Nicolas Jacon
email: nicolas.jacon@univ-reims.fr

Cédric Lecouvey
email: cedric.lecouvey@lmpt.univ-tours.fr

CORES OF ARIKI-KOIKE ALGEBRAS

We study a natural generalization of the notion of cores for l-partitions attached with a multicharge s ∈ Z l : the (e, s)-cores. We rely them both to the combinatorics and the notion of weight defined by Fayers. Next we study applications in the context of the block theory for Ariki-Koike algebras.

Introduction

Let F be a field of characteristic p ≥ 0. Let l and n be positive integers and s := (s 1 , . . . , s l ) ∈ Z l . Fix η ∈ F * . The Ariki-Koike algebra FH s n (η) associated with this datum is the unital associative F-algebra with a presentation by:

• generators: T 0 , T 1 ,..., T n-1 ,

• relations:

T 0 T 1 T 0 T 1 = T 1 T 0 T 1 T 0 , T i T i+1 T i = T i+1 T i T i+1 (i = 1, ..., n -2)
,

T i T j = T j T i (|j -i| > 1), (T 0 -η s1 )(T 0 -η s2 )...(T 0 -η s l ) = 0, (T i -η)(T i + 1) = 0 (i = 1, ..., n -1)
.

Let e ≥ 2 be minimal such that 1 + η + . . . + η e-1 = 0 in F so that e ∈ {2, 3, . . .} ∪ {∞}. If p = 0, e is the order of η as a root of unity. If p > 0, we have e = p if and only if η = 1.

The Ariki-Koike algebra, also called Hecke algebra of the complex reflection group G(l, 1, n), has been intensively studied during the last past decades. It is in relation with various important objects (e.g. rational Cherednik algebras, quantum groups, finite reductive groups etc.) and has a deep representation theory. Recently, the interest on these algebras have even grew up thanks to the introduction of the quiver Hecke algebras which has strengthened their relations with the theory of quantum groups and has allowed the study of their graded representation theory.

When l = 1, the Ariki-Koike algebra is nothing but the Hecke algebra of the symmetric group (when η = 1, it is isomorphic to the group algebra of the symmetric group over F). In the case where e is a prime number, the representation theory of this algebra presents strong analogies with the modular representation theory of the symmetric groups (in characteristic e): both structures admit a class of remarkable finite dimensional modules indexed by the set of partitions of n: the Specht modules. The simple modules are indexed by the set of e-regular partitions and the decomposition matrices, which control their representation theories, can be connected using an adjustment matrix. Using these decomposition matrices, one can obtain a natural partition of the set of Specht modules into smaller subsets called blocks. To each block, one can also associate another notion: the weight. Roughly speaking, this positive integer measures how "complicated" this block is. Remarkably, one can describe the blocks and the weights quite easily using well known combinatorial notions. In particular, the most "simple" blocks, the blocks with weight 0, can be described explicitly: they are singletons consisting of a unique Specht module labeled by an e-core partition. Moreover, any block with a given weight w may be obtained from a simple block by adding w times e-hooks to its associated e-core partition. Importantly, all these properties still make sense when e is an arbitrary positive integer (strictly greater than 1) on the side of the Hecke algebra.

2010 Mathematics Subject Classification: 20C08,20C20,05E15 When l > 1, one can also define analogues of Specht modules. They are now indexed by the set of l-partitions of n. The simple modules are then naturally indexed by certains generalizations of e-regular partitions which depend on s: the Uglov l-partitions. A notion of weight has also been provided by Fayers in [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF] which generalizes the case l = 1. Thanks to this definition, many properties known in the case l = 1 have been extended to the general case l ∈ N. In particular in [START_REF] Lyle | Blocks of cyclotomic Hecke algebras[END_REF], Lyle and Mathas have given a necessary and sufficient condition for two Specht modules for being in the same block. However, the generalization of e-core partitions and a generalization of the above process of adding e-hooks were missing in this picture (even if, as explained in §4.1, a non explicit definition of core multipartitions has been given by Fayers in [START_REF] Fayers | Simultaneous core multipartitions[END_REF]).

The aim of this paper is to study in details the (e, s)-core l-partitions, as introduced in a recent paper by the authors [START_REF] Jacon | Keys and Demazure crystals for Kac-Moody algebras[END_REF]. We show that this notion gives the right generalization of the e-core partitions: they correspond to the elements with weight 0 (with respect to Fayers definition of weight), and all l-partitions with a given weight may be obtained from them by adding analogues of e-hooks. As a consequence, we obtain a direct and simple generalization of what happen in the case l = 1. The only difference with this latter case is that, in our definition, the core of an l-partition associated with a multicharge is also a multipartition associated with a multicharge but, this last multicharge may be different from the initial one. To do this, the strategy is to show that essentially all the theory can be derived from the case l = 1 by introducing a weight-preserving map, defined by Uglov, from the set of (e, s)-core l-partitions to the set of e-core partitions.

The paper will be organized as follows. We first recall the definition of our main object of study: the (e, s)-cores and provide some of their combinatorial properties. The third part studies the weights of the l-partitions as defined by Fayers. We show how this notion can be interpreted in the theory of Fock spaces and computed via a combinatorial procedure detailed in our last section. This section will also explore some consequences of our results and will explain how our approach can simplify the block theory for Ariki-Koike algebras.

Generalized cores and abaci

In this section, after recalling certain classical combinatorial definitions regarding the partitions, we introduce the notion of (e, s)-core multipartition. Then we use abaci to associate to each (e, s)-core a certain core partition. This section will be purely combinatorial.

Partitions and multipartitions.

A partition is a nonincreasing sequence λ = (λ 1 , • • • , λ m ) of nonnegative integers. One can assume this sequence is infinite by adding parts equal to zero. The rank of the partition is by definition the number |λ| = 1≤i≤m λ i . We say that λ is a partition of n. By convention, the unique partition of 0 is the empty partition ∅.

More generally, for l ∈ Z >0 , an l-partition λ of n is a sequence of l partitions (λ 1 , . . . , λ l ) such that the sum of the ranks of the λ j is n. The number n is then called the rank of λ and it is denoted by |λ|. The set of l-partitions is denoted by Π l . The nodes of λ are by definition the elements of the Young diagram of λ:

[λ] := {(a, b, c) | a ≥ 1, c ∈ {1, . . . , l}, 1 ≤ b ≤ λ c a } ⊂ Z >0 × Z >0 × {1, . . . , l}.
(in the case of partition, the third coordinate which is always equal to 1 will be sometimes omitted.) Each l-partition will be identified with its Young diagram. We say that a node of λ is removable when one can remove it from the Young diagram of λ and still get the Young diagram of an l-partition µ. In this case, this node is called an addable node for µ.

Example 2.1. For l = 2, the 2-partition ((4), (2, 1)) of 7 is identified with its Young diagram:

, Now let us come back to the case l = 1 (we refer to [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF] for details). Let e ∈ N >1 . A rim e-hook (or simply an e-hook) of a partition λ is a connected subset of the rim of λ with exactly e nodes and which can be removed from λ to obtain another partition µ as in the following example.

× × × × × × × × ×
Starting from above, we can successively remove three rim 3-hooks (indicated with the symbol × above) By definition an e-core is a partition which does not admit any rim e-hook. The set of e-core partitions is denoted by C(e). If λ is an arbitrary partition, the e-weight ω e (λ) of λ is the number of consecutive e-hooks which can be removed from λ before obtaining an e-core, which is then denoted by Core e (λ). These notions are well-defined since both ω e (λ) and Core e (λ) do not depend on the order in which the rim e-hooks are removed from λ.

Example 2.3. Keeping the above example, we obtain ω 3 (λ) = 3 and Core 3 (λ) = (3, 1).

Let s = (s 1 , . . . , s l ) ∈ Z l . This is called a multicharge (a charge if l = 1). For an l-partition λ = (λ 1 , . . . , λ l ), one can associate to each node (a, b, c) of the Young diagram its residue ba + s c + eZ ∈ Z/eZ. The set of residues will be identified with {0, . . . , e -1}. If i ∈ Z/eZ, we denote by c e,s i (λ) the number of nodes with residue i in the l-partition. We moreover denote C e,s (λ) := (c 0 (λ), . . . , c e-1 (λ)).

Example 2.4. For l = 2, s = (0, 1) and e = 3 the residues of the nodes of the 2-partition ((4), (2, 1)) of 7 are given as follows:

0 1 2 0 , 1 2 0
Here we have C e,s ((4), (2, 1)) = (3, 2, 2).

2.2.

Abaci. The notion of abacus is convenient for reading the weight of a partition and obtaining its e-core. Let s ∈ Z. An abacus is a subset A of Z such that -i ∈ A and i / ∈ A for all i large enough. In a less formal way, each i ∈ A corresponds to the position of a black bead on the horizontal abacus which is full of black beads on the left and empty on the right. One can associate to λ and s ∈ Z an abacus L s (λ) such that k ∈ A if and only if there exists j ∈ N such that k = λ jj + s (Note that λ is assumed to have an infinite number of zero parts). Given an abacus L, one can easily find the unique partition λ and the integer s ∈ Z such that L s (λ) = L. Indeed, each part corresponds to a black bead of the abacus with length given by the number of empty positions at its left, the integer s is equal to x + 1 where x is the position of the rightmost black bead in the abacus obtained after sliding all the black beads as much as possible at the right in L.

Example 2.5. Let us take the partition λ := (5, 4, 2, 1, 1) and s = 0. The associated abacus L 0 (λ) may be represented as follows, where the positions at the right of the dashed vertical line are labelled by the non negative integers:

To L s (λ), one can associate an e-tuple of abacus L e s (λ) := (L 0 , . . . , L e-1 ). This is done as follows: for each black bead in position k in L s (λ), we write k = q.e + r where q ∈ Z and r ∈ {0, . . . , e -1} and we set a black bead in position q of the abacus L r . To picture this, write first the abacus L 0 , then immediately above the abacus L 1 and so on, so that all the beads associated with the entry 0 of each abacus appear in the same vertical line.

