A formula for distribution trace characters on nilpotent Lie groups
Authors: Corwin L. , Greenleaf F. P. 
Published: June 2010

The distribution trace character

θ
π

{\theta _\pi }

of an irreducible representation

π
\pi

on a simply connected nilpotent Lie group

N
N

is described as a superposition of unitary characters on certain subgroups, in terms of the canonical objects introduced by R. Penney [8]. For

l
∈

n

∗

l \in {\mathfrak {n}^*}

, define

h

1

(
l
)
=

{\mathfrak {h}_1}(l) =

smallest ideal containing the radical

r

(
l
)

\mathfrak {r}(l)

, and

h

k
+
1

(
l
)
=

h

1

(
l

|

h

k

(
l
)
)

{\mathfrak {h}_{k + 1}}(l) = {\mathfrak {h}_1}(l|{\mathfrak {h}_k}(l))

. These subalgebras terminate in a subordinate subalgebra

h

∞

(
l
)

{\mathfrak {h}_\infty }(l)

after finitely many steps. If

H
∞

=
exp
⁡
(

h

∞

)

{H_\infty } = \exp ({\mathfrak {h}_\infty })

,

X

∞

=
(

e

2
π
i
l

)
∘
log
⁡

|

H
∞

{\mathcal {X}_\infty } = ({e^{2\pi il}}) \circ \log |{H_\infty }

, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper X Subscript normal infinity Baseline comma upper H Subscript normal infinity Baseline right-parenthesis dot n equals left-parenthesis script upper X Subscript normal infinity Baseline dot n comma upper H Subscript normal infinity Baseline dot n right-parenthesis">

(

X

∞

,

H
∞

)
⋅
n
=
(

X

∞

⋅
n
,

H
∞

⋅
n
)

({\mathcal {X}_\infty },{H_\infty }) \cdot n = ({\mathcal {X}_\infty } \cdot n,\;{H_\infty } \cdot n)

, where

H
∞

⋅
n
=

n

−
1

H
∞

n

{H_\infty } \cdot n = {n^{ - 1}}{H_\infty }n

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript normal infinity Baseline dot n left-parenthesis h Superscript prime Baseline right-parenthesis equals script upper X Subscript normal infinity Baseline left-parenthesis n h prime n Superscript negative 1 Baseline right-parenthesis on upper H Subscript normal infinity Baseline dot n">

X

∞

⋅
n
(

h
′

)
=

X

∞

(
n

h
′

n

−
1

)

 on 

H
∞

⋅
n

{\mathcal {X}_\infty }\cdot n(h’) = {\mathcal {X}_\infty }(nh’{n^{ - 1}}){\text { on }}{H_\infty } \cdot n

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle theta Subscript pi Baseline comma phi mathematical right-angle equals integral Underscript upper K Subscript normal infinity Baseline minus upper N Endscripts mathematical left-angle left-parenthesis script upper X Subscript normal infinity Baseline comma upper H Subscript normal infinity Baseline right-parenthesis dot n comma phi mathematical right-angle d ModifyingAbove n With dot">

⟨

θ
π

,
ϕ

⟩

=

∫

K
∞

∖
N

⟨

(

X

∞

,

H
∞

)
⋅
n
,
ϕ

⟩

d

n
˙

\left \langle {{\theta _\pi },\phi } \right \rangle = {\smallint _{{K_\infty }\backslash N}}\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle d\dot n

, where the pair

(

X

∞

,

H
∞

)
⋅
n

({\mathcal {X}_\infty },{H_\infty }) \cdot n

is regarded as the tempered distribution
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle left-parenthesis script upper X Subscript normal infinity Baseline comma upper H Subscript normal infinity Baseline right-parenthesis dot n comma phi mathematical right-angle equals integral Underscript upper H Subscript normal infinity Baseline Endscripts script upper X Subscript normal infinity Baseline left-parenthesis h right-parenthesis phi left-parenthesis n Superscript negative 1 Baseline h n right-parenthesis d h">

⟨

(

X

∞

,

H
∞

)
⋅
n
,
ϕ

⟩

=

∫

H
∞

X

∞

(
h
)
ϕ
(

n

−
1

h
n
)
d
h

\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle = {\smallint _{{H_\infty }}}{\mathcal {X}_\infty }(h)\phi ({n^{ - 1}}hn)dh

, and where

k

∞

=
{
X
∈

n

:
l
[

h
∞

,
X
]
=
0
}

{\mathfrak {k}_\infty } = \{ X \in \mathfrak {n}:l[{h_\infty },X] = 0\}

gives the stabilizer of the pair

(

X

∞

,

H
∞

)

({\mathcal {X}_\infty },{H_\infty })

. The integral over

K
∞

∖
N

{K_\infty }\backslash N

is absolutely convergent for any Schwartz function

ϕ
\phi

on

N
N

.