Comptes Rendus Mathématique

Miloud Mihoubi

Said Taharbouchet

Comptes Rendus
Mathématique
Miloud Mihoubi and Said Taharbouchet
Polynomials with real zeros via special polynomials
Volume 359, issue 1 (2021), p. 57-64.
<https://doi.org/10.5802/crmath.147>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
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Comptes Rendus
Mathématique
2021, 359, n 1, p. 57-64
https://doi.org/10.5802/crmath.147
Theory of functions / Théorie des fonctions
Polynomials with real zeros via special
polynomials
Polynômes à racines réelles via des polynômes spéciaux
MiloudMihoubia and Said Taharbouchet∗, a
a RECITS Laboratory, Faculty of Mathematics, USTHB, P.O. Box 32, El Alia 16111,
Bab-Ezzouar, Algiers, Algeria
E-mails: mmihoubi@usthb.dz, staharbouchet@usthb.dz
Abstract. In this paper, we use particular polynomials to establish some results on the real rootedness of
polynomials. The considered polynomials are Bell polynomials and Hermite polynomials. To cite this article:
M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021).
Résumé. Dans ce papier, nous utilisons des polynômes particuliers pour établir quelques résultats sur les
polynômes à racines réelles. Les polynômes considérés sont des polynômes de Bell et des polynômes de
Hermite. Pour citer cet article : M. Mihoubi and S. Taharbouchet, C. R. Acad. Sci. Paris, Ser. I 340 (2021).
Manuscript received 3rd November 2019, accepted 6th November 2020.
1. Introduction
The real rootedness of polynomials has attracted researchers great interest. One of the reasons is
that any polynomial of real zeros implies the log-concavity and the unimodality of its coefficients,
which appear in various fields of mathematics, see [4, 13].
LetRZbe the set of all polynomials having only real zeros andNRZbe the set of all polynomials
having only non-positive real zeros. In a previous paper, we showed that for any polynomial f ,
if f (Bx) ∈ RZ, then (Bx)r f (Bx) ∈ RZ, where Bx is the generalized Bell umbra [2]. Motivated by
this work, we use in this paper the umbrae of special polynomials to give new results on the real
rootedness of polynomials. The tools used here are classical umbral calculus, theorem 1 of Wang
and Yeh [14] and Rolle’s theorem.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
58 Miloud Mihoubi and Said Taharbouchet
To begin, let (Pn(x)) be a sequence of polynomials with degPn = n and P0(x) = 1 and
let Pnx be the umbra defined by P
n
x := Pn(x). The conditions degPn = n and P0(x) = 1 show
that the system of polynomials
{P0 (x) ,P1 (x) ,P2 (x) , . . . }
is a free system and ensure that any polynomial with degree n can be expressed in terms of
1,Px, . . . , Pnx . We consider here some sequences of polynomials (Pn(x)) satisfying P0(x) = 1,
degPn = n and a relation of the form
Pn+1 (x) = (u1x +u2)Pn (x)+ (v1x + v2) d
d x
Pn (x) , (1)
for some known real numbers u1,u2, v1 and v2.
For more information about classical umbral calculus, see [2, 7, 11, 12].
We use below the notations
(x)0 := 1, (x)n := x (x −1) · · · (x −n +1) , n ≥ 1,
〈x〉0 := 1, 〈x〉n := x (x +1) · · · (x +n −1) , n ≥ 1.
The paper is organized as follows: in the following Section 2 we give the main results and in the
last Section 3 we present some of their applications.
2. Real-rooted polynomials via special polynomials
Let Bn(x) and Hn(x) be, respectively, the nth Bell and Hermite polynomials defined by∑
n≥0
Bn (x)
t n
n!
= exp(x (exp(t )−1)) ,
∑
n≥0
Hn (x)
t n
n!
= exp(2xt − t 2) ,
and let Bx and Hx be the umbrae defined respectively by
Bnx =Bn (x) and Hnx =Hn (x) .
These polynomials satisfy the following differential recurrence relations
Bn+1 (x)−xBn (x) = x d
d x
Bn (x) ,
Hn+1 (x)−2xHn (x) =− d
d x
Hn (x) ,
which can be written as
Bn+1x −xBnx = x
d
d x
Bnx , H
n+1
x −2xHnx =−
d
d x
Hnx .