Example 2.6. For the partition (5.4.2.1.1) and e = 3, we get the following:

For each runner, sliding one black bead from right to left is equivalent to remove an e-rim hook in the associated partition. As a consequence, after performing this procedure as much as possible, we obtain an e-abacus which can be transformed (by reversing the previous procedure) into an abacus representing the e-core of λ. The number of moves of the black beads gives the e-weight of λ.

Example 2.7. If we do the above procedure for our example, we obtain:

The associated abacus is whose associated partition is (3, 1) and we have ω 3 (λ) = 3 as in Example 2.3.

Last, we will need an additional notation. For two abaci L and L ′ , we write L ⊂ L ′ if we have the following property: for each black bead in position i of the abacus L, there is a black bead in position i in L ′ .

Let now consider s ∈ Z l and an l-tuple of abaci (L s1 , . . . , L s l ). This l-abacus is, as above, conveniently pictured as follows: take first the abacus L s1 and then just above the abacus L s2 and so on, so that all the beads in position 0 of each abacus appear in the same vertical line.

Definition 2.8. Under the above notations, we say that the l-tuple of abaci (L s1 , . . . , L s l ) is (e, s)-complete if:

(

1) l = 1 and L s1 (λ 1 ) ⊂ L s1+e (λ 1 ), (2)
or l > 1 and

L s1 (λ 1 ) ⊂ L s2 (λ 2 ) ⊂ . . . ⊂ L s l (λ l ) ⊂ L s1+e (λ 1 ).
To a multicharge s ∈ Z l and an l-partition λ, is associated its l-abacus defined as the l-tuple (L s1 (λ 1 ), . . . , L s l (λ l )). It can be pictured exactly as above and will be called the (e, s)-abacus of λ. In fact, it does not depend on e but we have chosen here a notation similar to the notion of (e, s)-core below.

Example 2.9. Let s = (0, 3) and e = 4. We consider the 2-partition ((4, 1, 1), (1, 1)). Its associated (e, s)-abacus (L 0 (4.1.1), L 1 (1.1)) can be represented as follows:

2.3. The notion of (e, s)-cores. The notion of (e, s)-core has been introduced in [5, Def. 5.7] (the definition below is slightly different but it is an easy exercice to show the equivalence). This is a generalization of the notion of e-core partitions in the context of l-partitions associated with a multicharge. First let us introduce the notion of reduced (e, s)-core: Definition 2.10. Assume that s ∈ Z l then we say that the l-partition λ is a reduced (e, s)-core if its (e, s)-abacus

(L s1 (λ 1 ), . . . , L s l (λ l )) is (e, s)-complete.
To give a first study of this notion, let us introduce the following two sets:

A l e := {(s 1 , . . . , s l ) ∈ Z l | ∀(i, j) ∈ {1, . . . , l}, i < j, 0 ≤ s j -s i ≤ e}, A l e := {(s 1 , . . . , s l ) ∈ Z l | ∀(i, j) ∈ {1, . . . , l}, i < j, 0 ≤ s j -s i < e}.
Proposition 2.11. Assume that s ∈ Z l then if λ is a reduced (e, s)-core, we have s ∈ A l e . Proof. Assume that (L s1 (λ 1 ), . . . , L s l (λ l )) is (e, s)-complete then for each i = 1, . . . , l -2, we have L s i (λ i ) ⊂ L si+1 (λ i+1 ) which implies that s i+1 ≥ s i . We also have L s l ⊂ L s1+e and this implies that s l ≤ s 1 + e. This concludes the proof.

We now give the definition of our main object of interest. Let s ∈ Z l and e ∈ N >0 , denote by s := (s ′ 1 , . . . , s ′ l ) ∈ {0, . . . , e -1} l the multicharge such that s ′ i ≡ s i (mod e). Then we define σ s ∈ S l to be the unique permutation such that

s ′ σs(1) ≤ s ′ σs(2) ≤ . . . ≤ s ′ σs(l)
with the additional property that if s σs(i) = s σs(i+1) for i ∈ {1, . . . , l -1} then σ s (i) < σ s (i + 1). We set:

(1) s σs := (s ′ σs(1) , s ′ σs(2) , . . . , s ′ σs(l) ).

We then clearly have s σs ∈ A l e .

Definition 2.12. Let s ∈ Z l , we say that the l-partition λ is a (e, s)-core if the l-partition λ σs := (λ σs (1) , . . . , λ σs(l) ) is a reduced (e, s σs )-core. We denote by C l (e, s) the set of all (e, s)-cores.

As already noted in the previous paragraph, for l = 1, the (e, s)-core are exactly the e-cores. Thus, the set C 1 (e, s) does not depend on s ∈ Z and is exactly given by the set of e-cores C(e). One can also easily see that if λ is a (e, s)-core, each component λ j is an e-core.

Remark 2.13. Assume that s ∈ A l e and there exists i ∈ {1, . . . , l -1} such that s i = s i+1 . Then if λ is a reduced (e, s)-core we must have λ i = λ i+1 .

We need to check that the reduced (e, s)-cores are always (e, s)-cores. This is clear if s ∈ A l e but not if s ∈ A l e \ A l e . So let us assume that s ∈ A l e but s / ∈ A l e and let λ be a reduced (e, s)-core, this implies that there exists j ∈ {2, . . . , l} such that s j = s j+1 = . . . = s l = s 1 + e. Then the abacus (L sj -e (λ l ), . . . , L s l -e (λ l ), L s1 (λ 1 ), . . . , L sj-1 (λ l-1 )) is (e, (s je, . . . , s le, s 1 , . . . , s j-1 ))-complete. By the above remark, we thus obtain λ j = . . . = λ l = λ 1 . We so conclude that in the case where s ∈ A l e , the (e, s)-cores are exactly the reduced (e, s)-cores.

Remark 2.14. The above definition can be formulated in terms of β-numbers and symbols (see [5, §5.1]), which gives an equivalent definition of the set of (e, s)-cores. We get that λ is a (e, s)-core if and only if

• for all c = 1, . . . , l -1 and j ∈ Z >0 , there exists i ∈ Z >0 such that λ σs(c) j

-j + s ′ σs(c) = λ σs(c+1) i -i + s ′ σs(c+1) ,
• for all j ∈ Z >0 , there exists i ∈ Z >0 such that

λ σs(l) j -j + s ′ σs(l) = λ σs(1) i -i + s ′ σs(1) + e.
Remark 2.15. As already noticed, the irreducible representations of the Ariki-Koike algebras associated with the datum (e, s) are naturally labeled by a distinguished set of l-partitions called Uglov l-partitions. In the particular case where s ∈ A l e , these l-partitions are called FLOTW l-partitions and it is easy to check that any (e, s)-core is then a FLOTW l-partition in the sense of [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF]Th. 5.8.5]. Now for an arbitrary choice of s, there is an explicit bijection between the set of FLOTW partitions associated with (e, s σs ) and the set of Uglov l-partitions associated with (e, s) (this bijection is described in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight Uv( sle)-modules of higher level[END_REF]). It is easy to see that this bijection restricted to the set of (e, s)-cores sends λ to λ σ -1 s . This implies that (e, s)-cores are always Uglov l-partitions. This fact has a representation theoretic meaning as we will see in the following.

Example 2.16. Let l = 2, e = 3 and s = (0, 1). Consider the 2-partition ((1, 1), (3, 1, 1)). With the above notation, we have σ = Id and s ′ = s = s σs . The associated 2-abacus is and we see that we here have a (e, s)-core. As a consequence, taking s = (10, 0), we have that the 2-partition ((3, 1, 1), (1, 1)) is a (e, s)-core. . We now show how to associate to a reduced (e, s)-core λ a certain e-core partition that we denote by τ e,s (λ) and conversely. This construction uses a map defined by Uglov [10, §4.1] (see also [11, §3.1] ) which associates a partition to any charged l-partition. We will be interested in the restriction of this map to the set of reduced (e, s)-cores.

Let λ be an l-partition. We consider the l-abacus (L s1 (λ 1 ), . . . , L s l (λ l )). Then we construct an associated 1-abacus as follows. For each c = 1, . . . , l and for each black bead in position k of the abacus L sc , we write k = q.e + r with q ∈ Z and r ∈ {0, . . . , e -1}. Then we set a black bead in our new abacus in position (lc)e + qel + r. We then define τ e,s (λ) to be the partition associated with this resulting abacus. We obtain a map τ e,s : Π l → Π 1 which will be called the Uglov map. Let us illustrate the computation of the Uglov map by two following examples.

Example 2.17. We resume Example 2.9. The above procedure gives the following abacus:

We thus get τ e,s (λ) = (5, 2, 2, 1, 1, 1).

Example 2.18. Let s = (0, 1, 2) and e = 4. We consider the 3-partition ((2), ( 1), (1, 1)), the associated 3-abacus (L 0 (2), L 1 (1), L 2 (1, 1)) can be written as:

The above procedure gives the following abacus:

We thus get τ e,s (λ) = (7, 4, 2, 2).

The map τ e,s is not surjective in general but it is clearly injective. Proof. This is clear by the characterization of e-cores with abaci in the last section.

