classification of irreducible admissible (g,K)-module for GL(3,R)
Is there a classification of irreducible admissible (g,K)-module for GL(3,R)?
For GL(2,R) we have principal series, discrete series and etc. Is there such a result for GL(3,R) or GL(n,R)?
Best Answer
For a general real reductive group, all irreducible admissible $({\mathfrak g},K)$-modules are quotients of parabolically-induced discrete series (or limits thereof) representations (where we allow "trivial" parabolic induction ($P=G$) for discrete series on the group). See Theorem 14.92 in Knapp's Representation Theory of Semisimple Groups. This is a refinement of the Langlands Classification (which replaces "discrete series" with "tempered"). Knapp's paper "Local Langlands Correspondence: the archimedean case", in volume 2 of Motives, PSPM 55, gives an explicit classification for $GL_n$ (over $\mathbb R$ and $\mathbb C$). Also see Moeglin's article "Representations of GL(n) over the Real Field" in Representation Theory and Automorphic Forms, PSPM 61.
For $GL_n(\mathbb R)$, we can say that given an irreducible admissible $({\mathfrak g},K)$-module $V$, there exists a parabolic subgroup $P=MN$ of $GL_n$ with block sizes either $1$ or $2$ (since $GL_n$ only has discrete series for $n=1$ or $2$), and a discrete series representation $\sigma$ of $M$, such that $V$ is isomorphic to the unique quotient of the $({\mathfrak g},K)$-module underlying ${\rm Ind}_P^G(\sigma,s)$, where $s$ is a tuple of complex parameters, one for each block in $M$. Further analysis can tell you when two induced representations give you the same $({\mathfrak g},K)$-module, and when the induced representation is irreducible.