Hence, for any polynomial f , these relations are equivalent to
Bx f (Bx)−x f (Bx) = x d
d x
f (Bx) , (2)
Hx f (Hx)−2x f (Hx) =− d
d x
f (Hx) . (3)
C. R. Mathématique, 2021, 359, n 1, 57-64
Miloud Mihoubi and Said Taharbouchet 59
2.1. Real-rooted polynomials via Wang–Yeh’s theorem
The results of this subsection are based on [14, Theorem 1] of Wang and Yeh and the principal
result is given by the following theorem.
Theorem 1. Let f be a polynomial and m be a positive integer. Then, if f (Bx) ∈RZ, there holds
f (Bx)
m∏
k=1
(Bx +ak x +bk ) ∈RZ, ak ∈R, bk ∈ [0,∞[ , k = 1, . . . , m,
and if f (Hx) ∈RZ, there holds
f (Hx)
m∏
k=1
(Hx +ak x +bk ) ∈RZ, ak ∈ [−2,∞[ , bk ∈R, k = 1, . . . , m.
Proof. Let g be a polynomial. Relation (2) implies(
Bx + g (x)−x
)
f (Bx) = g (x) f (Bx)+x d
d x
f (Bx) . (4)
The choice g (x) = (a1 +1)x +b1 in the last relation gives
f1 (Bx) = (Bx +a1x +b1) f (Bx) = ((a1 +1) x +b1) f (Bx)+x d
d x
f (Bx) .
For (a1,b1) ∈R× [0,∞[ apply [14, Theorem 1] of Wang and Yeh to deduce that f1(Bx) ∈RZ. So, by
repeating the same process, we conclude that
f2 (Bx) = (Bx +a2x +b2) f1 (Bx)
= (Bx +a2x +b2) (Bx +a1x +b1) f (Bx) ∈RZ,
and so on. The second result can be proved similarly on using the relation (3). 
Proposition 2. Let f be a polynomial such that f (Bx) ∈RZ. Then, for any polynomial g (x) ∈NRZ,
there holds
g (Bx +ax) f (Bx) ∈RZ, a ∈R,
and, when g (x) ∈RZ and f (Hx) ∈RZ, there holds
g (Hx +ax) f (Hx) ∈RZ, a ≥−2.
Proof. Let g (x) =
m∏
k=1
(x+bk ). Then, for the choices a1 = ·· · = am = a in Theorem 1, we obtain the
desired results. 
Proposition 3. Let g be a polynomial. Then, if g (x) ∈NRZ, then there holds
g (Bx) ∈RZ,
and if g (x) ∈RZ, then there holds
g (Hx) ∈RZ.
Proof. For a1 = ·· · = am = 0 and f (x) = x + b0, b0 > 0, in Theorem 1 and by setting g (x)
=
m∏
k=0
(x +bk ), we obtain g (Bx) ∈RZwhen g (x) ∈NRZ, and g (Hx) ∈RZwhen g (x) ∈RZ. 
Remark 4. Since for any non-negative integer k we have [1]
(Bx)k = xk , 〈Bx〉k = Lk (x) and (Bx −1)k = (−1)k Dk (1−x) ,
Proposition 3 shows that if one of the polynomials
m∑
k=0
g1 (m,k) (x)k ,
m∑
k=0
g2 (m,k)〈x〉k ,
m∑
k=0
g3 (m,k) (x −1)k
C. R. Mathématique, 2021, 359, n 1, 57-64
60 Miloud Mihoubi and Said Taharbouchet
has only non-positive real zeros, then its correspondent polynomial given between the polyno-
mials
m∑
k=0
g1 (m,k) x
k ,
m∑
k=0
g2 (m,k)Lk (x) ,
m∑
k=0
(−1)k g3 (m,k)Dk (1−x) ,
has also only real zeros, where
Ln (x) =
n∑
k=1
n!
k !
(
n −1
k −1
)
xk , n ≥ 1, L0 (x) = 1,
Dn (x) =
n∑
j=0
(
n
j
)
j ! (x −1)n− j
are, respectively, the Lah and derangement polynomials.
Remark 5. Proposition 3 shows that if the polynomial g (x) =
m∑
k=0
g (m,k)xk has only non-positive
real zeros, then the polynomial
g (Bx) =
m∑
k=0
g (m,k)Bk (x)
has only real zeros, and if the polynomial g (x) has only real zeros, then the polynomial
g (Hx) =
m∑
k=0
g (m,k)Hk (x)
has only real zeros.