Proposition 2.20. The map

τ l e : {(λ, s) | s ∈ A l e , λ ∈ C l (e, s)} → {(λ, s) | s ∈ Z, λ ∈ C 1 (e)} (λ, s) → (τ e,s (λ), 1≤i≤l s i ) is bijective.
Proof. First, the map is well defined. Indeed, assume that λ ∈ C l (e, s) with s ∈ A l e . Then λ satisfies the property in Definition 2.8 (2) but this implies that the partition τ e,s (λ) satisfies (1) of Definition 2.8. We deduce that it is an e-core as desired. Now let us prove that the map is bijective. Let s ∈ Z and λ ∈ C 1 (e). Then we have an associated (e, s)-abacus associated with this datum and by construction, there exists a unique λ ∈ Π l and s ∈ A l e such that τ e,s (λ) = λ and 1≤i≤l s i = s. It thus suffices to prove that λ ∈ C l (e, s). But it follows from the fact that its (e, s)-abacus is complete because λ is a e-core. We now give two important results showing remarkable links between λ and τ e,s (λ). The first one compare the number of nodes in the two Young diagrams with a given residue. (τ e,s (∅)) = c e,s i (λ) + l.c e,s 0 (λ) Proof. We will argue by induction on the rank of λ. If this rank is 0 then λ is the empty l-partition and the result is trivial. Assume now that λ is an l-partition of rank n > 0. Let µ be an l-partition of rank n -1 which is obtained from λ by removing a removable i-node for some i ∈ Z/eZ. Assume first that i = 0(mod e). Then it is easy to see that τ e,s (µ) is obtained from τ e,s (λ) by removing a removable i-node. As a consequence, we have c (e,s) j

(τ e,s (λ)) = c (e,s) j

(τ e,s (µ)) and c e,s j (λ) = c e,s j (µ) if j is different from i modulo eZ. Thus, we get c (e,s) i (τ e,s (λ)) = c (e,s) i (τ e,s (µ)) + 1 and c e,s i (λ) = c e,s i (µ) + 1. So the formula is still true by induction.

Assume now that i = 0(mod e). In this case, we still have c e,s j (λ) = c e,s j (µ) if j = 0 and c e,s 0 (λ) = c e,s 0 (µ) + 1. Now, we need to see how τ e,s (λ) is obtained from τ e,s (µ). The node that we add to µ to obtain λ corresponds to a black bead in the abacus of τ e,s (λ) and to another in the abacus of τ e,s (µ). Let us denote by m the number of black beads between theses two positions (not including these two) in the abacus (the number is the same in both abaci). Then τ e,s (λ) is obtained by removing a part of length x > 0 ending by a node with residue e -1, adding one node to the m parts above and adding one part of length x + l.em + 1 which ends with a node with residue 0. This thus consists in x + l.e + 1 consecutive nodes. More precisely, to obtain τ e,s (λ) from τ e,s (µ) we add l + 1 nodes with residue 0, and l nodes of residue j for all j = 0. Thus we obtain c (τ e,s (∅)) = c e,s i (µ) + l.c e,s 0 (µ) which permits to conclude.

Recall the notation C e,s (λ) introduced in Subsection 2.1 for the multiset of residues of a multipartition. Proof. This directly follows from the previous proposition.

Last, we will need a useful property which permits to compare the number of removable and addable i-nodes of λ and τ e,s (λ). To do this, we denote by M s i (λ) the number of addable nodes of λ minus the number of removable nodes of λ. Proposition 2.24. For all λ ∈ Π l , s ∈ A l e and i ∈ Z/eZ, we have:

M s i (λ) = M s i (τ e,s (λ)) if i = 0, M s i (τ e,s (λ)) + l -1 if i = 0. Proof.
First, consider a partition λ and a charge s and write its associated 1-abacus. Let i ∈ Z/eZ. Let x ∈ Z be such that x ≡ i(mod e). Note that each black bead in the abacus corresponds to a part λ i of the partition λ (the position of this bead being given by λ ii + s).

• If we have a black bead in position x and a black bead in position x -1, this does not correspond to any removable nor addable i-node. • If we have a black bead in position x and no black bead in position x -1, this does correspond to one removable i-node. • If we have no black bead in position x and a black bead in position x -1, this does correspond to one addable i-node.

Of course no black bead in position x and no black bead in position x -1 means that we have no associated addable or removable i-node.

Let us fix r << 0 and let us now consider all the black beads in position greater (or equal) than r.e in the abacus, for each i ∈ Z/eZ, write B r i (λ, s) the number of such black beads in position x in the abacus with x ≡ i(mod e). This number is finite by assumption. The above discussion shows that:

M s i (λ) = B r i-1 (λ, s) -B r i (λ, s) if i = 0 B r i-1 (λ, s) -B r i (λ, s) + 1 if i = 0
(the last equality comes from the fact that we have a black bead in position r.e -1). Now let (λ, s) ∈ Π l × Z l . We fix again r << 0, by the discussion above, for each c ∈ {1, . . . , l} and i ∈ Z/eZ, we have:

M sc i (λ sc ) = B r i-1 (λ sc , s c ) -B r i (λ sc , s c ) if i = 0 B r i-1 (λ sc , s c ) -B r i (λ sc , s c ) + 1 if i = 0
By construction, we obtain for all i ∈ Z/eZ

B lr i (τ e,s (λ), s) = c=1,...,l B r i (λ sc ).
As in addition, we also have:

M s i (λ) = c=1,...,l M sc i (λ sc ),
we can conclude.

Example 2.25. Let us illustrate the proof with the 2-partition ((4, 1, 1), (1, 1)) and the multicharge (0, 3) of Example 2.17 (here e = 4). The Young diagram with its residues is:

  0 1 2 3 3 2 , 3 2
 
We have seen that τ e,s ((4, 1, 1), (1, 1)) = (5, 2, 2, 1, 1, 1) with s = 0 + 3 = 3. Thus the associated Young diagram with residues is:

3 0 1 2 3 2 3 1 2 0 3 2
On the one hand, we have M (0,3) 0 ((4, 1, 1), (1, 1)) = 3 and M 3 0 (5, 2, 2, 1, 1, 1) = 2. On the other hand, we get M (0,3) 1

((4, 1, 1), (1, 1)) = M 3 1 (5, 2, 2, 1, 1, 1) = 2, M (0,3) 2 ((4, 1, 1), (1, 1)) = M 3 2 (5, 2, 2, 1, 1, 1) = -2 and M (0,3) 3 ((4, 1, 1), (1, 1)) = M 3 3 (5, 2, 2, 1, 1, 1) = -1.

Cores and weights for Ariki-Koike algebras

In this section, we review the notion of weight for Ariki-Koike algebras as introduced by Fayers in [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF]. To avoid a possible confusion with the notion of weight for the type A affine Kac-Moody algebra, Fayers weights will be refereed as core-weights in the sequel. We will notably interpret them in the representation theory of the type A affine Kac-Moody algebra.

3.1. Block weights for Ariki-Koike algebras and relations with Fock spaces. The block weight of an l-partition for a given multicharge is defined in [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF] as follows.

Definition 3.1. Let s ∈ Z l , e ∈ Z >0 and λ ∈ Π l , then the block (e, s)-weight (or simply block weight) of λ is p (e,s) (λ) = 1≤i≤l c e,s si (λ) -

1 2 i∈Z/eZ (c e,s i (λ) -c e,s i-1 (λ)) 2 .
Remark 3.2. From this definition, it is immediate to see that, under the notation of (1), we have for all λ ∈ Π l , p (e,s) (λ) = p (e, s σs ) (λ σs )

We can thus again restrict ourselves to the case s ∈ A l e . This notion of block weight has a natural interpretation in the representation theory of Kac-Moody algebras that we shall now make explicit. Consider the Kac-Moody algebra g of type A

(1) e-1 . Let h be a Q-vector space with basis {h 0 , . . . , h e-1 , D}. Let {Λ 0 , . . . , Λ e-1 , δ} be the dual basis with respect to the pairing:

., . :

h * × h → Q defined by: Λ i , h j = δ ij , Λ i , D = δ, h i = 0, δ, D = 1 (0 ≤ i, j ≤ e -1)
.

The Λ i with 0 ≤ i ≤ e -1} are called the fundamental weights. The simple roots α i with 1 ≤ i ≤ e -1 are the elements of h * defined by:

α i := -Λ i-1 + 2Λ i -Λ i+1 + δ i,0 δ
where the subscript have to be understood modulo e. For 0 ≤ i, j ≤ e -1, we denote by a ij the coefficient of Λ j in α i . Then the matrix A := (a ij ) 0≤i,j≤e-1 is the Cartan matrix of sl e . As (Λ 0 , α 0 , . . . , α e-1 ) is a basis of h * , one can define a symmetric non degenerate bilinear form on h * by setting:

(α i , α j ) = a ij , (Λ 0 , α i ) = δ i,0 , (Λ 0 , Λ 0 ) = 0 (0 ≤ i, j ≤ e -1}.
We then derive

(Λ i , α j ) = δ i,j , (δ, α i ) = 0 (0 ≤ i, j ≤ e -1)
,

where δ = α 0 + . . . + α e-1 is the null root. We have (δ, δ) = 0 and (δ, Λ i ) = 1 for all 0 ≤ i ≤ e -1.

Let now consider v an indeterminate and write U v ( sl e ) for the quantum affine algebra of type A

e-1 . This is an algebra over Q(q) with generators e i , f i , t ±1 i (0 ≤ i ≤ e -1) and δ, the relations will be omitted (see [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF]Def. 6.1.3]). Fix s ∈ Z l and consider the associated Fock space

F s := λ∈Π l Q(v)λ
with basis the l-partitions. There is a simple U v ( sl e )-action on F s (depending on s) which endows it with the structure of an integrable U v ( sl e )-module (see [START_REF] Uglov | Canonical bases of higher level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF]Th. 2.1]). In particular, this means that F s is the direct sum of its weight subspaces. The elements of the basis λ ∈ Π l are weight vectors whose weights can easily be calculated as follows:

α e,s (λ) := -∆ s δ + Λ s1 + . . . + Λ s l - 0≤i≤e-1 c e,s i (λ)α i ,
where Λ s := Λ s1 + . . . + Λ s l and

∆ s := 1 2 1≤i≤l ( s 2 i e -s i ) -( s ′2 i e -s ′ i ) ,
with s ′ i is the representant modulo e of s i in {0, 1, . . . , e -1}. Then we set:

λ (e,s) := (α e,s (λ), α e,s (λ)) 2 , Λ s = (Λ s , Λ s ) 2 , so that ∅ (e,s) = 1 2 (-∆ s δ + Λ s , -∆ s δ + Λ s ) = -∆ s l + Λ s .
Example 3.3. For l = 1 and s = 0 we have ∆ s = 0 and Λ s = 0 so that ∅ (e,0) = 0.

