# 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.