In order to talk abou representations of A fixed Lie Group using
atlas we need to talk about general parameters and more concretely
we will talk about everything between the discrete series and the
minimal principal series representations and how to uderstand their
parameters. Later we will talk about composition series, character
formulas and Kazhdan-Lusztig Polynomials.
Recall that we had a parameter
p determined by a triple
p=(x, lambda, nu), where
so the Cartan involution is determined by
So that each term in the expression of the infinitesimal character
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
The point is that the definition of equivalence is chosen so that we have the following
There are canonical bijections between
- Parameters for \(G\)
- Standard modules for \(G\)
- 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 and a cuspidal Parabolic subgroup \(P\). 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.