On the Moore Formula of Compact Nilmanifolds
Let G be a connected and simply connected two-step nilpotent Lie group and Γ a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation IndΓG(1)....
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2009 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2009
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149106 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Moore Formula of Compact Nilmanifolds / H. Hamrouni // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 12 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860226814570397696 |
|---|---|
| author | Hamrouni, H. |
| author_facet | Hamrouni, H. |
| citation_txt | On the Moore Formula of Compact Nilmanifolds / H. Hamrouni // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let G be a connected and simply connected two-step nilpotent Lie group and Γ a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation IndΓG(1). Extending then the Abelian case.
|
| first_indexed | 2025-12-07T18:20:12Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 062, 7 pages
On the Moore Formula of Compact Nilmanifolds
Hatem HAMROUNI
Department of Mathematics, Faculty of Sciences at Sfax,
Route Soukra, B.P. 1171, 3000 Sfax, Tunisia
E-mail: hatemhhamrouni@voila.fr
Received December 17, 2008, in final form June 04, 2009; Published online June 15, 2009
doi:10.3842/SIGMA.2009.062
Abstract. Let G be a connected and simply connected two-step nilpotent Lie group and Γ
a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the
sense of Moore, of irreducible unitary representations involved in the decomposition of the
quasi-regular representation IndG
Γ (1). Extending then the Abelian case.
Key words: nilpotent Lie group; lattice subgroup; rational structure; unitary representation;
Kirillov theory
2000 Mathematics Subject Classification: 22E27
1 Introduction
Let G be a connected simply connected nilpotent Lie group with Lie algebra g and suppose G
contains a discrete cocompact subgroup Γ. Let RΓ = IndG
Γ (1) be the quasi-regular representation
of G induced from Γ. Then RΓ is direct sum of irreducible unitary representations each occurring
with finite multiplicity [3]; we will write
RΓ =
∑
π∈(G:Γ)
m(π,G,Γ, 1)π.
A basic problem in representation theory is to determine the spectrum (G : Γ) and the multi-
plicity function m(π,G,Γ, 1). C.C. Moore first studied this problem in [7]. More precisely, we
have the following theorem.
Theorem 1. Let G be a simply connected nilpotent Lie group with Lie algebra g and Γ a lattice
subgroup of G (i.e., Γ is a discrete cocompact subgroup of G and log(Γ) is an additive subgroup
of g). Let π be an irreducible unitary representation with coadjoint orbit OG
π . Then π belongs to
(G : Γ) if and only if OG
π meets g∗Γ = {l ∈ g∗, 〈l, log(Γ)〉 ⊂ Z} where g∗ denotes the dual space
of g.
Later R. Howe [4] and L. Richardson [12] gave independently the decomposition of RΓ for
an arbitrary compact nilmanifold. In this paper, we pay attention to the question wether the
multiplicity formula
m(π,G,Γ, 1) = #[OG
π ∩g∗Γ/Γ] ∀π ∈ (G : Γ)
required in the Abelian context, still holds for non commutative nilpotent Lie groups (we write
#A to denote the cardinal number of a set A). In [7], Moore showed the following inequality
m(π,G,Γ, 1) ≤ #[OG
π ∩g∗Γ/Γ] ∀π ∈ (G : Γ), (1)
where Γ is a lattice subgroup of G, and produced an example for which the inequality (1) is
strict. More precisely, he showed that
m(π,G,Γ, 1)2 = #[OG
π ∩g∗Γ/Γ] ∀π ∈ (G : Γ) (2)
mailto:hatemhhamrouni@voila.fr
http://dx.doi.org/10.3842/SIGMA.2009.062
2 H. Hamrouni
in the case of the 3-dimensional Heisenberg group and Γ a lattice subgroup. The present paper
aims to show that every connected, simply connected two-step nilpotent Lie group satisfies
equation (2). We present therefore a counter example for 3-step nilpotent Lie groups.
2 Rational structures and uniform subgroups
In this section, we summarize facts concerning rational structures and uniform subgroups in
a connected, simply connected nilpotent Lie groups. We recommend [2] and [9] as a references.
2.1 Rational structures
Let G be a nilpotent, connected and simply connected real Lie group and let g be its Lie
algebra. We say that g (or G) has a rational structure if there is a Lie algebra gQ over Q such
that g ∼= gQ ⊗ R. It is clear that g has a rational structure if and only if g has an R-basis
{X1, . . . , Xn} with rational structure constants.
