Introduction on \(K\) orbits on \(G/B\)

In order to explain how atlas uses the KGB machinery we need to discuss the theory a little bit. KGB is the heart of the theory on which the software is based and it takes some time to get used to understanding how it works.

Previously we focused on the minimal principal series representations of a Lie group, which are defined using a character on the maximally split Cartan subgroup. In the coming sections we will talk about the discrete series representations which are at the other end of the spectrum of representations in the sense that they are based on a compact Cartan subgroup of G and with some rather different features. After that, we will talk about the more general representations defined using characters of intermediate Cartan subgroups and which are in a way a combination of the two first constructions. So, understanding these separately will help us see the general case.

Recall that a parameter is a triple \(p=(x,\lambda, \nu)\), where \(x \in K\backslash G / B\) and which determines an involution \(\theta _x\) of the Cartan subgroup.

\[\lambda \in(X^* +\rho )/(1-{\theta }_x)X^*\]

\[\nu \in {X}_{\mathbb Q} ^* /(1+{\theta }_x ) X_{\mathbb Q}^*\cong (X_{\mathbb Q} ^*)^{-\theta _x}\]

So we often see \(\nu\) as being an element in the space on the right hand side. But we can also think of it as being in the quotient space.

So the infinitesimal character can be written as

\[\gamma =\frac{1+\theta _x}{2}\lambda + \frac{1-\theta _x }{2}\nu\]

\[=\frac{1+\theta _x}{2}\lambda +\nu\]

since \(\nu\) is normally fixed by \(1-\theta_x\).

Recall that roughly the \(KGB\) element x determines a \(G(\mathbb R)\)-conjugacy class of Cartan subgroups \(H\) and we denote by \(\theta _x\) the Cartan involution of \(H(\mathbb R)\)

Previously we focused on the case when \(\theta _x\) is acting by \(-Id\) This corresponded to the split Cartan subgroup \(H(\mathbb R)={\mathbb R}^{*n}\)

In that case the parameter \(\lambda \in (X^* + \rho )/2X^*\) gives a character of

\[H(\mathbb R)^{\theta _x} ={(\mathbb Z)/2\mathbb Z}^n\]

So we get a character of the compact piece of the Cartan subgroup.

Now \(\nu \in {X}_{\mathbb Q} ^*\) and we had constructed the induced representation

\[Ind_B ^G (\chi \otimes \nu),\]

where \(\chi=\rho -\lambda \in X^*\), since \(\lambda \in X^* + \rho\).

So, we have a well defined pairing between \(X^*\) and \(X_*\):

\[\chi(exp(2\pi i\mu ^{\vee}))=exp(2\pi i(<\rho -\lambda ,\mu^{\vee}>),\]

with \(\mu^{\vee}\in \frac{1}{2}X_*\).

This pairing is the value of the character on the compact part of the Cartan subgroup and it equals \(\pm 1\)

\(KGB\) elements

In order to talk about all the representations of our group, we fix once and for all a Cartan subgroup \(H\) included in a Borel subgroup \(B\). This avoids having to work with different Cartan subgroups and roots and different choices of identifications. We will be working with the same Cartan subgroup all the time.

The choice of Cartan subgroup does not matter. We can think of \(H\) as close to diagonal and \(B\) upper triangular.

Now let us take a base point element \(x_b \in G\), so that \(x_b ^2 \in Z(G)\). Attached to \(x_b\) is the involution \(\theta=int(x_b)\).

By Cartan’s theory of real forms this gives a real form, \(G(\mathbb R)\), of \(G\). Denote by \(K=G^{\theta}\), the complexified maximal compact subgroup of \(G(\mathbb R)\).

In other words the real form obtained by the involution is the one whose complexified maximal compact subgroup is \(K\).

We are interested in the following space

\[K\backslash G/B=\{K-\text{orbits on the flag variety G/B }\}\]

\[=\{K \text{-conjugacy classes of Borel subgroups of G }\}.\]

That is \(G/B\) is a complex projective variety and \(K\) is an algebraic group acting on it with finitely many orbits. And \(G/B\) is isomorphic to the space of Borel subgroups of \(G\) and the \(K\)-orbits are the Borel subgroups up to conjugacy by \(K\).

Parametrization Theorem

The finitenes of the \(K\) orbits on the flag variety is what makes it easy for atlas to work with them. In fact, we will use the following result

\[K\backslash G/B \leftrightarrow \{x\in Norm_G (H)|x{\backsim }_G x_b\}/H\]

As we said, this is a finite set. The map is given as follows:

If \(x=gx_b g^{-1}\) then associate to x the element \([g^{-1}Bg]\), the \(K\)-conjugacy class of the Borel subgroup.

Now we check the map is well defined on \(K\)-conjugcy classes of Borel subgroups:

\[x=(gk)x_b (gk)^{-1} \mapsto [k^{-1}(g^{-1}Bg)k]=[gBg]\]

On the right hand side the brackets mean \(K\)-conjugacy class. So, by conjugating by \(K\) we get the same conjugacy class of Borel subgroups.

Parameter Set

Denote the above set by

\[\mathcal X =\{x\in Norm_G (H) | x{\backsim }_G x_b\}/H\]

(Note: this set is really denoted \(\mathcal X [x_b]\) in other sources and \(\mathcal X\) is the collection of all the sets for all the base points; which give all strong real forms of \(G\). The above set only has to do with one strong real form.

NOTE: FOR MORE INFORMATION ON STRONG REAL FORMS SEE: “Algorithms for Representation Theory of Real Reductive Groups”, by Adams and du Cloux, in www.liegroups.org/papers/.

Since this set is finite, it makes sense to have an atlas command KGB that will give a finite list of the elements in \(\mathcal X\)

\[KGB \leftrightarrow \{x_0 , \dots ,x_{n-1} \}\]