There is an easy way to calculate λ (e,s) . The proof is in fact contained in [START_REF] Yvonne | Bases canonique d'espaces de Fock de niveau supérieur[END_REF]Lemme 4.13] and is similar to [START_REF] Lascoux | Hecke algebras at roots of unity and crystal bases of quantum affine algebras[END_REF]Prop 8.1]. We give it below for the convenience of the reader. Proposition 3.4. Let s ∈ Z l and let λ ∈ Π l . Assume that µ ∈ Π l is such that one can add an addable i-node to µ to obtain λ. Then we have µ (e,s)λ (e,s) = M s i (µ) -1 where M s i (µ) is the number of addable nodes of µ minus the number of removable nodes of µ. Proof. Under the above notation, we have that:

µ (e,s)λ (e,s) = (1/2) ((α e,s (µ), α e,s (µ) -(α e,s (µ)α i , α e,s (µ)

-α i )) = (1/2) (2(α e,s (µ), α i ) -(α i , α i )) = (α e,s (µ), α i ) -1
Now, by the previous definition of the weight α e,s , we have α e,s (µ) = 0≤i≤e-1 a i Λ i + dδ if and only if δ.µ = dµ and t i µ = v ai µ. As by [10, Th. 2.1], it is known that t i µ = v M s i (µ) µ, we can conclude.

It is now easy to compute the block weight p (e,s) .

Proposition 3.5. Let s ∈ Z l and let λ ∈ Π l . We have λ (e,s) = ∅ (e,s)p (e,s) (λ).

Proof. We can write:

(α e,s (λ), α e,s (λ

)) = (-∆ s δ + Λ s - 0≤i≤e-1 c e,s i (λ)α i , -∆ s δ + Λ s - 0≤i≤e-1 c e,s i (λ)α i ) = 2 ∅ (e,s) -2 0≤i≤e-1 c e,s i (λ)(Λ s , α i ) + 0≤i,j≤e-1 c e,s i (λ)c e,s j (λ)(α i , α j ) = 2 ∅ (e,s) -2 1≤i≤l c e,s si (λ) + 0≤i,j≤e-1 (-c e,s i (λ)c e,s i-1 (λ) + 2c e,s i (λ) 2 -c e,s i (λ)c e,s i+1 (λ)) = 2 ∅ (e,s) -2 1≤i≤l c e,s si (λ) + 0≤i,j≤e-1 (c e,s i (λ) -c e,s i-1 (λ)) 2
= 2 ∅ (e,s) -2p (e,s) (λ).

Combining these two propositions leads to:

Proposition 3.6. Let s ∈ Z l and λ ∈ Π l . Assume that µ ∈ Π l is such that one can add an addable i-node to µ to obtain λ. Then we have p (e,s) (λ)p (e,s) (µ) = M s i (µ) -1. The above proposition will be a crucial ingredient in the proof of one of our main results in the next section.

3.2. Computation of weights. We here want to prove the following theorem. It mainly asserts that the block weight for an l-partition associated with a multicharge can always been computed in terms of the usual block weight for a partition. This result uses the map τ e,s defined in the previous section only for the multicharge in A where s = 1≤i≤l s i Proof. We argue by induction on the rank n of λ. Assume that n = 0. Then p (e,s) (λ) = 0 and by Proposition 2.19, τ e,s (λ) is an e-core so its weight is equal to 0. Assume now that n > 0. Let µ be an l-partition obtained from λ by deleting a removable i-node for some i ∈ Z/eZ. By Proposition 3.6, we get p (e,s) (λ)p (e,s) (µ) = M s i (µ) -1. Now we have two cases to consider.

• Assume that i = 0 then τ e,s (µ) is also obtained from τ e,s (λ) by deleting a removable i-node. We have p (e,s) (τ e,s (λ))p (e,s) (τ e,s (µ)) = M s i (τ e,s (µ)) -1. We can thus conclude by induction using Proposition 2.24 • Assume that i = 0 then to simplify the notation write µ := τ e,s (µ) and λ := τ e,s (λ). We assume that the removable node corresponds to a black bead of the 1-abacus of τ e,s (λ) in position x. By hypothesis, there is no black bead in position x -(l -1)e -1, because τ e,s (λ) and τ e,s (µ) have the same abacus except that the black beads between position x and x -(l -1)e -1 are exchanged (and so are the empty position in the remaining one). Again, we will consider two cases:

-Assume that there is no black bead in position x -(l -1)e. Then one can consider ν the partition defined by the abacus obtained by moving the bead in position x from the abacus of λ to the position x -(l -1)e. Its weight p (e,s) (ν) is equal to p (e,s) (λ) -(l -1) because ν is obtained from λ by removing l -1 hooks from λ. Now we have by Proposition 3.6:

p (e,s) (ν) -p (e,s) (µ) = M s 0 (µ) -1. We conclude that p (e,s) (λ) -p (e,s) (µ) = M s 0 (µ) + l that is, by Proposition 2.24 p (e,s) (λ) -p (e,s) (µ) = M s 0 (µ) -1.
-Assume that there is no black bead in position x -(l -1)e. Then we proceed in the opposite way: we define ν to be the partition obtained from λ by moving the bead in position x -(l -1)e to the position x -(l -1)e. Then µ is obtained from ν by moving the bead in position x to the position x -(l -1)e (which consists in removing (l -1) e-hooks). We conclude exactly as in the previous case.

What can we do in the case where s / ∈ A l e ? In fact, one can use the procedure in §2.3 and associate to λ and s a multicharge s σs ∈ A l e and a multipartition λ σs . It is clear from the definition that: p (e,s) (λ) = p (e, s σs ) (λ σs ) which thus gives an effective way to compute the block weight in all cases.

Remark 3.8. A (maybe more direct) proof might also be obtained using Proposition 2.22 but the above one has the advantage to avoid cumbersome computations.

Further remarks and applications

In this section, we show how our main results simplify the block theory for Ariki-Koike algebras. In particular we show the relations of our work with some results by Fayers. By the definitions of cores and block weights, one can assume that s ∈ A l e in this section. However, we will try to explain how all our results can be adapted to the general case s ∈ Z l . Proof. Let λ ∈ Π l , by Theorem 3.7, we have:

p (e,s) (λ) = p (e, 1≤i≤l si) (τ e,s (λ)) so λ is of block weight 0 if and only if τ e,s (λ) is of block weight 0. Now, we know that the e-cores are exactly the partitions with block weight 0 and we can thus conclude thanks to Proposition 2.20. and apply our procedure, one can check that we obtain the same core.

Remark 4.6. When l=1 and given an e-core λ, one can obtain directly all the partitions in a fixed block with a given core weight w by adding w hooks to λ while we stay in the set of partitions. This process is less direct if l > 1. Let λ be a (e, s)-core. We can assume that s ∈ A l e . Then if we perform w "inverse" elementary moves on its l-abacus, we obtain an l-partition µ associated with a multicharge s ′ and the core of µ in FH s ′ n (η) is (λ, s). Now, still starting from the e-core, if we do w other "inverse" elementary moves on its l-abacus, one may obtain another l-partition ν but also another multicharge s ′′ . Thus µ and ν will be in the same block of FH e,s n (η) if and only if s ′ = s ′′ . This means, one can obtain all the l-partitions in a fixed block of FH s ′ n (η) as in level 1 except we have to keep only those with associated multicharge s ′ .

4.2. Multipartitions of small (block) weights. As already noted in [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF], in level l > 1, each block of block weight 0 contains exactly one (e, s)-core and thus is a simple block, as in the case l = 1. This implies in particular that the Specht modules labeled by these l-partitions are irreducible and that they coincide with their projective cover. This shows that the (e, s)-cores are always Uglov l-partitions. This is consistent with remark 2.15.

In [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF]Th. 4.4], Fayers has given a description of the blocks of block weight 1. Using our approach, we here give an explicit characterization of these blocks. When l = 1, such blocks always contain exactly e partitions. We will see that when l > 1, this will depend on the multicharge we choose. Let v ∈ Z l and consider an l-partition µ with block weight 1. The core af µ is the same as the core of the l-partition µ σv associated with the multicharge v σv ∈ A l e . This means that we can in fact assume that v ∈ A l e . Now the l-abacus of a l-partition µ with block weight 1 for the multicharge v can be derived from a reduced (e, s)-core λ where s ∈ A l e by performing one inverse elementary operation on the abacus of λ (that is by inversing the procedure described in §4.1). This consists in moving a black bead in position x from a runner i ∈ {2, . . . , l} to the position x of the runner i -1, or from the runner 1 in position x to the runner l in position x + e, if possible.

All the l-partitions µ of weight 1 are then obtained as follows:

• For all i ∈ {1, . . . , l -1}, if s := (v 1 , . . . , v i-1 -1, v i + 1, . . . , v l ) is such that s ∈ A l e , they are obtained from a (e, s)-core λ by doing one inverse elementary operation in its abacus from the runner i to the runner i -1. By definition of our notion of core, we can exactly do v i + 1 -(v i-1 -1) inverse elementary operations between the runner i -1 and the runner i. Thus, we have exactly v iv i-1 + 2 multipartitions obtained from a given such core and they are all of the same rank |λ| + v i+1v i + 1.

• If s := (v 1 + 1, . . . , v l -1) is such that s ∈ A l e , they are obtained from a (e, s)-core λ by doing one inverse elementary operation in its abacus from the runner 1 to the runner l. We have exactly v 1v l + 2 + e multipartitions obtained from a given such core and they are all of the same rank |λ| + v 1v l + e + 1.

Remark 4.7. By [START_REF] Lyle | Blocks of cyclotomic Hecke algebras[END_REF], the procedure described in this paper also gives the description of the blocks for affine Hecke algebras of type A.

Remark 4.8. It is likely that the results of this paper may be used to study the block theory for the cyclotomic Hecke algebras of type G(r, p, n). Besides, Theorem 3.7 gives a correspondence between (e, s)-core and ecores which could induce similarities between blocks of Ariki-Koike algebras and blocks of Hecke algebras of type A. We will come back to these questions in future works.

4.3.

Examples. We end this section with an example of computation of block weights and cores. We here take n = 4, e = 4 and s = (0, 1). Here is a table giving the block weight and the core of each 2-partition.

Example 2 . 2 .