Remark 6. From the above remarks, it follows that the linear transformations
(x)k → xk ,
〈x〉k → Lk (x) ,
(x −1)k → (−1)k Dk (1−x) ,
xk →Bk (x)
mapNRZ onto RZ and xk →Hk (x) maps RZ onto RZ.
For the choice g (x) = xq ( f (x))n−1 in Proposition 3 we obtain:
Corollary 7. Let f be a polynomial. If f (x) ∈NRZ, then there hold
Bqx
(
f (Bx)
)n ∈RZ, n = 1,2, . . . , q = 0,1, . . . ,
and, if f (x) ∈RZ, then there holds
Hqx
(
f (Hx)
)n ∈RZ, n = 1,2, . . . , q = 0,1, . . . .
Proposition 8. Let f be a polynomial such that f (Bx) ∈ RZ. Then, for any non-negative integers
r1, . . . , rp , there holds
〈Bx +a1x〉r1 · · ·
〈
Bx +ap x
〉
rp
f (Bx) ∈RZ, a1, . . . , ap ∈R.
Furthermore, if r1 + ·· · + rp ≥ 1, and if g (x) ∈NRZ, there holds
〈Bx〉r1 · · · 〈Bx〉rp g (Bx +ax) ∈RZ, a ∈R.
Proof. For the first part of the proposition choice in Proposition 2 g (x) to be 〈x〉r1 and replace a
by a1 to obtain
f1 (Bx) := 〈Bx +a1x〉r1 f (Bx) ∈RZ, a1 ∈R.
Now, choice in Proposition 2 the polynomial g (x) to be 〈x〉r2 and replace a by a2 and f by f1 to
obtain
f2 (Bx) := 〈Bx +a2x〉r2 f1 (Px) ∈RZ, a2 ∈R,
C. R. Mathématique, 2021, 359, n 1, 57-64
Miloud Mihoubi and Said Taharbouchet 61
and so on. For the second part of the proposition, by choosing a1 = ·· · = ap = 0 and f (x) = x + i
for such integer i ∈ {r1, . . . , rp }, the first part of this proposition proves that
〈Bx〉r1 · · · 〈Bx〉rp ∈RZ,
with r1+ ·· · +rp ≥ 1 and n ≥ 1. So, the choice f (x) = 〈x〉r1 · · · 〈x〉rp in the first part of Proposition 2
completes the proof. 
One can prove similarly the following Proposition 9.
Proposition 9. Let f be a polynomial such that f (Hx) ∈ RZ. Then, for any non-negative integers
r1, . . . , rp , there hold
(Hx +a1x)r1 · · ·
(
Hx +ap x
)
rp
f (Hx) ∈RZ, a1 ≥−2, . . . , ap ≥−2,
〈Hx +a1x〉r1 · · ·
〈
Hx +ap x
〉
rp
f (Hx) ∈RZ, a1 ≥−2, . . . , ap ≥−2.
Furthermore, if r1 + ·· · + rp ≥ 1,n ≥ 1, and g (x) ∈RZ, there hold
(Hx)r1 · · · (Hx)rp g (Hx +ax) ∈RZ, a ≥−2,
〈Hx〉r1 · · · 〈Hx〉rp g (Hx +ax) ∈RZ, a ≥−2.
2.2. Real-rooted polynomials via Rolle’s theorem
The results of this subsection are based on Rolle’s theorem and the principal result is given by the
following Theorem 10.
Theorem 10. Let a1, . . . , am ,b1, . . . , bm be real numbers and let f be a polynomial. Then, if
f (Bx) ∈RZ, there hold
f (Bx)
m∏
k=1
(
Bx +ak x2 +bk x
) ∈RZ, a1 ≤ 0, . . . , am ≤ 0.
Proof. Let n be the degree of the polynomial f .
The choice g (x) = (a1x +b1 +1)x in (4) gives(
Bx +a1x2 +b1x
)
f (Bx)
= x exp
(
−a1 x
2
2
− (b1 +1) x
)
d
d x
(
exp
(
a1
x2
2
+ (b1 +1) x
)
f (Bx)
)
.
