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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)?

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  • $\begingroup$ There is no etc. for GL(2,R)... $\endgroup$
    – Marc Palm
    Commented Dec 10, 2012 at 12:30

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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.

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    $\begingroup$ Which of these are are cohomological? $\endgroup$
    – David Loeffler
    Commented Oct 5, 2011 at 20:53
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    $\begingroup$ David, you might want to look at Speh's "Unitary representations of Gl(n,R) with non-trivial (g,K)-cohomology", or Vogan and Zuckerman "Unitary representations with non-zero cohomology", or, for more general statements, Vogan and Knapp "Cohomological Induction and Unitary Representations". $\endgroup$
    – B R
    Commented Oct 5, 2011 at 22:12
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    $\begingroup$ Dear David, For tempered representations, they are cohomological precisely when they are inductions of discrete series from the "fundamental parabolic", i.e. from the maximal parabolic whose Levi admits discrete series. So for $GL_3(\mathb R)$, the tempered reps. that will be cohomological are those that are inductions of $GL_2(\mathbb R)$-discrete series from a $(2,1)$ parabolic. (This is somewhere in Borel--Wallach.) Regards, Matt $\endgroup$
    – Emerton
    Commented Oct 6, 2011 at 4:57

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