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)....
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irk-123456789-1491062019-02-20T01:27:54Z On the Moore Formula of Compact Nilmanifolds Hamrouni, H. 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. 2009 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149106 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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. |
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Hamrouni, H. On the Moore Formula of Compact Nilmanifolds Symmetry, Integrability and Geometry: Methods and Applications |
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Hamrouni, H. |
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Hamrouni, H. |
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On the Moore Formula of Compact Nilmanifolds |
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On the Moore Formula of Compact Nilmanifolds |
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On the Moore Formula of Compact Nilmanifolds |
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On the Moore Formula of Compact Nilmanifolds |
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On the Moore Formula of Compact Nilmanifolds |
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on the moore formula of compact nilmanifolds |
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Інститут математики НАН України |
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2009 |
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On the Moore Formula of Compact Nilmanifolds / H. Hamrouni // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 12 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT hamrounih onthemooreformulaofcompactnilmanifolds |
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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.
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(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
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