Case a1 < 0. The application of Rolle’s theorem on the function
T (x) = exp
(
a1
x2
2
+ (b1 +1) x
)
f (Bx)
shows that the polynomial f1(Bx) = (Bx+a1x2+b1x) f (Bx), which is of degree n+2, vanishes n−1
times between the zeros of f (Bx). But since lim
x→±∞T (x) = 0 it results that f1(Bx) vanishes also on
]−∞, x1[ and on ]xn ,∞[, where x1 and xn are, respectively, the minimum and the maximum zeros
of f (Bx). Hence, the polynomial f1(Bx) vanishes n+1 times, the missing zero must be necessarily
real.
Case a1 = 0: When b1 + 1 = 0 the Theorem 10 is evident, otherwise, the application of Rolle’s
theorem on the function
T (x) = exp((b1 +1) x) f (Bx)
shows that the polynomial f1(Bx) = (Bx + b1x) f (Bx) vanishes n − 1 times between the zeros of
f (Bx). But since lim
x→+∞T (x) = 0 or limx→−∞T (x) = 0 it results that f1(Bx) vanishes also on ]−∞, x1[ or
]xn ,∞[. Hence, the polynomial f1(Bx) which is of degree n +1 vanishes n times, the missing zero
must be necessarily real.
C. R. Mathématique, 2021, 359, n 1, 57-64
62 Miloud Mihoubi and Said Taharbouchet
To obtain the first part of the Theorem 10, one can replace f (Bx) by f1(Bx) and proceed as
above, and so on. 
Proposition 11. Let a be a real number and let f , g be polynomials with deg g = m. Then, if
f (Bx) ∈RZ and g (x) ∈RZ, we get
xm g
(
Bx
x
+ax
)
f (Bx) ∈RZ, a ≤ 0.
Proof. The proposition 11 follows form Theorem 10 by taking a1 = ·· · = am = a ≤ 0 and by setting
g (x) =
m∏
k=1
(x +bk ). 
Remark 12. Let q,r1 ≤ ·· · ≤ rp be non-negative integers not all null.
For f (x) = (x)r1 · · · (x)rp xq we get
f (Bx) = (Bx)rp · · · (Bx)rp Bqx = xrp Bq
(
x;r1, · · · , rp
) ∈RZ
and for g (x) =
m∑
k=0
g (m,k)xk ∈RZ and a = 0 in Proposition 11 we get
xm−rp g
(
Bx
x
)
f (Bx) =
m∑
k=0
g (m,k) xm−kBk+q
(
x;r1, · · · , rp
) ∈RZ,
where Bn(x;r1, · · · , rp ) is the nth(r1, · · · , rp ) − Bell polynomial introduced by Mihoubi and
Maamra [8, 10], see also [1].
In other words, for 0 ≤ k ≤ m, the linear transformation
xk → xm−kBk+q
(
x;r1, . . . , rp
)
, q + r1 + ·· · + rp ≥ 1,
map the set RZ on to RZ.
3. Some applications
In this section, we present some applications of the above results.
Corollary 13. Let G be a simple graph with n vertices and let
P (G , x) =
n∑
k=χ(G)
αk (G) (x)k
be its chromatic polynomial such that P (G , x) ∈RZ. Then, the polynomial
P (G ,−Bx) =
n∑
k=χ(G)
(−1)k αk (G)Lk (x)
has only real zeros, where χ(G) is the chromatic number of G.
Proof. The chromatic polynomial P (G , x) can also be written as
P (G , x) = x
n−1∑
k=1
(−1)k tk (G) (x −1)k−1 ,
where t1(G), . . . , tn−1(G) are non-negative integers, see [6]. This shows that the zeros of the
polynomial P (G , x) are not in ]−∞,1[−{0}. Then, the zeros of the polynomial P (G ,−x) are non-
positive real numbers. Since [1]
(Bx)k f (Bx) = xk f (Bx +k) and 〈Bx〉k = Lk (x) , k = 0,1,2, . . . ,
then, by Proposition 3, it follows that P (G ,−Bx) ∈RZ, with
C. R. Mathématique, 2021, 359, n 1, 57-64
Miloud Mihoubi and Said Taharbouchet 63
P (G ,−Bx) =
n∑
k=χ(G)
(−1)k αk (G)〈Bx〉k =
n∑
k=χ(G)
(−1)k αk (G)Lk (x) .

Example 14. For q = 0, f (x) = x in Corollary 7, we get(
f (Bx)
)n =Bn (x) ∈RZ, ( f (Hx))n =Hn (x) ∈RZ, n = 1,2, . . . .
