Generalized Formal Degree

For a real or p-adic connected reductive group G, Harish-Chandra introduced the Plancherel measure on the tempered dual Ĝtemp and founded the Plancherel formula [7, 18] relating functions on G to functions on Ĝtemp. While the Plancherel measure is equal to the formal degree for square-integrable representations, there has been no similar interpretation for tempered but nonsquare-integrable representations because formal degree is not defined for them yet. This paper intends to generalize formal degree to all unitary representations. The traditional theory of formal degree deals only with square-integrable representations of G on Hilbert spaces because it involves an integral of matrix coefficients that diverges for nonsquare-integrable representations. Let Z be the center of G, and π an irreducible unitary action of G on a Hilbert space Vπ . When π is square-integrable,