-dimensional, we denote by the dual module of and by the contravariant dual of (see Section II.2.12 in Reference Jan03). For we denote by , the space of -weight and call its non-zero elements weight vectors of weight We write . for the Borel subgroup of corresponding to and define for Finally, we write . for the Weyl module of highest weight and for the simple module of highest weight .

For we denote by , the derived subgroup of the Levi subgroup of corresponding to the simple roots a simply connected semisimple algebraic group. For , we write , for the set .

### 2.2. The hyperalgebra and its infinitesimal subalgebra

Instead of working with the group schemes and directly, we will be using the hyperalgebra of and its infinitesimal subalgebra, which will enable us to carry out explicit constructions of weakly maximal vectors in tensor products in Section 3. Let be the complex simple Lie algebra with root system let ,

be a Chevalley basis of and denote by the universal enveloping algebra of .

In the following, we will write and instead of and for the images of the divided powers in and we abbreviate , by .

Recall that is a Hopf with comultiplication, counit and antipode given by -algebra

respectively, for elements

As shown in Section II.1.12 in Reference Jan03, *distribution algebra* of

The infinitesimal subalgebra

## 3. Weakly maximal vectors in tensor products

Let us begin with the definition of a weakly maximal vector.

Weakly maximal vectors have previously been considered by J. Brundan and A. Kleshchev in the proof of Theorem 3.3 in Reference BK99, where they were called weakly primitive vectors. Our Lemma 3.4 below was inspired by a computation in the aforementioned proof.

In this section, we prove some results about weakly maximal vectors in tensor products of