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I have a couple of elementary question about the limit of discrete series representation (=LDS) of $GL(2, \mathbb{R})$. I find the topic fairly confusing.

Write $I(\mu_1, \mu_2, s)$ the prinicpal series and $M :I(\mu_1, \mu_2, s) \rightarrow I(\mu_1, \mu_2, -s)$ for the intertwiner. Here $\mu_j$ is either the trivial character or the sign character of $\mathbb{R}^\times$.

  1. The reps $I(sign, 1,0)$ and $I(1, sign,0)$ are the LDS?

  2. Is the LDS a proper subquotient?

  3. Is the LDS stable under character twist by $sign \circ \det$?

  4. What does the intertwiner $M:I(sign, 1,0) \rightarrow I(1, sign,0)$ do?

  5. Why do we call it LDS, when it is simply a discrete series representation?

  6. Is there a good reference for this topic simply for $GL(2, \mathbb{R})$?

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up vote 3 down vote accepted

Edited, in response to questioner's follow-up: In general, limits of discrete series are not whole principal series, but subquotients. But in the present example, it is possible to get the impression that the "odd/ramified" principal series $s=0$ (with intertwining $s\rightarrow -s$ normalization) is the "limit of discrete series", since, unlike the unramified principal series, there are no "lower" $K$-types to be obviously left out.

It may be somewhat simpler to look at $SL(2,\mathbb R)$ rather than $GL(2)$, since the former has a unique "odd" principal series for given complex parameter $s$, while $GL(2)$ has three. In any case, the diagonal $-1,1$ element in $GL(2)$ is not so much the distinguisher of "odd" principal series.

Rather, the odd/even business is already visible in $SL(2,\mathbb R)$ in the diagonal $\pm 1_2$ lying in the intersection $P\cap K$ of the standard (upper-triangular) parabolic and the standard maximal compact $SO(2)=SO(2,\mathbb R)$. That is, the odd principal series induce the character $\pmatrix{a & * \cr 0 & a^{-1}}\rightarrow |a|^{2s-1}\cdot \hbox{sgn}(a)$.

The terminology is inherited because, at the Lie algebra level, and in some models, the LDS is indeed "the same as" a genuine discrete series. However, its (matrix) coefficient functions are not in $L^2(G)$, so it doesn't qualify on that account.

I think Lang's old book on $SL(2,\mathbb R)$ talks about such details. Gelfand-Graev-PiatetskiShapiro's book on automorphic functions does, and usefully juxtaposes the p-adic analogues of things. I'd think Knapp's "Repn ... by example" would do these examples. Varadarajan's small gray volume "Harmonic analysis on ... (semi-simple? reductive?) groups" emphasizes such things, and gives computational details.

The sort of computation necessary to understand the intertwiners directly is also illustrated, although in the even principal series case, in some notes of mine: Similar devices will succeed for odd principal series.

Edit: specifically in response to the questioner's follow-up, directly computing the effect of natural intertwining operators for $SL(2,\mathbb R)$, I seem to find that at $s=0$ (in the questioner's normalization, so the functional equation is $s\rightarrow -s$) the "positive" $K$-types are the kernel, and the "negative" $K$-types are the image. So the "limit of discrete series" seems not to be the whole principal series.

But this leaves something to be done when we return to $GL(2,\mathbb R)$: specifically, the element $\varepsilon=\pmatrix{-1 & 0 \cr 0 & 1}$ interchanges the non-zero $\pm$ $SO(2)$-types. If we too-glibly presume that the repn space for the $SL(2,\mathbb R)$ principal series is the same as that for $GL(2,\mathbb R)$, this seems to say that the $s=0$ odd/ramified principal series is irreducible, since the positive and negative $K$-types are now bound to each other.

Yet the direct computation of intertwinings (which are also $GL(2,\mathbb R)$ integertwinings, given by the same integrals!) say there is a kernel, and that the image is proper. How to reconcile this?

The only possibility is that the multiplicities of the $SO(2)$-types in the ramified $GL(2,\mathbb R)$ principal series are not $1$, but are $2$. (Here I'm thinking of the odd principal series that maps to itself under the intertwining. I've not thought through the analogous issues for the other two ramified principal series for $GL(2,\mathbb R)$.)

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Thank you for your answer. But what puzzles me mostly, is how it can be a proper subrepresentation if the restriction to $O(2)$ are the same as for the ramified principal series representation. No $K$-type is substracted. – Jul 22 '12 at 16:23
I see your point: for SL(2), "half" the K-types appear, and for GL(2), the diag(-1,1) (does it?) puts the two pieces together, so the "full" LDS has all the K-types of the whole (odd/ramified) principal series. Now I am a bit interested to do the computation of the intertwinings, because I suspect that the self-intertwinings in the odd/ramified case have kernel(s) consisting of "half" the K-types. This should be ascertainable in finite time. – paul garrett Jul 22 '12 at 16:31
I supsect this because missing $K$-types would allow the construction of pseudo coefficients for the LDS, and allow trace formulas for weight one forms. Theorem 11.15 in Knightly, Li - "traces of Hecke operators" page 164 together with the list on page 186 gives the result: No K type is missing. But they argue with the Lie algebra, so I do not see what the intertwiner is doing. In Bump "automorphic reps", the intertwiner is computed on the level of $O(2)$ types. My whole problem is how to interpret all this information, and what is so special about the LDS to not to call it PS. – Jul 22 '12 at 17:04
To reiterate some of what I added to the "answer": this is a situation in which the "slight" differences between the categories of repns matter much more than usual. My sketch indicates that, whichever category we're in, the LDS is not the whole principal series. Also, again, there is the non-$L^2$-ness of matrix coefficient functions. – paul garrett Jul 22 '12 at 18:01

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