Also, since Bn(x) has non-negative coefficients, it follows that its zeros are non-positive and by
Proposition 3, we get
Bn (Px) =
n∑
k=0
{
n
k
}
Pk (x) ∈RZ,
Hn (Px) =
[n/2]∑
k=0
(−1)k 2
n−2k n!
k ! (n −2k)!Pn−2k (x) ∈RZ,
where [x] is the largest integer ≤ x, {nk} is the (n,k)th Stirling number of the second kind, and,
Pnx =Pn(x) is any one of the polynomials Bn(x) and Hn(x).
Example 15. For f (x) = x + y we get f (Bx) = f (Hx) = x + y ∈RZ and via Corollary 7, we obtain
Bqx
(
f (Bx)
)n = n∑
k=0
(
n
k
)
yn−kBk+q (x) ∈RZ, y ≥ 0,
Hqx
(
f (Hx)
)n = n∑
k=0
(
n
k
)
yn−kHk+q (x) ∈RZ, y ∈R.
Example 16. From Proposition 8 it follows, for any real number a and any polynomial f such
that f (Bx) ∈RZ, that (Bx +ax)n−2 f (Bx) ∈RZ, n ≥ 2. Then, for a =−1, f (Bx) = (Bx − x)2 = x we get
the known result [3]
Vn (x) := (Bx −x)n =
n∑
k=0
(
n
k
)
(−x)n−k Bk (x) =
n∑
k=0
{
n
k
}2↑
xk ∈RZ,
where
{n
k
}2↑ is the number of ways to partition the set {1, . . . ,n} into k blocs without singletons.
More generally, for f (Bx) = (Bx +ax)2 = x((a +1)2x +1) we get
Vn, a (x) := (Bx +ax)n =
n∑
k=0
Bn,k (a +1,1,1, . . . ) xk ∈RZ.
We note here that
∑
n≥0
Vn, a(x)
t n
n! = exp(x(e t − 1+ at )), where Bn,k (a1, a2, . . .) are the partial Bell
polynomials, see [5, 9].
Example 17. For rp = max{r1, . . . , rp }, a1 = ·· · = ap = 0 and f (x) = xn in the first identity of
Proposition 8, it follows that the polynomial
Ln;r1, ...,rp (x) :=
n+r1+···+rp−1∑
k=0
(−1)k
{
n + r1 + ·· · + rp
k + rp
}
r1, ...,rp
Lk (x)
C. R. Mathématique, 2021, 359, n 1, 57-64
64 Miloud Mihoubi and Said Taharbouchet
has only real roots. Indeed, from [1] we can write
〈Bx〉r1 · · · 〈Bx〉rp Bnx
= (−1)n+r1+···+rp (−Bx)r1 · · · (−Bx)rp (−Bx)n
= (−1)n+r1+···+rp
n+r1+···+rp−1∑
k=0
{
n + r1 + ·· · + rp
k + rp
}
r1, ...,rp
(−Bx)k
= (−1)n+r1+···+rp
n+r1+···+rp−1∑
k=0
(−1)k
{
n + r1 + ·· · + rp
k + rp
}
r1, ...,rp
〈Bx〉k
= (−1)n+r1+···+rp
n+r1+···+rp−1∑
k=0
(−1)k
{
n + r1 + ·· · + rp
k + rp
}
r1, ...,rp
Lk (x)
= (−1)n+r1+···+rp Ln;r1, ...,rp (x) ,
where
{n
k
}
r1, ...,rp
is the
(
r1, . . . , rp
)
-Stirling numbers of the second kind introduced by Mihoubi
and Maamra [8, 10].
References
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On cogrowth function of algebras and its logarithmical gap

Wenran Liu

Comptes Rendus
Mathématique
Wenran Liu
Chern characters in equivariant basic cohomology
Volume 359, issue 1 (2021), p. 1-5.
<https://doi.org/10.5802/crmath.14>
© Académie des sciences, Paris and the authors, 2021.
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Comptes Rendus
Mathématique
2021, 359, n 1, p. 1-5
https://doi.org/10.5802/crmath.14
Differential geometry / Géométrie différentielle
Chern characters in equivariant basic
cohomology
Wenran Liua, b
a SIAE Civil Aviation University of China, 2898 Jinbei Street Dongli District, 300300
Tianjin, China
b IMAG Université de Montpellier, Place Eugène Bataillon 34090 Montpellier, France
E-mail: wrliu@cauc.edu.cn
Abstract. The purpose of this Note is to establish a geometric realization of the cohomological isomorphism
in the case of a transversely oriented Killing foliation on a compact smooth manifold through equivariant
basic Chern characters.
