Square integrable representations and a Plancherel theorem for parabolic subgroups

Abstract

Let G be a semisimple Lie group with Iwasawa decomposition G = K A N G\, = \,KAN . Let g 0 = f 0 + a + n {\mathfrak {g}_0} \,=\, {\mathcal {f}_0} \,+\, \mathfrak {a} \,+\, \mathfrak {n} be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type A 1 {A_1} or A 2 {A_2} . The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the A 1 {A_1} case for the solvable subgroup NA by combining Mackey’s “Little Group” method with an idea due to C. C. Moore: we find an operator D, defined on the C ∞ {C^\infty } functions on NA with compact support, such that \[ ϕ ( e ) = ∫ ( N A ) ∧ tr ( D π ( ϕ ) ) d μ ( π ) \phi (e) = \int _{{{(NA)}^\wedge }} {{\text {tr}}} (D\pi (\phi ))d\mu (\pi ) \] where ( N A ) ∧ {(NA)^ \wedge } is the unitary dual, e is the identity, and μ \mu is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the A 1 {A_1} case, D involves fractional powers of the Laplace operator and hence is not a differential operator.