A Plancherel formula for idyllic nilpotent Lie groups

Abstract

ABSTRACT. A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups A,, C¡, G2.Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of G ¡N. To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective A

On Kirillov's character formula.

“…Then we found, that in order, that we have (3) J <p eKp(e) (p) dv = $<f> ((J(e))'p) exp ( -/(<% p)) dv o ((J(e))' Stands for the transpose of J(e)} for an arbitrary choice of e e g and of φ e & (G), it is sufficient, but not necessary, that there exist a polarization which is also an ideal, for one (and hence for all) element of 0, or even that T be quasi-equivalent to a representation, induced by a one-dimensional character of a connected invariant subgroup of G (cf. Then we found, that in order, that we have (3) J <p eKp(e) (p) dv = $<f> ((J(e))'p) exp ( -/(<% p)) dv o ((J(e))' Stands for the transpose of J(e)} for an arbitrary choice of e e g and of φ e & (G), it is sufficient, but not necessary, that there exist a polarization which is also an ideal, for one (and hence for all) element of 0, or even that T be quasi-equivalent to a representation, induced by a one-dimensional character of a connected invariant subgroup of G (cf.…”

“…1. Assume, that ^4eg 3 -g 4 , 5 e 9 4 -9 5 > ^6 e 9 5 ; ^re is then a basis {e l9 e 29 e 3 4 + be s + ce 6 . Lemma 5.…”

“…Suppose, that S(Q) ^ 5. By s (g) <^5 we obtain R(l, e) = (1/240) ad(e) (ad (l))3 e +[/,/(/, e)] where /(/, *) s (1/720) (ad(/)) 3^ + (1/180) ad(/) (ad(e)) 2 /. We recall (cf [8],.…”

Cross sections and an explicit formula for the Plancherel measure on a nilpotent Lie group

“…For f6g*, the irreducible representation n s associated with f is constructed as follows. Carlton [1] gives a formula for/~ when G is idyllic. The functional f induces a character ~s on the Lie subgroup M s of G with m s as Lie algebra by the formula: ~s(expX) = e2~i<s, x~ (X ~ ms).…”