 22 Let λ := (5, 4, 2, 1, 1) and e = 3. The Young diagram of λ is:

2. 4 .

 4 Uglov map. Let s ∈ A l e

Proposition 2 .

 2 19. Let s ∈ A l e , then τ e,s (∅) is an e-core.

Remark 2 .

 2 21. If we consider s / ∈ A l e and a (e, s)-core λ then we have τ e,s (λ) / ∈ C 1 (e) in general.

Proposition 2 .

 2 22. Let λ ∈ Π l and s ∈ A l e . Set s = 1≤i≤l s i . For all i = 0, 1, . . . , e -1, we have: c (e,s) i (τ e,s (λ))c (e,s) i

1 .

 1 (e,s) i (τ e,s (λ)) = c (e,s) i (τ e,s (µ)) + l if i = 0 and c (e,s) 0 (τ e,s (λ)) = c (e,s) 0 (τ e,s (µ)) + l + Now we have by induction for all i ∈ {0, . . . , e -1}: c (e,s) i (τ e,s (µ))c (e,s) i

Corollary 2 .

 2 23. Let λ ∈ Π l , µ ∈ Π l and s ∈ A l e . We have C e,s (τ e,s (λ)) = C e,s (τ e,s (µ)) ⇐⇒ C e,s (λ) = C e,s (µ).

Theorem 3 . 7 .

 37 Let s ∈ A l e and λ ∈ Π l . We have:p (e,s) (λ) = p (e,s) (τ e,s (λ))

4. 1 .

 1 Cores of multipartitions. Let us start with an easy corollary of Theorem 3.7.

Corollary 4 . 1 .

 41 Assume s ∈ A l e . Then, the reduced (e, s)-core are exactly the elements of block weight 0.

If we take s ∈ Z l , then we have already noticed that:

p (e,s) (λ) = p (e, s σs ) (λ σs ).

Since in addition λ σs is a reduced s σs -core if and only if λ is a (e, s)-core, we conclude that in the general case, the (e, s)-cores are exactly the elements of core weight 0.

In [START_REF] Fayers | Simultaneous core multipartitions[END_REF], Fayers has also introduced a notion of core for an l-partition associated with a multicharge. His definition is the following one. Let s ∈ A l e . Then an l-partition λ is a (e|s)-core if there is no other l-partition µ such that C e,s (λ) = C e,s (µ). In fact this coincides with our notion of (e, s)-cores. Indeed, by the results in [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF], the (e|s)-core multipartitions are exactly the elements of weight 0 (see [START_REF] Fayers | Simultaneous core multipartitions[END_REF]Rem 2.3.1]) which are exactly the (e, s)-cores by the above corollary. In other words, Definition 2.8 thus reveals the combinatorial structure of the (e|s)-cores introduced by Fayers. Let us explain the consequences concerning the block theory of Ariki-Koike algebras and especially, the similarities and the differences with the case l = 1 that is, the case of the symmetric group.

Let FH s n (η) be the Ariki-Koike algebra as defined in the introduction. The representation theory of FH s n (η) is controlled by its decomposition matrix which we now briefly define. For all l-partition λ; one can associate a certain finite dimensional FH s n (η)-module S λ called a Specht module. For each M ∈ Irr(FH s n (η)), we have the composition factor [S λ : M ]. The matrix:

is the decomposition matrix. By definition, two l-partitions λ and µ lie in the same block if there exists a sequence (M 1 , . . . , M r ) of simple FH s n (η)-modules and a sequence of l-partitions (λ 1 , . . . , λ r+1 ) with λ 1 = λ, λ r+1 = µ and for all i ∈ {1, . . . , r}, we have [S λi : M i ] = 0 and [S λi+1 : M i ] = 0. When l = 1, we know that two partitions are in the same block if and only if they have the same e-core and that their common weight is the number of e-hooks that can be removed to obtain this e-core. For l > 1, a criterion has been provided by Lyle and Mathas [START_REF] Lyle | Blocks of cyclotomic Hecke algebras[END_REF] but it does not consist in any notion of hook or cores. It asserts that λ and µ are in the same block of FH s n (η) if we have C e,s (λ) = C e,s (µ). Let λ be an l-partition of rank n. To describe the blocks, one can restrict ourselves to the case s ∈ A l e (as usual the general case is derived by using the transformations in §2.3). We consider the (e, s)-abacus (L s1 , . . . , L s l ) of λ. An elementary operation on this abacus is defined as a move of one black bead from one runner of the abacus to another satisfying the following rule.

(1) If this black bead is not in the top runner, then we can do such an elementary operation on this black bead only if there is no black bead immediately above (that is in the same position on the runner just above). In this case, we slide the black bead from its initial position, in a runner i, to the runner i + 1 located above in the same position. The resulting l-abacus corresponds to an l-partition of rank ns i+1 + s i -1. Indeed, when we add a black bead in the runner i + 1 the rank becomes n + Ns i+1 -1 for a certain integer N and when we remove a black bead from the runner i in the same position, the rank becomes

(2) If this black bead is in the top runner in position x, then we can do such an elementary operation only if there is no black bead in position xe on the lowest runner. In this case, we slide the bead to the position xe of the lowest runner. As above, the rank of the resulting l-partition is n -

Note that, after this procedure, the resulting multicharge associated with the l-abacus may not be in A l e but this is not a problem: we can still perform it in the resulting abacus. At the end, by construction, we obtain an l-abacus (L v1 , . . . , L v l ) satisfying:

This abacus is complete. Thus, by Proposition 2.11, this corresponds to an l-partition µ and a multicharge v ∈ A l e such that µ is a reduced (e, v)-core. Definition 4.2. The core of the l-partition λ associated with a multicharge s is the pair (µ, v) attached to λ and s by the previous procedure. Doing an elementary operation on the (e, s)-abacus of λ as above is equivalent to remove one e-hook on the Young diagram of τ e,s (λ). As a consequence, by Theorem 3.7 and by the definition of the Uglov map, p (e,s) (λ) is the number of elementary operations we have made in this process to obtain our final abacus. The rank of the multipartition can also been computed thanks to the above remarks. The fact that this does not depend on the order in which the elementary operations are performed follows from the case l = 1. Remark 4.3.

(1) Take a black bead in the runner i of the l-abacus of λ in position x such that there is no black bead in position x-e in the same runner. Then one can always perform a series of l elementary operations (as defined in the previous procedure) to obtain the same abacus except that the black bead in position x moves to the position x-e of the same runner. Indeed, let us denote by b 1 the bead in position x in runner i, and consider all the beads b 2 , . . . , b k in position x and runner i 2 , . . . , i k with i < i 2 . . . < i k .

Consider also the beads b k+1 , . . . , b r in position xe and runner i k+1 , . . . , i r with i k+1 . . . < i r < i.

Then we can slide the bead b r in position xe in runner i, and then slide the bead b r-1 to the position previously occupied by b r and so on. At the end, we obtain the desired abacus and we have made l elementary operations to do that. In this case, the rank of the resulting l-partition is equal to

The l-partition so obtained is just the l-partition λ where a rim e-hook has been removed in λ i . This is thus consistent with our result. Nevertheless, this shorter hook removal procedure does not suffice to produce the core of λ for it can only yield a sequence of l cores, that is a multicore. ( 2) In [START_REF] Fayers | Weights of multipartitions and representations of Ariki-Koike algebras[END_REF], the notion of multicore is used instead of our notion of core. From an arbitrary l-partition λ, one can indeed associate another l-partition, with a smaller block weight, which may be seen as an "intermediate" between the given l-partition and its e-core in the sense of Definition 4.2. To do this, we can simply take the e-core of each partition or apply a sequence of elementary operations as we have just explained. We have already seen that the (e, s)-cores are multicores but the converse is not true in general.

They are in the same block because C e,s (λ) = C e,s (µ). Now the 3-abacus of λ is

To determine its core, we perform the above procedure and we obtain the following 3-abacus:

the associated (e, s)-core is the 3-partition ((1), ∅, ∅) together with the multicharge (0, 2, 2) and the weight is 8 because we perform 8 moves of beads to obtain this core. Now if we consider µ whose 3-abacus is (∅; (0, 1)) 2 Note that the core of the blocks of block weight 1 are always associated with the same multicharge, which is v = (0, 3) and there is two different cores which gives 3 = v 2v 1 elements in the same block in both cases. The rank of this core is the n -(s 2s 1 + 1) = 2. This is consistent with the results of the previous section.

Let us consider now the multicharge s := (0, 1, 3) with e = 4 and n = 4. Then

• we have seven (e, s)-cores: (∅, (3, 1), (∅, (1), (1, 1, 1)), ((1), (2, 1), ∅), ((2), (1, 1), ∅), (∅, (2), (1, 1)), ((1, 1), ∅, (2)), ((1), ∅, (2, 1)). • We have three blocks of clock weight 1 which are:

-{((2, 1), ∅, (1)), ((1, 1), ( 1), ( 1)), (∅, (1, 1, 1), ( 1))} with (((1), ∅, (1)), (-1, 2, 3)) as a core.

-{((3), (1), ∅), ((1, 1), ( 1), ( 1)), (∅, (2, 2), ∅)} with (∅, (1, 1), ∅), (-1, 2, 3)) as a core.

-{(∅, (1), (3)), (2, 1, 1), ((2, 1), (1), ∅)} with ((1), (1), ∅), (1, 1, 2)) as a core. Again, this is consistent with the results of the previous section.


