What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:58:50Z http://mathoverflow.net/feeds/question/111564 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111564/what-is-the-plancherel-measure-for-textrmsl-3-mathbbq-p What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$? anstei 2012-11-05T14:59:33Z 2012-11-06T07:14:32Z <p>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.</p> http://mathoverflow.net/questions/111564/what-is-the-plancherel-measure-for-textrmsl-3-mathbbq-p/111625#111625 Answer by Paul Broussous for What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$? Paul Broussous 2012-11-06T07:14:32Z 2012-11-06T07:14:32Z <p>This is not a complete answer but only some hints that could help you.</p> <p>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 :</p> <p>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.</p> <p>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. </p> <p>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. </p>