Let g have a fixed rational structure given by gQ and let h be an R-subspace of g. Define
hQ = h ∩ gQ. We say that h is rational if h = R-span {hQ}, and that a connected, closed
subgroup H of G is rational if its Lie algebra h is rational. The elements of gQ (or GQ = exp(gQ))
are called rational elements (or rational points) of g (or G).
2.2 Uniform subgroups
A discrete subgroup Γ is called uniform in G if the quotient space G/Γ is compact. The
homogeneous space G/Γ is called a compact nilmanifold. A proof of the next result can be
found in Theorem 7 of [5] or in Theorem 2.12 of [11].
Theorem 2 (the Malcev rationality criterion). Let G be a simply connected nilpotent Lie
group, and let g be its Lie algebra. Then G admits a uniform subgroup Γ if and only if g admits
a basis {X1, . . . , Xn} such that
[Xi, Xj ] =
n∑
k=1
cijkXk, ∀ 1 ≤ i, j ≤ n,
where the constants cijk are all rational. (The cijk are called the structure constants of g relative
to the basis {X1, . . . , Xn} .)
More precisely, we have, if G has a uniform subgroup Γ, then g (hence G) has a rational
structure such that gQ = Q-span {log(Γ)}. Conversely, if g has a rational structure given by
some Q-algebra gQ ⊂ g, then G has a uniform subgroup Γ such that log(Γ) ⊂ gQ (see [2] and [5]).
If we endow G with the rational structure induced by a uniform subgroup Γ and if H is a Lie
subgroup of G, then H is rational if and only if H ∩ Γ is a uniform subgroup of H. Note that
the notion of rational depends on Γ.
2.3 Weak and strong Malcev basis
Let g be a nilpotent Lie algebra and let B = {X1, . . . , Xn} be a basis of g. We say that B is
a weak (resp. strong) Malcev basis for g if gi = R-span {X1, . . . , Xi} is a subalgebras (resp. an
ideal) of g for each 1 ≤ i ≤ n (see [2]).
Let Γ be a uniform subgroup of G. A strong or weak Malcev basis {X1, . . . , Xn} for g is said
to be strongly based on Γ if
Γ = exp(ZX1) · · · exp(ZXn).
Such a basis always exists (see [5, 2, 6]).
On the Moore Formula of Compact Nilmanifolds 3
A proof of the next result can be found in Proposition 5.3.2 of [2].
Proposition 1. Let Γ be uniform subgroup in a nilpotent Lie group G, and let H1 $ H2 $
· · · $ Hk = G be rational Lie subgroups of G. Let h1, . . . , hk−1, hk = g be the corresponding
Lie algebras. Then there exists a weak Malcev basis {X1, . . . , Xn} for g strongly based on Γ and
passing through h1, . . . , hk−1. If the Hj are all normal, the basis can be chosen to be a strong
Malcev basis.
2.4 Lattice subgroups
Definition 1 ([7]). Let Γ be a uniform subgroup of a simply connected nilpotent Lie group G,
we say that Γ is a lattice subgroup of G if log(Γ) is an Abelian subgroup of g.
In [7], Moore shows that if a simply connected nilpotent Lie group G satisfies the Malcev
rationality criterion, then G admits a lattice subgroup.
We close this section with the following proposition [1, Lemma 3.9].
Proposition 2. If Γ is a lattice subgroup of a simply connected nilpotent Lie group G = exp(g)
and {X1, . . . , Xn} is a weak Malcev basis of g strongly based on Γ, then {X1, . . . , Xn} is a Z-basis
for the additive lattice log(Γ) in g.
3 Main result
We begin with the following definition.
Definition 2. Let G be a connected, simply connected nilpotent Lie group which satisfies the
Malcev rationality criterion, and let g be its Lie algebra.
(1) We say that G satisfies the Moore formula at a lattice subgroup Γ if we have
m(π,G,Γ, 1)2 = #[OG
π ∩g∗Γ/Γ], ∀π ∈ (G : Γ)).
(2) We say that G satisfies the Moore formula if G satisfies the Moore formula at every lattice
subgroup Γ of G.
Examples.
(1) Every Abelian Lie group satisfies the Moore formula.
(2) The 3-dimensional Heisenberg group satisfies the Moore formula (see [7, p. 155]).
The main result of this paper is the following theorem.
Theorem 3. Every connected, simply connected two-step nilpotent Lie group satisfies the Moore
formula.
