Introduction

Recall that we had a parameter p determined by a triple p=(x, lambda, nu), where

\[x\in K\backslash G/B \rightarrow \theta _x\]

so the Cartan involution is determined by x

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

So that each term in the expression of the infinitesimal character

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

is well defined.

Now we will talk a bit more about what the parameters mean. And there is a notion of equivalence of parameters. For more details you can go to

http://www.liegroups.org/papers/equivalenceOfParameters.pdf

The point is that the definition of equivalence is chosen so that we have the following

Theorem

There are canonical bijections between

  1. Parameters for \(G\)
  2. Standard modules for \(G\)
  3. Irreducible representations of \(G\)

Moreover, the bijections are given as follows:

\[1\rightarrow 2: p\rightarrow I(p)=Ind_P^G (\sigma ).\]

The full induced from \(\sigma =\) (limit of) discrete series This is called the standard module with parameter p.

\[2\rightarrow 3: Ind_P ^G (\sigma ) \rightarrow J\]

where \(J\) is the unique irreducible quotient of the standard module, which always exists in this setup.

Hence

\[1\rightarrow 3: p\rightarrow J(p)\]

the unique irreducible quotient of \(I(p)\)

Composition series and character formulas

Now, if we have a standard module, which is a full induced representation, it can be reducible and therefore it has a composition series, which is given by Kazhdan-Lusztig theory (KL), as a formal linear combination of irreducible modules. Conversely, given an irreducible, we have, via KL, a character formula that gives the irreducible as a formal linear combination of standard modules. The coefficients of these combinations are the KL polynomials

This is what we will do the rest of the chapter.