Ma Feng

Maryam Darabi

Emilie Tubeuf

Aurélie Canicio

Marie Lhomme

Eric Frisdal

Sandrine Lanfranchi-Lebreton

Lucrèce Mathéron

Fabiana Rached

Maharajah Ponnaiah

Lucrèce Matheron

Carlos V D Serrano

Raul D Santos

Fernando Brites

Gerard Bolbach

Emmanuel Gautier

Thierry Huby

Alain Carrie

Eric Bruckert

Maryse Guerin

Philippe Couvert

Philippe Giral

Philippe Lesnik

Wilfried Le Goff

Isabelle Guillas

PhD Anatol Kontush
email: anatol.kontush@sorbonne-universite.fr

Free cholesterol transfer to

Keywords: lipoprotein lipase, atherosclerosis, postprandial lipid metabolism, apolipoproteins, chylomicrons, HDL function, HDL metabolism, LCAT, lipoprotein metabolism, lipid transfer proteins

Introduction

Low levels of plasma high-density lipoprotein-cholesterol (HDL-C) represent a well-established cardiovascular risk factor. 1,2 However, therapeutic raising of HDL-C on a background of statin treatment does not exert expected beneficial effects on cardiovascular disease (CVD). 3 Furthermore, large-scale epidemiological studies reveal that both cardiovascular and overall mortality are paradoxically increased at extremely high HDL-C levels, [4][5][6] an observation which might account for the negative results of HDL-C-raising trials. It has therefore been suggested that it is not HDL-C itself that is causatively related to atheroprotection but rather a cardioprotective HDL function, which cannot always be reliably estimated through the assay of HDL-C. [START_REF] Rader | The not-so-simple HDL story: Is it time to revise the HDL cholesterol hypothesis?[END_REF] The major atheroprotective function of HDL is presently thought to involve cholesterol efflux from arterial wall cells with its subsequent transport to the liver for excretion in a process of reverse cholesterol transport (RCT). [START_REF] Rader | The not-so-simple HDL story: Is it time to revise the HDL cholesterol hypothesis?[END_REF] Such "HDL flux hypothesis" is based on negative associations of CVD with in vitro measurements of cellular cholesterol efflux from lipid-loaded macrophages. [START_REF] Bhatt | HDL Cholesterol Efflux Capacity: Cardiovascular Risk Factor and Potential Therapeutic Target[END_REF] Cholesterol efflux from tissue macrophages however provides only a small contribution to HDL-C levels if any [START_REF] Rosenson | Cholesterol efflux and atheroprotection: advancing the concept of reverse cholesterol transport[END_REF] and can therefore hardly explain the complex non-linear association between HDL-C and cardiovascular risk.

Intravascular HDL metabolism is intimately linked to that of triglyceride-rich lipoproteins (TGRL), a phenomenon which is frequently manifested as a negative correlation between plasma levels of HDL-C and triglycerides. [START_REF] Nordestgaard | Triglycerides and cardiovascular disease[END_REF][START_REF] Rosenson | Genetics and causality of triglyceride-rich lipoproteins in atherosclerotic cardiovascular disease[END_REF] Given that elevated triglyceride levels are increasingly accepted as a cardiovascular risk factor, [START_REF] Nordestgaard | Triglycerides and cardiovascular disease[END_REF][START_REF] Rosenson | Genetics and causality of triglyceride-rich lipoproteins in atherosclerotic cardiovascular disease[END_REF] low HDL-C has been proposed to represent a biomarker of elevated concentrations of atherogenic TGRL remnants, [START_REF] Nordestgaard | Triglycerides and cardiovascular disease[END_REF] a clinically attractive concept which however lacks mechanistic insight. In the circulation, HDL and TGRL interact via multiple metabolic pathways a most studied of which includes heteroexchange of core lipids mediated by cholesteryl ester transfer protein (CETP). [START_REF] Chapman | Cholesteryl ester transfer protein: at the heart of the action of lipid-modulating therapy with statins, fibrates, niacin, and cholesteryl ester transfer protein inhibitors[END_REF] A key pathway linking HDL and TGRL involves transfer to HDL of surface remnants generated during TGRL lipolysis by lipoprotein lipase (LPL). [START_REF] Nikkila | Plasma high-density lipoprotein concentration and subfraction distribution in relation to triglyceride metabolism[END_REF] This frequently overlooked process delivers high amounts of free cholesterol (FC) to HDL and constitutes a quantitatively major source of circulating HDL-C, accounting for up to 50% of its variation. [START_REF] Magill | Relationships between the metabolism of high-density and very-low-density lipoproteins in man: studies of apolipoprotein kinetics and adipose tissue lipoprotein lipase activity[END_REF] Indeed, intestinal cholesterol secretion within chylomicrons is comparable to net cholesterol efflux from all extra-hepatic tissues combined. [START_REF]Intestinal absorption of triglyceride and cholesterol. Dietary and pharmacological inhibition to reduce cardiovascular risk[END_REF][START_REF] Schwartz | Lipoprotein cholesteryl ester production, transfer, and output in vivo in humans[END_REF] Mechanistically, FC acquirement by HDL can be primarily mediated by small-sized complexes containing phospholipid (PL) and apolipoprotein (apo) A-I, the major HDL protein and potent biological detergent. [START_REF] Pownall | The unique role of apolipoprotein A-I in HDL remodeling and metabolism[END_REF][START_REF] Huang | Apolipoprotein A-I structural organization in highdensity lipoproteins isolated from human plasma[END_REF] Remarkably, high amounts of apoA-I are synthesised by the intestine to be incorporated into chylomicrons and postprandially transferred to HDL. [START_REF] Bisgaier | Intestinal synthesis, secretion, and transport of lipoproteins[END_REF] To explain the relationship between HDL-C levels and cardiovascular risk, we propose a reverse remnant cholesterol transport (RRT) hypothesis which involves acquirement by HDL of surface remnants of TGRL upon lipolysis with subsequent transport of remnant-derived cholesterol to the liver in a pathway which originates in the intestine with the secretion of apoA-I on chylomicrons followed by their transport to plasma via lymph. As a corollary, we suggest that plasma concentration of HDL-C is an imperfect static measure of cholesterol flux through this dynamic pathway.

To assess this hypothesis, we developed an in vitro fluorescent assay to evaluate the capacity of HDL to acquire FC from TGRL during lipolysis by LPL. The assay was applied to several groups of subjects markedly differing in HDL-C levels. Our findings reveal decreased FC transfer from TGRL to HDL in subjects with both low and extremely high HDL-C, a finding which may link HDL-associated cardiovascular risk to triglyceride metabolism and account for the U-shape relationship of HDL-C and CVD.

Methods

Study populations

Four groups of subjects markedly differing in HDL-C levels were studied together with corresponding control groups. Male patients presenting with ST segment elevation acute myocardial infarction (AMI; n=22) were recruited at the Heart Institute-InCor University of Sao Paulo Medical School Hospital (São Paulo, Brazil) before initiation of any treatment within no later than 24h of clinical presentation in the Emergency Room. Healthy non-smoking normolipidemic male subjects (n=8) were recruited as controls.

Treatment-naïve patients with well-controlled Type 2 diabetes (T2D; males and postmenopausal women, n=17) were recruited from the Ramon Carrillo Centre (La Matanza, Buenos Aires, Argentina). Healthy non-smoking normolipidemic subjects (n=16) were recruited from the same geographical area to constitute the control group. Subjects with high (1.81 to 2.59 mmol/L [70 to 100 mg/dL]; n=20) and extremely high (>2.59 mmol/L [>100 mg/dL]; n=20) HDL-C levels together with healthy normolipidemic control volunteers (n=20) were recruited at the La Pitié-Salpetrière University Hospital (Paris, France).

Measurement of HDL capacity to acquire FC during TGRL lipolysis by LPL

Given that fluorometric measurements are distinguished by high sensitivity and specificity, introduction in the assay of a fluorescent FC derivative provides excellent means for evaluating FC transfer across lipoproteins. Importantly, selective labelling of TGRL with fluorescent FC implies its absence from HDL at baseline, thereby strongly enhancing its relative enrichment in HDL over time as compared to that of unlabelled FC. We therefore labelled TGRL with fluorescent 23-(dipyrrometheneboron difluoride)-24norcholesterol (Avanti Polar Lipids, Alabaster, AL, USA), also known as TopFluor® cholesterol (TopF) and BODIPY-cholesterol, and evaluated its transfer to HDL during LPL-induced lipolysis.

To isolate TGRL (d<1.019 g/ml), a single-step ultracentrifugation was used. HDL was employed in the form of ultracentrifugally isolated HDL (UC-HDL) or as apoB-depleted plasma. To label human TGRL with TopF, TGRL was added to lipoprotein-deficient plasma (LPDP) at the LPDP:TGRL ratio of 1:100 by volume. The mixture was filtered and TopF was added as a chloroformic solution followed by overnight incubation at 37°C under gentle stirring. Labelled TGRL was separated from unbound fluorescent lipid using filtration through a PD-10 Sephadex column. Triglyceride concentration in purified labelled TGRL was measured by photometry and TopF fluorescence was registered at the excitation/emission wavelengths of 500/525 nm using Gemini Microplate Reader (Molecular Devices, USA) to verify the labelling.

To evaluate the transfer of TopF, Tris buffer (0.4 M, pH 8) was mixed on ice with required amounts of TopF-labelled TGRL and HDL, and LPL from Pseudomonas sp. or from bovine milk (both from Sigma, France) was added to start lipolysis at 37°C. At the end of the incubation, the reaction mixture was placed on ice and apoB precipitant containing phosphotungstic acid and MgCl 2 was added. Following incubation at room temperature and centrifugation, HDL-containing supernatant was aspirated, filtered and transferred to a black microplate for fluorescence reading. Fluorescence of the standard TGRL sample alone was also measured at a triglyceride concentration employed in the assay and fluorescent values measured in HDL were expressed as a percentage of fluorescence of such standard sample. In some experiments, inhibitors for CETP (torcetrapib, 25 µM) and lecithin-cholesterol acyltransferase (LCAT) (iodoacetamide, 750 µM) were used.

To characterise the transfer of TopF to HDL obtained from clinical plasma samples, TGRL and apoBdepleted plasma were used at final concentrations of 0.34 mmol/L (30 mg triglyceride/dL) and 3.3%, respectively. Assay conditions (lipoprotein concentrations, incubation time, etc.) were chosen to model those of postprandial TGRL lipolysis in humans. A reference apoB-depleted plasma sample obtained from one healthy normolipidemic control plasma aliquoted and stored at -80°C was included in each series of measurements and all values obtained in clinical samples were normalised to that observed in the reference plasma.

