I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex special linear groups and on $p$-adic general linear groups. Any help would be appreciated.
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$\begingroup$ What do you need it for? Are you interested in it for its own sake? Do you know the Plancherel measure for $SL(2)$ or $GL(2)$? A wild guess is that beautiful formulas are only available for Hecke algebras associated to distinguished strata and that in principle there is an algorithm because the characters and representations of SL(N) have been classified. $\endgroup$– Marc PalmCommented Nov 6, 2012 at 13:05
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$\begingroup$ Also see Paul Garrett's answer here: mathoverflow.net/questions/70801/… $\endgroup$– Marc PalmCommented Nov 6, 2012 at 13:06
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$\begingroup$ In short, even Paul Garrett has no reference for the Plancherel of GL(n), which is easier than SL(n). $\endgroup$– Marc PalmCommented Nov 6, 2012 at 13:10
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$\begingroup$ I'm studying the local parameters of $\textrm{SL}_3(\mathbb{Z})$-automorphic Hecke eigenfunctions. We already have a measure $\mu_T$ satisfying $lim_{T \to \infty} \int_{\textrm{SL}_3(\mathbb{Q}_p)-\textrm{spectrum}} \chi_\rho(\pi) d \mu_T(\pi) = \alpha_p(\rho)$ and I'm looking for the limiting measure $\lim_{T \to \infty} \mu_T$. A natural candidate is the Plancherel measure. $\endgroup$– ansteiCommented Nov 6, 2012 at 15:25
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$\begingroup$ What is a $SL_3(\mathbb{Z})$-automorphic Hecke eigenfunction? You do not mean a $SL_3(\mathbb{Z}_p)$-biinvariant function, because then everything is fairly easy and I can give the answer. $\endgroup$– Marc PalmCommented Nov 6, 2012 at 15:35
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1 Answer
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This is not a complete answer but only some hints that could help you.
Bushnell, Kutzko and Henniart have shown, for a general reductive group, that the restriction of the Plancherel measure to each block of the Bernstein decomposition may be computed via isomorphisms of Hecke algebras :
Bushnell, Colin J.; Henniart, Guy; Kutzko, Philip C. Types and explicit Plancherel formulæ for reductive $p$-adic groups. On certain $L$-functions, 55–80, Clay Math. Proc., 13, Amer. Math. Soc., Providence, RI, 2011.
In the following paper, Bushnell and Kutzko show that for certain blocks of ${\rm SL}(N)$the Hecke algebra is isomorphic to a Iwahori Hecke algebra for some ${\rm SL}(N')$ over another field.
Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. I. Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 261–280.
answered Nov 6, 2012 at 7:14
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$\begingroup$ Thank you, I will look up these references and see what I can get from those. $\endgroup$– ansteiCommented Nov 6, 2012 at 13:12