Plancherel measure

In mathematics , the Plancherel measure is an important concept introduced by Harish-Chandra in the representation theory of groups .

definition

Be a real reductive group . Consider the regular representation (by left and right multiplication) of on , i.e. the vector space of the functions that can be integrally integrated with respect to the hair measure . Then there is an integral decomposition ${\displaystyle G}$ ${\displaystyle G\times G}$ ${\displaystyle H=L^{2}(G,\mu _{G})}$

${\displaystyle H=\int _{\widehat {G}}H_{\omega }d\mu _{\widehat {G}}(\omega ),}$

where the dual group (i.e. the group of equivalence classes of irreducible representations of ) and is. ${\displaystyle {\widehat {G}}}$ ${\displaystyle G}$ ${\displaystyle H_{\omega }=\omega \otimes \omega ^{*}}$

The measure defined by this decomposition on the dual group is the Plancherel measure . The decomposition and thus the Plancherel measure were explicitly described by Harish-Chandra. In particular, it was proven that the carriers of the lower space of the tempered representations is contained. ${\displaystyle {\widehat {G}}}$ ${\displaystyle d\mu _{\widehat {G}}}$ ${\displaystyle d\mu _{\widehat {G}}}$

literature

• Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", Acta Mathematica, 116 (1): 1–111