Résumé. L’objet de cette Note est d’établir une réalisation géométrique de l’isomorphisme cohomologique
dans le cas d’un feuilletage de Killing transversalement orienté sur une variété compacte à travers les
caractères de Chern basiques équivariants.
Manuscript received 9th September 2019, revised 18th October 2019, accepted 20th December 2019.
1. Introduction
Let (M ,F ) be a smooth compact manifold equipped with a Riemannian foliation F . Let E =
E+ ⊕ E− be a foliated vector bundle over M and DE+b : C∞F -bas(M ,E+) → C∞F -bas(M ,E−) be a
transversely elliptic basic differential operator on basic sections. Thus, we can consider its basic
index Indexb(D
E+
b ) ∈Z, and a natural question emerges: how to obtain a cohomological formula
as Atiyah–Singer index theorem for this invariant. Some (partial) responses have been proposed
in the literature for this problem [1, 3, 5]. In the 1990s, El Kacimi proposed to tackle this problem
by using Molino’s theory. Let us recall the main lines of this proposal. To each transversely
oriented Riemannian foliation (M ,F ) of codimension q , Molino associated an oriented manifold
W equipped with an action of the orthogonal group SO(q). In this setting, the space of the leaf
closures of the foliation F exhibits a natural identification with the quotient W /SO(q). In [3], El
Kacimi constructed a SO(q)-equivariant bundle E over W . Brüning, Kamber and Richardson [1]
studied the basic Dirac operator DE+b over a basic Clifford bundle. An SO(q)-transversely elliptic
operator DE is associated such that Indexb(D
E+
b ) is equal to the index of D
E when restricted
to invariant sections. Leveraging this valuable identity, we investigate the acquisition of the
Indexb(D
E+
b ) information from that of D
E . And in turn, to obtain an index formula by results
in [9].
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
2 Wenran Liu
Let H•(M ,F ) be the basic de Rham cohomology of the foliated manifold (M ,F ). If (M ,F ) is a
Killing foliation (a priori Riemannian), there is an abelian Lie algebra a, which acts transversely on
(M ,F ). The orbit of the leaves of the foliation F under the action of a is exactly the leaf closure.
Let H•a(M ,F ) be the a-equivariant basic de Rham cohomology with polynomial coefficients.
Goertsches and Töben [4] proved a cohomological isomorphism H•a(M ,F ) ' H•so(q)(W ). Duflo
and Vergne extended the equivariant cohomology to the case of C∞-coefficients in [2]. In this
Note, we first present an assumption for the existence of a basic connection and proceed to
introduce the definition of the equivariant basic Chern character. Then, we establish a geometric
realization of the cohomological isomorphism through equivariant basic Chern characters.
2. Equivariant basic Chern character
We use the notation presented in Section 1 in the following definitions.
Definition 1. A foliated principal bundle (P,FP ) is a principal bundle P equipped with a foliation
FP such that
(1) FP is invariant under the action of the structure group of P, and
(2) the projection of FP on M is F and at each point of P, the projection is an isomorphism.
Let (E ,FE ) be a foliated hermitian bundle over (M ,F ). In order to define the basic Chern
character Ch(E ,FE ) ∈ H•(M ,F ), (E ,FE ) must be associated with a foliated principal bundle
(P,FP ). Letω be a connection on P . We denote by X(FP ) the Lie algebra of the vector fields on M
that are tangent to FP . IfX(FP ) belongs to the kernel ofω,ω is said to be adapted to FP . By [6], an
adapted connection always exists. Ifω is invariant with respect to X(FP ) under the Lie derivative,
i.e., LZω= 0,∀ Z ∈X(FP ), it is a basic connection. That is the case for which (P,FP ) is a principal
F -bundle, introduced in [3, 6]. However, in general, the existence of a basic connection cannot
be guaranteed; a counterexample is presented in [7, Section 4.1.1].
Example 2. Let M = S1 ×S1 and P = M ×S1. The coordinates are denoted by (x, y, z) ∈ M ×S1,
and the foliation F on M is given by ∂/∂x. Let f ∈ C∞(S1) be a non-constant real function. The
foliation FP is generated by V =∂/∂x+ f (y)∂/∂z.