Postprandial aortic cholesterol accumulation in mice

Postprandial aortic accumulation of FC was evaluated in vivo after gavage with 100 µCi of [ 3 H]cholesterol administrated with olive oil to three groups of mice markedly differing in HDL-C levels, including low-HDL-C apoA-I knock-out mice, high-HDL-C human apoA-I transgenic mice and control wild-type mice. The animals were euthanized 2h after the gavage, their aortas removed and weighted, and specific radioactivity measured. HDL capacity to acquire TopF from TGRL during LPL-induced lipolysis was evaluated in apoB-depleted EDTA plasma obtained from every mouse following overnight fast several weeks before the gavage experiment.

A detailed description of the methods is provided in Supplementary Material.

Results

HDL isolated by ultracentrifugation or apoB depletion readily acquired TopF upon lipolysis of TopFlabelled TGRL (Fig. 1; Supplemental Figure 1). The transfer was time-dependent and typically saturated after 1-2h, except for the highest concentration of UC-HDL (Fig. 1, A andC). Dose-dependence of the transfer was characterised by an increase with increasing UC-HDL concentrations (Fig. 1,B), while maximal TopF accumulation in HDL isolated by apoB depletion was observed at intermediate concentrations of apoB-depleted plasma, decreasing at higher concentrations (Fig. 1,D). This dosedependence was confirmed by measurements of unlabelled native FC (Supplemental Fig. 2), revealing a correlation coefficient of 0.71 (p<0.001) between the two methods and demonstrating the validity of our fluorescence-based approach. In addition, when HDL was re-isolated from the same reaction mixtures by apoB precipitation and by UC, TopF fluorescence readings in such HDL were strongly correlated (r=0.95, p=0.014), further validating our approach. When V max and K d for the transfer of unlabelled FC were calculated, they revealed maxima at intermediate concentrations of HDL (Fig. 2,A), consistent with the results obtained using fluorescent FC. As a consequence, concentration of apoB-depleted plasma of 3.3% and UC-HDL concentration of 4 mg protein/dL, the both corresponding to the initial concentration-dependent sections of the dose-dependent curves (Fig. 1, B and D), were chosen to compare individual plasma samples for the capacity of their HDL to acquire FC. Initial characterisation of the assay revealed good reproducibility of the measurement of TopF transfer to normolipidemic apoB-depleted plasma, with intra-and inter-assay coefficients of variation of 5.1% and 8.8%, respectively (n=10). The assay was initially employed to obtain insight into mechanisms of FC transfer from TGRL to HDL during LPL-mediated lipolysis using inhibitors. Both inhibition of CETP using torcetrapib and inhibition of LCAT using iodoacetamide markedly (1.8-and 2.1-fold, respectively) enhanced the accumulation of TopF in UC-HDL (Fig. 2,B). Torcetrapib equally increased the accumulation of TopF when apoB-depleted plasma was used as a source of HDL (from 49 ±11% to 74 ±11 in the absence and presence of 25 µM torcetrapib, respectively, in 10% apoB-depleted plasma; n=4, p<0.05), modifying the dose-dependence of the transfer in such a way that its decrease at high concentrations of apoB-depleted plasma was abolished.

The assay was subsequently applied to several groups of patients markedly differing in plasma HDL-C.

Patients with AMI featured reduced HDL-C (-31%, p<0.01; Supplemental Table 1) relative to healthy normolipidemic controls. The capacity of HDL to acquire TopF was decreased in AMI patients by -46% (controls, 103±12% vs. patients, 57±28%, p<0.001; Fig. 3,A). This decrease was thereby more pronounced as compared to that in HDL-C levels. The capacity of HDL to acquire TopF was positively correlated with HDL-C in this low HDL-C group (r=0.49, p<0.05). By contrast, accumulation of TopF in UC-HDL isolated from AMI plasma was similar to that observed in UC-HDL from controls when the lipoproteins were compared on a protein concentration basis (Supplemental Fig. 3,A). To evaluate whether between-group differences were specifically related to the transfer of fluorescent cholesterol, we measured the accumulation of unlabelled native FC in HDL from a subpopulation of AMI and control subjects. The transfer of FC was strongly (6.5-fold, p<0.01) reduced in AMI relative to control HDL (Supplemental Fig. 3,B) and was correlated with measurements of TopF-cholesterol (r=0.73, p=0.005).

Treatment-naïve patients with well-equilibrated T2D, an established cardiovascular risk factor, were characterised by a typical lipid profile involving reduced HDL-C (-21%, p<0.05) and elevated triglycerides (+94%, p<0.05) as compared to healthy normolipidemic controls (Supplemental Table 2).

The capacity of HDL to acquire FC was decreased in T2D patients by -24% (controls, 96±28% vs. patients, 72±33%, p<0.05; Fig. 3,A). This decrease was similar to that in HDL-C levels.

As compared to healthy, normolipidemic controls, subjects with high HDL-C levels (from 1.81 to 2.59 mmol/L [70 to 100 mg/dL]) were characterised by elevated HDL-C (+34%, p<0.001; Supplemental Table 3). Despite such marked increase, the capacity of HDL to acquire FC did not differ between the groups (controls, 97±22% vs. patients, 94±33%, Fig. 3,A), thereby diverging from the HDL-C assay.

In subjects with extremely high HDL-C concentrations (>2.59 mmol/L [>100 mg/dL]), HDL-C was elevated almost 2-fold (+92%) relative to controls (Supplemental Table 4). Despite such pronounced increase, the capacity of HDL to acquire FC was diminished by -20% in this group (controls, 101±20% vs. patients, 81±27%, p<0.05; Fig. 3,A), further diverging from HDL-C. As a result, no correlation between HDL capacity to acquire TopF and plasma HDL-C was observed across all the populations studied (r=0.10, p=0.29; n=123). Such absence of linear correlation was related to a complex non-linear dependence between TopF transfer and HDL-C observed across both individual subjects (Fig. 3,B) and studied populations (Fig. 3,C), which resembled an inverse U-shape. It is of note that no difference in the capacity of HDL to acquire TopF was observed between the four control groups, validating our study design (Fig. 3,A).

To evaluate potential determinants of the reduced FC transfer to HDL, plasma activities of CETP and LCAT were assessed in the patients using fluorometric kits. The only significant difference observed relative to controls involved reduced CETP activity in AMI patients (-50%, p<0.01; Supplemental Fig. 4).

Given that subjects with low and high HDL-C levels greatly differ in their HDL particle profile with large HDL prevailing at high HDL-C, [START_REF] Kontush | Antioxidative activity of HDL particle subspecies is impaired in hyperalphalipoproteinemia: relevance of enzymatic and physicochemical properties[END_REF] we evaluated as to whether HDL particle subpopulations differed in their capacity to acquire FC upon TGRL lipolysis by LPL. We found that large, light HDL2 particles were up to 3.8-fold less efficient acceptors for TopF relative to small, dense HDL3 at a fixed protein concentration of 4 mg/dL (Supplemental Fig. 5), equivalent to 2.8-fold per particle difference.

The relationship of the assay with CVD was initially evaluated on the basis of published mortality data.

When cardiovascular and overall mortality data obtained from large-scale CANHEART HDL study 4

and

Copenhagen City Heart Study 6 were recalculated for the mean HDL-C levels observed in the five populations studied by us (AMI, T2D, subjects with high and extremely high HDL-C, and pooled controls) and plotted against HDL-C, a U-shape relationship was observed as reported 4,6 (Supplemental Fig. 6,A). In striking contrast, when HDL-C was replaced by the mean values of HDL capacity to acquire TopF, linear negative relationships with both cardiovascular and overall mortality were obtained (r 2 from 0.79 to 0.97, p from 0.043 to 0.002), indicative of the significant relationship of mortality with the assay outcome (Supplemental Fig. 6,B).

Pathophysiological relevance of the assay was further evaluated in vivo using murine models displaying marked differences in circulating HDL-C levels, specifically wild-type mice, dyslipidemic low HDL-C apoA-I -/-mice and high HDL-C mice transgenic for human apoA-I. As expected, HDL-C concentrations were reduced by -82% in apoA-I knock-out mice and elevated by +72% in human apoA-I transgenic animals (Fig. 4,A). Expectedly, CETP activity was nearly absent from plasma of all mice and LCAT activity was greatly reduced relative to human plasma (Supplemental Fig. 7), with apoA-I knock-out animals featuring the lowest LCAT activity.

The capacity of HDL to acquire FC measured upon TGRL lipolysis by LPL in vitro using TopF was reduced by -38% (p<0.01) in apoA-I knock-out mice but was not modified in human apoA-I transgenic animals (Fig. 4,B). When all the mice received an oral gavage of 3 H-cholesterol together with olive oil, postprandial accumulation of 3 H-cholesterol in the aorta was significantly (6.1-fold) enhanced 2h after the gavage in apoA-I knock-out relative to wild-type mice, while no difference between wild-type and human apoA-I transgenic mice was found (Fig. 4,C). As a consequence, the capacity of HDL to acquire FC in vitro was significantly and negatively correlated with the aortic accumulation of 3 H-cholesterol in vivo across all mice, attesting this functional parameter as a negative metric of postprandial atherosclerosis (Fig. 4,D).

Discussion

In the present study, we evaluated our RRT hypothesis that FC transfer to HDL upon TGRL lipolysis by LPL underlies the complex relationship between HDL-C levels and cardiovascular risk. We provide evidence, using human subjects and animal models markedly differing in HDL-C concentrations, that this process links triglyceride metabolism to atherosclerosis and may account for the relationship between HDL-C and CVD.

