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<p><a id="X845E3A7E87C93239" name="X845E3A7E87C93239"></a></p>
<div class="ChapSects"><a href="chap4.html#X845E3A7E87C93239">4 <span class="Heading">Real nilpotent orbits</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A6BB7967FE7ABA4">4.1 <span class="Heading">Nilpotent orbits in real Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8424BB44791EAA48">4.1-1 NilpotentOrbitsOfRealForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A05B2957A625D85">4.1-2 RealCayleyTriple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X804830757E5971E9">4.1-3 WeightedDynkinDiagram</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BE2BD6B79367FC8">4.2 <span class="Heading">The real Weyl group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8196CAF57F4CD8C7">4.2-1 RealWeylGroup</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Real nilpotent orbits</span></h3>

<p><a id="X7A6BB7967FE7ABA4" name="X7A6BB7967FE7ABA4"></a></p>

<h4>4.1 <span class="Heading">Nilpotent orbits in real Lie algebras</span></h4>

<p><strong class="pkg">CoReLG</strong> 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, <strong class="pkg">CoReLG</strong> will first read the database of nilpotent orbits.</p>

<p><a id="X8424BB44791EAA48" name="X8424BB44791EAA48"></a></p>

<h5>4.1-1 NilpotentOrbitsOfRealForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentOrbitsOfRealForm</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> 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 <var class="Arg">L</var>. For this function to work, <var class="Arg">L</var> must be defined over <var class="Arg">SqrtField</var>.</p>


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

<p><a id="X7A05B2957A625D85" name="X7A05B2957A625D85"></a></p>

<h5>4.1-2 RealCayleyTriple</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealCayleyTriple</code>( <var class="Arg">o</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">o</var> is a nilpotent orbit constructed by <code class="func">NilpotentOrbitsOfRealForm</code> (<a href="chap4.html#X8424BB44791EAA48"><span class="RefLink">4.1-1</span></a>) of a simple real Lie algebra. This function returns a real Cayley triple <var class="Arg">[ f, h, e ]</var> corresponding to the orbit <var class="Arg">o</var>. The third element <var class="Arg">e</var> is a representative of the orbit.</p>


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

<p><a id="X804830757E5971E9" name="X804830757E5971E9"></a></p>

<h5>4.1-3 WeightedDynkinDiagram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WeightedDynkinDiagram</code>( <var class="Arg">o</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">o</var> is a nilpotent orbit constructed by <code class="func">NilpotentOrbitsOfRealForm</code> (<a href="chap4.html#X8424BB44791EAA48"><span class="RefLink">4.1-1</span></a>) 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 <var class="Arg">o</var> lies.</p>

<p><a id="X7BE2BD6B79367FC8" name="X7BE2BD6B79367FC8"></a></p>

<h4>4.2 <span class="Heading">The real Weyl group</span></h4>

<p><a id="X8196CAF57F4CD8C7" name="X8196CAF57F4CD8C7"></a></p>

<h5>4.2-1 RealWeylGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealWeylGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealWeylGroup</code>( <var class="Arg">L</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra with Cartan subalgebra <var class="Arg">H</var>. (If <var class="Arg">H</var> is not given, then <var class="Arg">CartanSubalgebra(L)</var> will be taken.) This function returns the real Weyl group <span class="SimpleMath">N_G(H)/C_G(H)</span> associated with <var class="Arg">H</var>, where <span class="SimpleMath">G</span> is the connected component of the group of real points of the complex adjoint group of <var class="Arg">L</var>. 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.</p>


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