Invariant forms on irreducible modules of simple algebraic groups

Let G be a simple linear algebraic group over an algebraically dosed field K of characteristic p >= 0 and let V be an irreducible rational G-module with highest weight A. When is self-dual, a basic question to ask is whether V has a non-degenerate G-invariant alternating bilinear form or a non degenerate G-invariant quadratic form. If p not equal 2, the answer is well known and easily described in terms of A. In the case where p = 2, we know that if is self-dual, it always has a non-degenerate G-invariant alternating bilinear form. However, determining when V has a non-degenerate G-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and A is a fundamental highest weight omega(i), and in the case where G is of type A(i) and lambda = omega(r) + omega(s) for 1

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

Let

G

be a simple linear algebraic group over an algebraically closed field

K

of characteristic

p \geq 0

. In this thesis, we investigate closed connected reductive subgroups

X < G

that contain a given distinguished unipotent element

u

G

. Our main result is the classification of all such

X

that are maximal among the closed connected subgroups of

G

. When

G

is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where

G

is simple of classical type, say

G = \operatorname{SL}(V)

G = \operatorname{Sp}(V)

, or

G = \operatorname{SO}(V)

. We begin by considering the maximal closed connected subgroups

X

G

which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where

X

is the stabilizer of a tensor decomposition of

V

. For

p = 2

and

X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)

, we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015). After the geometric subgroups, the maximal closed connected subgroups that remain are the

X < G

such that

X

is simple and

V

is an irreducible and tensor indecomposable

X

-module. The bulk of this thesis is concerned with this case. We determine all triples

(X, u, \varphi)

where

X

is a simple algebraic group,

u \in X

is a unipotent element, and

\varphi: X \rightarrow G

is a rational irreducible representation such that

\varphi(u)

is a distinguished unipotent element of

G

. When

p = 0

, this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015). In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups

X

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

u

which are contained in some proper parabolic subgroup of

G

. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when

u

is a regular unipotent element of

G

, no such

X

exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where

(G, p) = (C_2, 2)

, a connected reductive overgroup of a distinguished unipotent element of order

p

cannot be contained in a proper parabolic subgroup of

G

EPFL2018

Irreducibility of disconnected subgroups of exceptional algebraic groups

Soumaïa Nadia Ghandour

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

EPFL2008

Restriction of irreducible modules for Spin2+1(K) to Spin2(K)

Mikaël Cavallin

Let K be an algebraically closed field of characteristic

p\geq0

and let

Y=SPin_{2n+1}(K) (n\geq3)

be a simply connected simple algebraic group of type

B_n

over

K

. Also let

X

be the subgroup of type

D_n

, embedded in

Y

in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for

Y

. In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational

KY

-modules on which

X

acts with exactly two composition factors, completing the work of Ford in [12].

2017

Restriction of irreducible modules for Spin2+1(K) to Spin2(K)

Mikaël Cavallin

Let K be an algebraically closed field of characteristic

p\geq0

and let

Y=SPin_{2n+1}(K) (n\geq3)

be a simply connected simple algebraic group of type

B_n

over

K

. Also let

X

be the subgroup of type

D_n

, embedded in

Y

in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for

Y

. In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational

KY

-modules on which

X

acts with exactly two composition factors, completing the work of Ford in [12].

2017

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

Let

G

be a simple linear algebraic group over an algebraically closed field

K

of characteristic

p \geq 0

. In this thesis, we investigate closed connected reductive subgroups

X < G

that contain a given distinguished unipotent element

u

G

. Our main result is the classification of all such

X

that are maximal among the closed connected subgroups of

G

. When

G

is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where

G

is simple of classical type, say

G = \operatorname{SL}(V)

G = \operatorname{Sp}(V)

, or

G = \operatorname{SO}(V)

. We begin by considering the maximal closed connected subgroups

X

G

which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where

X

is the stabilizer of a tensor decomposition of

V

. For

p = 2

and

X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)

X < G

such that

X

is simple and

V

is an irreducible and tensor indecomposable

X

-module. The bulk of this thesis is concerned with this case. We determine all triples

(X, u, \varphi)

where

X

is a simple algebraic group,

u \in X

is a unipotent element, and

\varphi: X \rightarrow G

is a rational irreducible representation such that

\varphi(u)

is a distinguished unipotent element of

G

. When

p = 0

X

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

u

which are contained in some proper parabolic subgroup of

G

. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when

u

is a regular unipotent element of

G

, no such

X

(G, p) = (C_2, 2)

, a connected reductive overgroup of a distinguished unipotent element of order

p

cannot be contained in a proper parabolic subgroup of

G

EPFL2018