Spherical functions for sl(2,Q_p)

I kindly would like to ask you the following- I am refering to page 175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms ..."

My question to which I would kindly ask you to answer me is : Are there unitary representations of SL(2, Q_P) of type I (that is with spherical functions) that are not in the continuous series. Gelfand is ambiguous in the text by saying that he will give examples of such class 1 unitary representations in the continuous series. If there other type I unitary reps, in which series are they (the discrete series perhaps?).

What would be the trace formula for this class 1, non-principal series?

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Being type I and having spherical functions are different things. Please clarify your question. –  Marc Palm Jun 2 '14 at 9:59
Sorry by class I, I meant spherical. –  Florin Radulescu Jun 6 '14 at 22:20

The theorem of Borel-Casselman-Matsumoto from about 1976 shows, more generally, that any admissible representation of a $p$-adic reductive group with an Iwahori-fixed vector imbeds in an unramified principal series. In particular, an admissible representation with a spherical vector is in this class, that is, is a subrepresentation (and quotient) of an unramified principal series.

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Does this imply that every unitary spherical representation is continuously occurring in the left regular representation of p-adic group? –  Florin Radulescu Jun 5 '14 at 9:46
Yes... while noting that proving admissibility of (irreducible) unitaries of p-adic reductive groups is non-trivial. (Harish-Chandra and others reduced to the supercuspidal case, finished by Bernstein c. 1975.) –  paul garrett Jun 5 '14 at 12:30
So this also implies discrete (square integrable) unitary series contains no spherical? If this is true, I can not imagine how the discrete series then occurs in the L^2 space of the left regular representation of G? –  Florin Radulescu Jun 5 '14 at 13:23
Correct, discrete series do not have spherical vectors. Also, I'd say it is correct that it is non-trivial to see how discrete series occur in $L^2(G)$ or $L^2(G/Z)$, indeed. For $SL_2$, pairing with $O(1,1)$ (in effect) and Segal-Shale-Weil/oscillator constructs some of these. In effect, this is what is done in Gelfand-et-al, though I don't remember whether they really convincingly prove exhaustion. –  paul garrett Jun 5 '14 at 14:04
I've not carried out the computation myself, but I'd strongly expect that the discrete series repns produced from local oscillator repns for $SL(2)$ would admit essentially identical trace computations, via that presentation of them. –  paul garrett Jun 5 '14 at 15:49

Be careful that among the irreducible unitary reps also the trivial representation has this property. That's why Paul Garrett says "embeds" into a prinicpal series, so you get not only unramified unitary principal series, but also the trivial representations.

Paul Garrett doesn't address the trace formula, so I will try to do this:

Let $F$ be a local field, $B$ the Borel in $SL(2)$, and $K$ being a maximal compact subgroup of $SL(2,F)$ with $SL(2,F) = B(F) \cdot K$.

Let $\mu$ be a complex character and $\pi(\mu) = Ind_{B(F)}^{SL(2,F)} \mu$ be the normalized induction (=principal series), then $$tr\; \pi(\mu) (\phi) = \int\limits_{B(F)} \int_K \phi(k^{-1}bk) \mu(b) \Delta(b)^{1/2}\; dk \; db,$$ where the right Haar measures are related as $$\int_G d g = \int_B \int_K dkdb.$$ In your case, you could take $\phi$ being right and left $K$-invariant, hence can replace the $K$-integral by $vol(K)$. So yes, there is a strong analogy between p-adic and Lie groups.

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Thks a lot for answs. Still very puzzled if a spherical representation for SL(2,Q_p) in the non-principal series, has an associated character. Does the procedure which gives the character for a spherical in the principal series, diverge for representations not in the left regular. The procedure, would compute the "trace at g" by taking the trace of images of convolutors by characteristic functions of sets A, containing g, normalizing by Haar measure of A, and then taking the limit when A shrinks to g. Does this procedure diverge for non principal, since they are not in the Planch. series? –  Florin Radulescu Jun 10 '14 at 9:01
Being in the left regular correspond to being tempered, so you will not get the trivial representation nor the non-tempered principal series. But it does not matter, my computation assume $\mu$ general, not necessarily unitary. You probably means character seen as a function, which is locally integrable - not as a distribution. Yes you can do a similiar things whether as long as you assume admissible. –  Marc Palm Jun 10 '14 at 9:10
But are these characters for non-principal, complementary unitary spherical being computed (even as distribution)?- It is my impression that either in Sally, either Gelfand do not make the computation –  Florin Radulescu Jun 10 '14 at 9:17
I gave the formula for the distribution. It is general. If you want to find a function on $G$ corresponding to this, I can't tell where to look for explicit computations. My feeling is that finding some useful computational analogy can only done to some extent. $P$-adic Gamma factors do not look like real or complex Gamma-functions. –  Marc Palm Jun 10 '14 at 12:59