Before proving Theorem 3, we must review more of the Corwin–Greenleaf multiplicity for-
mula.
4 H. Hamrouni
3.1 The Corwin–Greenleaf multiplicity formula
Using the Poisson summation and Selberg trace formulas, L. Corwin and F.P. Greenleaf [1] gave
a formula for m(π,G,Γ, 1) that depended only on the coadjoint orbit in g∗ corresponding to π
via Kirillov theory. We state their formula for lattice subgroups. Let Γ be a lattice subgroup of
a connected, simply connected nilpotent Lie group G = exp(g). Let
g∗Γ = {l ∈ g∗ : 〈l, log(Γ)〉 ⊂ Z} .
Let πl be an irreducible unitary representation of G with coadjoint orbit OG
πl
⊂ g∗ such that
OG
πl
6= {l}. According to Theorem 1, we have m(πl, G,Γ, 1) > 0 if and only if OG
πl
∩g∗Γ 6= ∅, so
we will suppose this intersection is nonempty. The set OG
πl
∩g∗Γ is Γ-invariant. For such Γ-orbit
Ω ⊂ OG
πl
∩g∗Γ one can associate a number c(Ω) as follows: let f ∈ Ω and g(f) = ker(Bf ), where
Bf is the skew-symmetric bilinear form on g given by
Bf (X, Y ) = 〈f, [X, Y ]〉, X, Y ∈ g.
Since 〈f, log(Γ)〉 ⊂ Z then g(f) is a rational subalgebra. There exists a weak Malcev basis
{X1, . . . , Xn} of g strongly based on Γ and passing through g(f) (see [2, Proposition 5.3.2]). We
write g(f) = R-span {X1, . . . , Xs}. Let
Af = Mat
(
〈f, [Xi, Xj ]〉 : s < i, j ≤ n
)
. (3)
Then det(Af ) is independent of the basis satisfying the above conditions and depends only on
the Γ-orbit Ω. Set
c(Ω) =
(
det(Af )
)− 1
2 .
Then c(Ω) is a positive rational number and the multiplicity formula of Corwin–Greenleaf is
m(πl, G,Γ, 1) =
1, if g(l) = g,∑
Ω∈[OG
πl
∩g∗Γ/Γ]
c(Ω), otherwise. (4)
For details see [1].
Proof of Theorem 3. Let l ∈ OG
π ∩g∗Γ. The result is obvious if g(l) = g. Next, we suppose
that g(l) 6= g. Since G is two-step nilpotent Lie group then g(l) is an ideal of g, and hence
we have g(l) = g(f) for every f ∈ OG
π and OG
π = l + g(l)⊥ (see [2, Theorem 3.2.3]). On the
other hand, as l belongs to g∗Γ then g(l) is rational. By Proposition 5.3.2 of [2] there exists
a Jordan–Hölder basis B = {X1, . . . , Xn} of g strongly based on Γ and passing through g(l).
Set g(l) = R-span {X1, . . . , Xs}.
Then, for every Ω ∈ [OG
π ∩g∗Γ/Γ] and for every f ∈ Ω, we have
c(Ω) = det(Af )−
1
2 = det(Al)−
1
2 = c(Γ · l),
since f |[g,g] = l|[g,g]. It follows from (4) that
m(π,G,Γ, 1) = #[OG
π ∩g∗Γ/Γ] c(Γ · l). (5)
Next, we calculate #[OG
π ∩g∗Γ/Γ]. Let (t1, . . . , tn) ∈ Zn and f ∈ OG
π ∩g∗Γ. We have
(
exp(−t1X1) · · · exp(−tnXn)
)
· f = f +
n∑
i=s+1
n∑
j=s+1
tj〈f, [Xj , Xi]〉
X∗
i
= f +
n∑
i=s+1
n∑
j=s+1
tj〈l, [Xj , Xi]〉
X∗
i ,
On the Moore Formula of Compact Nilmanifolds 5
since f |[g,g] = l|[g,g]. It follows that
Γ · f = f +
n∑
j=s+1
Zej ,
where
ej =
n∑
i=s+1
〈l, [Xj , Xi]〉X∗
i , ∀ s < j ≤ n.
Let
L = OG
π ∩g∗Γ − f =
⊕
s<i≤n
ZX∗
i and L0 =
n∑
j=s+1
Zej .