Let Hol(M ,F ) be the holonomy groupoid of (M ,F ). If a vector bundle E is equipped with
a groupoid action of Hol(M ,F ), it is Hol(M ,F )-equivariant. Henceforth, we fix P = U (E) as
the unitary frame bundle of E . Naturally, P is also equipped with an action of Hol(M ,F ).
The groupoid action generates a foliation FP on P such that (P,FP ) is a foliated principal
bundle. Unfortunately, under these conditions, we also have a counterexample [7, Section 4.2.1].
Therefore, an assumption is necessary to progress further in our work. Let Hol(M ,F )T
T
be
the étale holonomy groupoid restricted on a complete transversal section T of (M ,F ). In [5],
Gorokhovsky and Lott defined its closure Hol(M ,F )T
T
. Respectively, these restricted bundles on
T are denoted by E |T and P |T . We state the assumption as follows.
Assumption 3. The groupoid action of Hol(M ,F )T
T
on E |T extends to a groupoid action of
Hol(M ,F )T
T
on E |T .
Under Assumption 3, we can construct a basic connection ωP on P and the basic Chern
character is well-defined. This result is detailed in Section 4. Let l (M ,F ) be the Lie algebra of
the transverse fields on (M ,F ). Assume that (M ,F ) is a Killing foliation. In this setting, the
leaf closure F is given by the action of a, i.e., a ·F = F , see [4]. We consider the Lie algebroid
of Hol(M ,F )T
T
. By [5], it is trivialisable to T × a. Let us consider the infinitesimal version
of the action of Hol(M ,F )T
T
on P |T . We obtain a transverse action a → l (P,FP )G (i.e., a Lie
C. R. Mathématique, 2021, 359, n 1, 1-5
Wenran Liu 3
homomorphism) and a commutative diagram of Lie homomorphisms, where G denotes the
structure group of P and (−)G denotes the G-invariant transverse fields.
l (P,FP )G

a
;;
// l (M ,F )
(1)
The diagram (1) is essential for the definition of the equivariant basic Chern character. Addi-
tionally, the basic connection ωP must be compatible with the transverse action a→ l (P,FP )G .
This compatibility is guaranteed because they all result from the action of Hol(M ,F )T
T
. Next, the
associated bundle E is considered, we have the following definition.
Definition 4. The a-equivariant basic Chern character associated with the vector F -bundle E is
defined by the following relation
Cha(E ,FE ) =
[
Cha(E ,FE , ,∇E ,Y )
] ∈ H∞a (M ,F )
where, Cha(E ,FE ,∇E ,Y ) = Tr
(
e−(R
E+µE (Y ))) with RE representing the curvature of ∇E and µE
representing the moment map on E and Y ∈ a.
3. Our main result
The main result presented in [7] is as follows.
Theorem 5. Each Hol(M ,F )-equivariant hermitian bundle satisfying Assumption 3 is a vector
F -bundle.
Theorem 6. Let (M ,F ) be a transversely oriented Killing foliation of codimension q and E →
(M ,F ) be a hermitian Hol(M ,F )-equivariant vector bundle. Under Assumption 3, we can asso-
ciate a SO(q)-equivariant vector bundle E → W to E such that we have a geometric realization of
cohomological isomorphism through equivariant basic Chern characters.
Ha(M ,F ) ' Hso(q)(W ), Cha(E ,FE ) ' Chso(q)(E ). (2)
4. Proof of Theorem 5
We use the notation presented in Section 2 in this proof. Evidently, the holonomy groupoid action
on E induces a corresponding action on P . Under Assumption 3, the Hol(M ,F )T
T
-action on
P |T extends to a Hol(M ,F )TT -action on P |T . It is enough to construct a basic connection on
P . We take an adapted connection ω on (P,FP ) and restrict it on P |T , denoted by ωT . By [5],
the groupoid Hol(M ,F )T
T
is proper. By [10], every proper groupoid admits a Haar system and
a “cut-off” function. The cross-product groupoid, denoted by GP , is Hol(M ,F )TT nPT . The key
step is to averageωT by GP . We lift a Haar system and a “cut-off” function on Hol(M ,F )TT to GP ,
denoted by µGP and by ϕGP . We define
(ω̂T )|p =
∫
γ∈(GP )p
γ∗(ωT |r (γ))ϕGP
(
r (γ)
)
dµGPp (γ), ∀ p ∈ PT . (3)
Then, by the holonomy of FP , we construct a connection ωP on P from ω̂T . This connection
is invariant with respect to X(FP ), so it is basic.
