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 Nilpotent orbits in real Theta groups
4.2-1 RealWeylGroup
4.2-2 CarrierAlgsForNilpOrbsInZGrading
4.2-3 CarrierAlgsForNilpOrbsInZmGrading

### 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 Nilpotent orbits in real Theta groups

Let \(L=\bigoplus_{i\in \mathbb{Z}_m} L_i\) be a real semisimple Lie algebra, where \(\mathbb{Z}_m=\mathbb{Z}/m\mathbb{Z}\) if \(m\geq 1\), and \(\mathbb{Z}_0=\mathbb{Z}\). We provide some functions which help to determine the \(G_0\)-orbits of nilpotent elements in \(L_1\), where \(G_0\) is the adjoint group of \(L_0\). An approach to compute these orbits is described in [DFG15]; the first step is to construct, up to \(G_0\)-conjugacy, all carrier subalgebras of \(L\). Functions `CarrierAlgsForNilpOrbsInZGrading` (4.2-2) and `CarrierAlgsForNilpOrbsInZmGrading` (4.2-3) do this in the case that \(L\) is a split real form of a complex simple Lie algebra.

##### 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 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.

##### 4.2-2 CarrierAlgsForNilpOrbsInZGrading

`‣ CarrierAlgsForNilpOrbsInZGrading`( type, rank, d ) | ( attribute )

Gives a record containing (up to conjugacy) the carrier algebras of the real theta group specified by the input: here type and rank are the type and rank of the simple real Lie algebra (split real form) where everything happens; d is a list of degrees of the simple roots, defining a \(\mathbb{Z}\) grading on the Lie algebra. The output is a record with the following entries: L the Lie algebra, grad the grading that was used (different format for Z-grading, L0 the 0-component of the grading, Hs the Cartan subalgebras of L0 that are used, cars the carrier algebras. Here cars is a list of lists; for each Cartan subalgebra of L0 there is one list: the first corresponds to the split Cartan subalgebra, and so has just the complex carrier algebras (which are also real), the other lists contain lists as well, for each complex carrier algebra (i.e., for each element of the first list) there is a list containing the real carrier algebras which are strongly \(H_i\)-regular, and over the complex numbers conjugate to the given complex carrier algebra. Furthermore, a carrier algebra is given by a record, containing the fields g0, gp (positive degree), and gn (negative degree).

    gap> ca:=CarrierAlgsForNilpOrbsInZGrading("G",2,[1,0]);
    rec(
     Hs  := [ <Lie algebra of dimension 2 over SqrtField>,
              <Lie algebra of dimension 2 over SqrtField> ],
      L  :=   <Lie algebra of dimension 14 over SqrtField>,
      L0 :=   <Lie algebra of dimension 4 over SqrtField>,
    cars :=  [
              [rec( g0 := [ v.13+(3)*v.14 ], gn := [ [ v.9 ] ], gp := [ [ v.3 ] ] )]
                ,[ [  ] ]
             ],
    grad := rec( g0 := [ v.2, v.8, v.13, v.14 ],
                   gn := [ [ v.7, v.9 ], [ v.10 ], [ v.11, v.12 ] ],
                   gp := [ [ v.1, v.3 ], [ v.4 ], [ v.5, v.6 ] ] ) )

##### 4.2-3 CarrierAlgsForNilpOrbsInZmGrading

`‣ CarrierAlgsForNilpOrbsInZmGrading`( type, rank, m0, str, num ) | ( attribute )

Gives a record (up to conjugacy) containing the carrier algebras of the real theta group specified by the input: here type and rank are the type and rank of the simple real Lie algebra (split real form) where everything happens; m0 is the order of the automorphism defining the grading, str is "inner" or "outer", depending on whether the automorphism is inner or not, num the num-th automorphism in the list FiniteOrderInnerAutomorpisms( type, rank, m0 ) or FiniteOrderOuterAutomorphisms( type, rank, m0, 2 ) which is used to define the grading. The output is as for `CarrierAlgsForNilpOrbsInZGrading` (4.2-2), with the exception that the record entry grad is a list with m0 entries, the i-th entry containing the basis for the i-th component of the grading.

    gap> gap> ca:=CarrierAlgsForNilpOrbsInZmGrading("G",2,2,"inner",1);
    rec(
     Hs := [ <Lie algebra of dimension 2 over SqrtField>,
             <Lie algebra of dimension 2 over SqrtField>,
             <Lie algebra of dimension 2 over SqrtField>,
             <Lie algebra of dimension 2 over SqrtField> ],
      L  :=  <Lie algebra of dimension 14 over SqrtField>,
      L0 :=  <Lie algebra of dimension 6 over SqrtField>,
    cars := [
             [rec( g0 := [ v.13+v.14 ], gn := [ [ v.11 ] ], gp := [ [ v.5 ] ] ),
              rec( g0 := [ v.13, v.14 ], gn := [ [ v.2, v.10 ] ],
                   gp := [ [ v.4, v.8 ] ] ),
              rec( g0 := [ v.13+(3/2)*v.14 ], gn := [ [ v.10 ] ],
                   gp := [ [ v.4 ] ] ),
              rec( g0 := [ v.1, v.7, v.13, v.14 ],
                   gn := [ [ v.8, v.9, v.10, v.11 ], [ v.12 ] ],
                   gp := [ [ v.2, v.3, v.4, v.5 ], [ v.6 ] ] ),
              rec( g0 := [ v.13, v.14 ],
                   gn := [ [ v.2, v.9 ], [ v.7 ], [ v.10 ], [ v.12 ], [ v.11 ] ],
                   gp := [ [ v.3, v.8 ], [ v.1 ], [ v.4 ], [ v.6 ], [ v.5 ] ] ) ],
             [ [  ], [  ], [  ], [  ], [  ] ], [ [  ], [  ], [  ], [  ], [  ] ],
             [ [  ], [  ], [  ], [  ], [  ] ] ],
     grad := [ [ v.1, v.6, v.7, v.12, v.13, v.14 ],
             [ v.2, v.3, v.4, v.5, v.8, v.9, v.10, v.11 ] ] )

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