Since g(l)∩R-span {Xs+1, . . . , Xn} = {0}, then the vectors es+1, . . . , en are linearly independent.
Therefore, L0 is a sublattice of L. It is well known that there exist εs+1, . . . , εn a linearly
independent vectors of g∗ and ds+1, . . . , dn ∈ N∗ such that
L =
⊕
s<i≤n
Zεi and L0 =
⊕
s<i≤n
diZεi.
Consequently, we have
#[OG
π ∩g∗Γ/Γ] = ds+1 · · · dn.
Let [εs+1, . . . , εn] be the matrix with column vectors εs+1, . . . , εn expressed in the basis (X∗
s+1,
. . . , X∗
n). From
L =
⊕
s<i≤n
ZX∗
i =
⊕
s<i≤n
Zεi,
we deduce that
[εs+1, . . . , εn] ∈ GL(n− s,Z).
On the other hand, let [es+1, . . . , en] (resp. [ds+1εs+1, . . . , dnεn]) be the matrix with column
vectors es+1, . . . , en (resp. ds+1εs+1, . . . , dnεn) expressed in the basis (X∗
s+1, . . . , X
∗
n). Since
L0 =
n∑
j=s+1
Zej =
⊕
s<i≤n
diZεi,
then there exists T ∈ GL(n− s,Z) such that
[es+1, . . . , en] = [ds+1εs+1, . . . , dnεn]T.
The latter condition can be written
tAl = [εs+1, . . . , εn]diag[ds+1, . . . , dn]T.
Form this it follows that
det(Al) = ds+1 · · · dn.
Consequently
#[OG
π ∩g∗Γ/Γ] = det(Al). (6)
Substituting the last expression (6) into (5), we obtain
m(π,G,Γ, 1)2 = #[OG
π ∩g∗Γ/Γ].
This completes the proof. �
6 H. Hamrouni
As a consequence of the above result, we obtain the following result.
Corollary 1. Let G be a connected, simply connected two-step nilpotent Lie group, let g be the
Lie algebra of G, and let Γ be a lattice subgroup of G. Let l ∈ g∗ such that the representation πl
appears in the decomposition of RΓ. Let Al as in (3). The multiplicity of πl is
m(πl, G,Γ, 1) =
{
1, if g(l) = g,
(det(Al))
1
2 , otherwise.
Remark 1. Note that in [10], H. Pesce obtained the above result more generally when Γ is a
uniform subgroup of G.
4 Three-step example
In this section, we give an example of three-step nilpotent Lie group that does not satisfy the
Moore formula. Consider the 4-dimensional three-step nilpotent Lie algebra
g = R-span {X1, . . . , X4}
with Lie brackets given by
[X4, Xi] = Xi−1, i = 2, 3,
and the non-defined brackets being equal to zero or obtained by antisymmetry. Let G be the
simply connected Lie group with Lie algebra g. The group G is called the generic filiform
nilpotent Lie group of dimension four. Let Γ be the lattice subgroup of G defined by
Γ = exp(ZX1)exp(ZX2)exp(ZX3)exp(6ZX4) = exp(ZX1 ⊕ZX2 ⊕ZX3 ⊕ 6ZX4).
Let l = X∗
1 . It is clear that the ideal m = R-span {X1, . . . , X3} is a rational polarization at l.
On the other hand, we have 〈l,m ∩ log(Γ)〉 ⊂ Z. Consequently, the representation πl occurs
in RΓ (see [12, 4]). Now, we have to calculate #[OG
πl
∩g∗Γ/Γ].
Following [2] or [8], the coadjoint orbit of l has the form
OG
πl
=
{
X∗
1 + tX∗
2 +
t2
2
X∗
3 + sX∗
4 : s, t ∈ R
}
.
On the other hand, it is easy to verify that
g∗Γ = Z-span
{
X∗
1 , . . . , X∗
3 ,
1
6
X∗
4
}
.
Therefore
OG
πl
∩g∗Γ =
{
X∗
1 + tX∗
2 +
t2
2
X∗
3 +
s
6
X∗
4 : s ∈ Z, t ∈ 2Z
}
.
Let
ft0,s0 = X∗
1 + t0X
∗
2 +
t20
2
X∗
3 +
s0
6
X∗
4 ∈ OG
πl
∩g∗Γ
and
γ = exp(rX2)exp(sX3)exp(6tX4) ∈ Γ.
