Let $G$ be a reductive p-adic group with a chosen Haar measure $dg$. The Plancherel measure is the measure $\mu$ on the set of (tempered) irreducible representations of $G$ such that for any locally constant compactly supported function $f$ on $G$ we have $$ f(e)=\int\limits_{\pi\in Irr(G)} Tr(\pi(f)dg) d\mu (\pi). $$

In particular, we may restrict $\mu$ to the set of tempered principal series representations, which a are parameterized by unitary characters of the maximal torus $T$ of $G$.

$\mathbf{Question:}$ Is there a good formula for the Mellin transform of this measure (this is a function on $T$)? I am mostly interested in the case $G=SL(2, F)$ for a local non-archimedian field $F$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.