GAP (CoReLG) - Chapter 4: Real nilpotent orbits

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4 Real nilpotent orbits
4.1 Nilpotent orbits in real Lie algebras
4.1-1 NilpotentOrbitsOfRealForm
4.1-2 RealCayleyTriple
4.1-3 WeightedDynkinDiagram
4.2 The real Weyl group
4.2-1 RealWeylGroup

### 4 Real nilpotent orbits

#### 4.1 Nilpotent orbits in real Lie algebras

CoReLG has a database of the nilpotent orbits of the real forms of the simple Lie algebras of ranks up to 8. When called the first time in a GAP session, CoReLG will first read the database of nilpotent orbits.

##### 4.1-1 NilpotentOrbitsOfRealForm

`‣ NilpotentOrbitsOfRealForm`( L ) | ( attribute )

Here L is a real form of a complex simple Lie algebra of rank up to 8. This function returns the list of nilpotent orbits (under the action of the adjoint group) of L. For this function to work, L must be defined over SqrtField.

    gap> L:= RealFormById( "F", 4, 3 );;
    gap> no:= NilpotentOrbitsOfRealForm( L );;
    #I CoReLG: read database of real triples ... done
    gap> no[1];
    <nilpotent orbit in Lie algebra>

##### 4.1-2 RealCayleyTriple

`‣ RealCayleyTriple`( o ) | ( attribute )

Here o is a nilpotent orbit constructed by `NilpotentOrbitsOfRealForm` (4.1-1) of a simple real Lie algebra. This function returns a real Cayley triple [ f, h, e ] corresponding to the orbit o. The third element e is a representative of the orbit.

    gap> L:= RealFormById( "F", 4, 2 );;
    gap> no:= NilpotentOrbitsOfRealForm( L );;
    gap> o:= no[10];
    <nilpotent orbit in Lie algebra>
    gap> t:=RealCayleyTriple(o);;
    gap> theta:= CartanDecomposition(L).CartanInv;
    function( v ) ... end
    gap> theta(t[1]) = -t[3];
    true
    gap> theta(t[2]) = -t[2];
    true
    gap> t[3]*t[1] = t[2];
    true

##### 4.1-3 WeightedDynkinDiagram

`‣ WeightedDynkinDiagram`( o ) | ( attribute )

Here o is a nilpotent orbit constructed by `NilpotentOrbitsOfRealForm` (4.1-1) of a simple real Lie algebra. This function returns the weighted Dynkin diagram of the orbit, which identifies its orbit in the complexification of the real Lie algebra in which o lies.

#### 4.2 The real Weyl group

##### 4.2-1 RealWeylGroup

`‣ RealWeylGroup`( L ) | ( function )

`‣ RealWeylGroup`( L, H ) | ( function )

Here L is a real semisimple Lie algebra with Cartan subalgebra H. (If H is not given, then CartanSubalgebra(L) will be taken.) This function returns the real Weyl group \(N_G(H)/C_G(H)\) associated with H, where \(G\) is the connected component of the group of real points of the complex adjoint group of L. The real Weyl group will be stored in the Cartan subalgebra, so that a new call to this function, with the same input, will return the real Weyl group immediately.

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