On the Moore Formula of Compact Nilmanifolds 7
We calculate
Ad∗(γ)ft0,s0 = X∗
1 + (t0 − 6t)X∗
2 +
(t0 − 6t)2
2
X∗
3 +
(s0
6
+ st0 + r − 6st
)
X∗
4 .
Then (see [8])
Ad∗(Γ)ft0,s0 =
{
X∗
1 + (t0 + 6t)X∗
2 +
(t0 + 6t)2
2
X∗
3 +
(s0
6
+ s
)
X∗
4 : s, t ∈ Z
}
= {ft0+6t,s0+6s : s, t ∈ Z} .
From this we deduce that #[OG
πl
∩g∗Γ/Γ] = 3 · 6 = 18, and hence
m(πl, G,Γ, 1)2 6= #[OG
πl
∩g∗Γ/Γ].
Therefore, the group G does not satisfy the Moore formula at Γ.
Acknowledgements
It is great pleasure to thank the anonymous referees for their critical and valuable comments.
References
[1] Corwin L., Greenleaf F.P., Character formulas and spectra of compact nilmanifolds, J. Funct. Anal. 21
(1976), 123–154.
[2] Corwin L.J., Greenleaf F.P., Representations of nilpotent Lie groups and their applications. Part I. Basic
theory and examples, Cambridge Studies in Advanced Mathematics, Vol. 18, Cambridge University Press,
Cambridge, 1990.
[3] Gelfand I.M., Graev M.I., Piatetski-Shapiro I.I., Representation theory and automorphic functions,
W.B. Saunders Co., Philadelphia, Pa.-London – Toronto, Ont. 1969.
[4] Howe R., On Frobenius reciprocity for unipotent algebraic group over Q, Amer. J. Math. 93 (1971), 163–172.
[5] Malcev A.I., On a class of homogeneous spaces, Amer. Math. Soc. Transl. 1951 (1951), no. 39, 33 pages.
[6] Matsushima Y., On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J.
2 (1951), 95–110.
[7] Moore C.C., Decomposition of unitary representations defined by discrete subgroups of nilpotent Lie groups,
Ann. of Math. (2) 82 (1965), 146–182.
[8] Nielsen O.A., Unitary representations and coadjoint orbits of low dimensional nilpotent Lie groups, Queen’s
Papers in Pure and Applied Mathematics, Vol. 63, Queen’s University, Kingston, ON, 1983.
[9] Onishchik A.L., Vinberg E.B., Lie groups and Lie algebras. II. Discrete subgroups of Lie groups and coho-
mologies of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences, Vol. 21, Springer-Verlag,
Berlin, 2000.
[10] Pesce H., Calcul du spectre d’une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339
(1993), 433–461.
[11] Raghunathan M.S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 68, Springer-Verlag, New York – Heidelberg, 1972.
[12] Richardson L.F., Decomposition of the L2 space of a general compact nilmanifolds, Amer. J. Math. 93
(1971), 173–190.
1 Introduction
2 Rational structures and uniform subgroups
2.1 Rational structures
2.2 Uniform subgroups
2.3 Weak and strong Malcev basis
2.4 Lattice subgroups
3 Main result
3.1 The Corwin-Greenleaf multiplicity formula
4 Three-step example
References
|
| id | nasplib_isofts_kiev_ua-123456789-149106 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:20:12Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hamrouni, H. 2019-02-19T17:25:42Z 2019-02-19T17:25:42Z 2009 On the Moore Formula of Compact Nilmanifolds / H. Hamrouni // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 12 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 22E27 https://nasplib.isofts.kiev.ua/handle/123456789/149106 Let G be a connected and simply connected two-step nilpotent Lie group and Γ a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation IndΓG(1). Extending then the Abelian case. It is great pleasure to thank the anonymous referees for their critical and valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Moore Formula of Compact Nilmanifolds Article published earlier |
| spellingShingle | On the Moore Formula of Compact Nilmanifolds Hamrouni, H. |
| title | On the Moore Formula of Compact Nilmanifolds |
| title_full | On the Moore Formula of Compact Nilmanifolds |
| title_fullStr | On the Moore Formula of Compact Nilmanifolds |
| title_full_unstemmed | On the Moore Formula of Compact Nilmanifolds |
| title_short | On the Moore Formula of Compact Nilmanifolds |
| title_sort | on the moore formula of compact nilmanifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149106 |
| work_keys_str_mv | AT hamrounih onthemooreformulaofcompactnilmanifolds |