C. R. Mathématique, 2021, 359, n 1, 1-5
4 Wenran Liu
5. Proof of Theorem 6
With the construction of the basic connection, we establish constructions and calculations
on principal bundles for the case in which vector bundles are always associated. By Molino’s
theory [8], the transversely oriented frame bundle M̃ of (M ,F ) is naturally equipped with
a tranversely parallelizable (T.P.) foliation F̃ . In addition, (M̃ ,F̃ ) is a principal F -bundle of
structure group SO(q) equipped with the Bott connection. The space of the leaf closures W =
M̃/F̃ is a manifold. By [4, Proposition 4.9] [2], the cohomological isomorphism is given by
H∞a (M ,F ) → H∞a×so(q)(M̃ ,F̃ ) and H∞a×so(q)(M̃ ,F̃ ) → H∞so(q)(W ). The first isomorphism is the
inclusion, and the second one is the equivariant Chern–Weil map CWso(q). We prove Theorem 6
with two steps below:
First step. Consider the pullback bundle P̃ of P over (M̃ ,F̃ ). We also lift the foliation FP and
the basic connection ωP to P̃ , denoted by FP̃ and ωP̃ , respectively. Under Assumption 3, the
action of Hol(M ,F )T
T
can be lifted to P̃ . The lifted groupoid action induces a transverse action
a→ l (P̃ ,FP̃ )G×SO(q). Analogous to Definition 4, the a× so(q)-equivariant basic Chern character
Cha×so(q)(Ẽ ,FẼ ) ∈ H∞a×so(q)(M̃ ,F̃ ) is well-defined where Ẽ is the pullback bundle of E . Naturally,
Cha×so(q)(Ẽ ,FẼ ) is the inclusion of Cha(E ,FE ) as we do the constructions by the "pullback"
approach.
Second step. The lifted foliation (P̃ ,FP̃ ) is T.P., just as (M̃ ,F̃ ). Again, by Molino’s theory, the
space of the leaf closure W̃ = P̃/FP̃ is a manifold. The key step is to prove that FP̃ is also
given by the action of a, i.e., a ·FP̃ = FP̃ . Furthermore, the foliation FP̃ has no torsion, see [7,
Proposition 5.3.2], which infers that G acts freely on W̃ . Thus, W̃ → W is a principal bundle (see
the diagram below).
(P̃ ,FP̃ )
ω̃
a×SO(q) $$
W̃ = P̃/FP̃
	 SO(q)
ω
xx
(M̃ ,F̃ ) //
/SO(q)

W = M̃/F̃
(P,FP )
ωP
a
// (M ,F )
As can be seen in the diagram, the connection ω̃ is a modification of the lifted connection by
adding a term of the action of a such that it is FP̃ -basic. By quotient, we have an SO(q)-invariant
connection ω on W̃ . Thus, E is the associated bundle W̃ ×G CN with G =U (N ) and N the rank of
E . Hence, the so(q)-equivariant Chern character Chso(q)(E ) ∈ H∞so(q)(W ) is well-defined.
We have explicitly calculated the results to show that Chso(q)(E ) = CWso(q)
(
Cha×so(q)(Ẽ ,FẼ )
)
.
We note that E and E have the same rank because E is the associated bundle. Therefore, the proof
of Theorem 6 is complete.
Acknowledgements
I am deeply grateful to Professor P.-E. Paradan and Professor M. Benameur for directing, support-
ing, and encouraging me during my thesis on this problem. Without their help, this paper would
never have been written. I am also deeply grateful to the Université de Montpellier for provid-
ing a four-year doctoral contract and an extraordinary research environment. I would also like to
C. R. Mathématique, 2021, 359, n 1, 1-5
Wenran Liu 5
thank A. Baldare and other colleagues for our regular discussions. I thank especially French Re-
public for everything given to me during my Masters and Doctoral studies, helping to ensure I’m
well prepared for my future.
This work was supported by a three-year doctoral contract and a one-year ATER at Université
de Montpellier.
This article was prepared while I had my position in Civil Aviation University of China (CAUC).
I would like to thank my current university for their hospitality.
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