To assess our hypothesis, we developed an original method to evaluate in vitro FC transfer from TGRL to HDL upon LPL-mediated lipolysis of TGRL labelled with TopF, a fluorescent derivative of FC. Earlier studies of material exchange between TGRL and HDL revealed that multiple lipid and protein components, including FC, PL and apolipoproteins, were transferred from TGRL to HDL upon LPLmediated lipolysis. [START_REF] Nikkila | Plasma high-density lipoprotein concentration and subfraction distribution in relation to triglyceride metabolism[END_REF][START_REF] Magill | Relationships between the metabolism of high-density and very-low-density lipoproteins in man: studies of apolipoprotein kinetics and adipose tissue lipoprotein lipase activity[END_REF][START_REF] Chung | Lipolytic remnants of human VLDL produced in vitro. Effect of HDL levels in the lipolysis mixtures on the apoCs to apoE ratio and metabolic properties of VLDL core remnants[END_REF][START_REF] Rutledge | Direct visualization of lipid deposition and reverse lipid transport in a perfused artery -Roles of VLDL and HDL[END_REF] Mechanisms of such exchange are defined by the removal of excess surface fragments from TGRL, which cannot adapt the structure of remnant particles possessing reduced hydrophobic core. Formation of surface TGRL remnants possibly involves spontaneous organisation of excessive surface lipids separated from TGRL into low-energy complexes which resemble micelles or liposomes. [START_REF] Chung | Liposome-like particles isolated from human atherosclerotic plaques are structurally and compositionally similar to surface remnants of triglyceride-rich lipoproteins[END_REF] ApoA-I, the major HDL protein present in high amounts on chylomicrons, [START_REF] Bisgaier | Intestinal synthesis, secretion, and transport of lipoproteins[END_REF] can play a central role in this process as a result of its distinct lipid-binding properties. [START_REF] Pownall | The unique role of apolipoprotein A-I in HDL remodeling and metabolism[END_REF][START_REF] Huang | Apolipoprotein A-I structural organization in highdensity lipoproteins isolated from human plasma[END_REF] Indeed, apoA-I avidly interacts with minimal amounts of lipids present in aqueous solutions, transforming them into highly organised lipoprotein complexes in the HDL density range. [START_REF] Pownall | The unique role of apolipoprotein A-I in HDL remodeling and metabolism[END_REF][START_REF] Huang | Apolipoprotein A-I structural organization in highdensity lipoproteins isolated from human plasma[END_REF] While apoA-I can solubilise excess surface lipids released from TGRL upon lipolysis, HDL PL may provide lipid surface required for their absorption. [START_REF] Chung | Lipolytic remnants of human VLDL produced in vitro. Effect of HDL levels in the lipolysis mixtures on the apoCs to apoE ratio and metabolic properties of VLDL core remnants[END_REF][START_REF] Chung | Liposome-like particles isolated from human atherosclerotic plaques are structurally and compositionally similar to surface remnants of triglyceride-rich lipoproteins[END_REF][START_REF] Schaefer | Transfer of human lymph chylomicron constituents to other lipoprotein density fractions during in vitro lipolysis[END_REF] Other structural HDL apolipoproteins, including apoA-II, apoE and apoM, can equally be expected to participate in lipid transfer during lipolysis, consistent with the presence of FC acceptor activity in HDL from apoA-I -/-mice.

As lipid transfer proteins play key roles in exchanging lipids across lipoproteins, these proteins may actively participate in the transfer of surface lipids upon LPL-mediated lipolysis. To assess this possibility, we employed torcetrapib, a specific CETP inhibitor, and observed enhanced transfer of FC upon inhibition. Such inhibitory role of CETP towards cholesterol removal from TGRL can be explained by its capacity to transfer cholesterol, in a form of cholesteryl ester (CE), from HDL back to TGRL. [START_REF] Chapman | Cholesteryl ester transfer protein: at the heart of the action of lipid-modulating therapy with statins, fibrates, niacin, and cholesteryl ester transfer protein inhibitors[END_REF] Indeed, HDL represents a major source of CE in human plasma, reflecting preferential association of LCAT with HDL. [START_REF] Schwartz | Lipoprotein cholesteryl ester production, transfer, and output in vivo in humans[END_REF] When FC is transferred from TGRL to HDL and esterified under the action of LCAT, a molecule of CE formed becomes a substrate for CETP and can be readily transferred back to TGRL in exchange for a molecule of TGRL-derived triglyceride. This mechanism can be operative for TopF as it is efficiently esterified by LCAT and as its ester is transferred by CETP in vitro. [START_REF] Sakurai | Development of a novel fluorescent activity assay for lecithin:cholesterol acyltransferase[END_REF][START_REF] Bisgaier | Use of fluorescent cholesteryl ester microemulsions in cholesteryl ester transfer protein assays[END_REF] If this mechanism is true and LCAT and CETP act along the same pathway of cholesterol removal from HDL, then inhibition of LCAT should enhance LPL-induced accumulation of TGRL-derived FC in HDL.

Consistent with this conclusion, we did observe accelerated accumulation of TopF upon LCAT inhibition by iodoacetamide, suggesting a multi-step mechanism of FC movement between HDL and TGRL upon LPL-induced lipolysis, which involves acquirement of FC by HDL with its subsequent esterification by LCAT followed by CETP-mediated exchange of the generated CE for triglyceride present in TGRL.

Importantly, kinetic data reveal that esterification of FC by LCAT occurs more slowly than both facilitated diffusion of CE mediated by CETP and FC transfer across lipoproteins, [START_REF] Schwartz | Lipoprotein cholesteryl ester production, transfer, and output in vivo in humans[END_REF][START_REF] Kosek | Binding affinity and reactivity of lecithin cholesterol acyltransferase with native lipoproteins[END_REF][START_REF] Paromov | Lipid transfer inhibitor protein defines the participation of high density lipoprotein subfractions in lipid transfer reactions mediated by cholesterol ester transfer protein (CETP)[END_REF] thereby constituting a rate-limiting step in the exchange of cholesterol between TGRL and HDL. Such CE transfer from HDL to TGRL might therefore be negligibly low at low concentrations of LCAT (and low concentrations of HDL, as LCAT is exclusively carried by HDL under our experimental conditions) but might greatly increase at high HDL concentrations, decreasing net cholesterol accumulation in HDL.

Reduction of FC concentration gradient between TGRL and HDL occurring at high HDL concentrations may additionally contribute to the latter effect. Consistent with this notion, the dose-dependence of FC transfer was characterised by direct relationship with HDL concentration at low concentrations of both UC-HDL and apoB-depleted plasma, most likely reflecting the presence in the assay of increasing amounts of acceptor HDL particles for TGRL-derived FC. However, such direct relationship evolved into an inversed dose-dependence at high concentrations of HDL when apoB-depleted plasma (possessing high CETP activity) was employed as the HDL source. Interestingly, the inverse relationship was not observed in apoB-depleted plasma in the presence of torcetrapib, or in UC-HDL (containing only minor amounts of CETP following UC [START_REF] Vaisar | Shotgun proteomics implicates protease inhibition and complement activation in the antiinflammatory properties of HDL[END_REF] ). In the latter experiment, the transfer of FC was directly related to HDL concentration throughout the whole concentration range studied, potentially reflecting low CETP activity in the samples.

Clinical value of the assay was assessed in four populations markedly differing in plasma HDL-C. The transfer of FC to HDL was reduced in low HDL-C patients with AMI, low HDL-C patients with T2D and subjects with extremely high HDL-C levels of >100 mg/dL relative to healthy normolipidemic controls.

When these data were plotted against individual HDL-C levels in all subjects, or against mean HDL-C levels in the groups, inverse U-shape relationships were observed which mirrored the U-shape relationship of overall and cardiovascular mortality with HDL-C concentrations recently reported in large-scale epidemiological studies. [4][5][6] As a consequence, the relationships of FC transfer to HDL with cardiovascular and overall mortality obtained from these studies were linear and negative, in clear contrast to the U-shape relationships observed for HDL-C. Albeit circumstantial, these data suggest that FC transfer to HDL upon TGRL lipolysis by LPL may underlie the U-shape relationship between HDL-C and CVD.

Importantly, FC transfer was unchanged in subjects with high HDL-C of 70 to 100 mg/dL and was only diminished at extremely high HDL-C levels of >100 mg/dL. It is worth noting that the high HDL-C groups expectedly featured a prevalence of female over male subjects, while the AMI group was only composed of males, rising a question of the role of gender for the observed differences. Each of the studied groups was however compared to a matched control group and no difference in the TopF transfer was observed between the four control groups despite their differences in the sex ratio, thereby arguing against the role of gender in the between-group effects observed by us.

According to the dose-dependences of HDL capacity to acquire FC from TGRL upon lipolysis, the decreases in this metric observed in low HDL-C patients with AMI and T2D may straightforwardly reflect low concentrations of acceptor HDL particles. Indeed, no difference in FC transfer to AMI and control HDL was found when the lipoproteins were compared on a protein concentration basis. Other metabolic alterations potentially underlying decreased FC transfer in both low and extremely high HDL-C subjects involve elevated CETP and/or LCAT activities, which were however not detected by us. Altered HDL particle profile typical of extremely high HDL-C states includes highly elevated levels of large, lipid-rich HDL and reduced concentrations of small, lipid-poor HDL particles. [START_REF] Kontush | Antioxidative activity of HDL particle subspecies is impaired in hyperalphalipoproteinemia: relevance of enzymatic and physicochemical properties[END_REF][START_REF] Asztalos | Apolipoprotein composition of HDL in cholesteryl ester transfer protein deficiency[END_REF] As our data document superior capacity of small vs. large HDL to acquire FC upon TGRL lipolysis (which is consistent with the superiority of such particles to efflux cellular cholesterol and to perform other biological functions [START_REF] Camont | Small, dense high-density lipoprotein-3 particles are enriched in negatively charged phospholipids: relevance to cellular cholesterol efflux, antioxidative, antithrombotic, anti-inflammatory, and antiapoptotic functionalities[END_REF] ), the decrease in this metric at extremely high HDL-C might reflect such alterations in the HDL particle profile.

To establish a direct link between lipolytic FC transfer from TGRL to HDL and development of CVD, we studied postprandial movement of radiolabelled cholesterol in three groups of mice markedly differing in HDL-C. Small amounts of [ 3 H]-cholesterol were consistently found in murine aortic tissue 2h following oral gavage with this compound. Such aortic accumulation of [ 3 H]-cholesterol was inversely associated with HDL-C concentrations, being highly elevated in low HDL-C apoA-I knock-out animals. In parallel, the capacity of murine HDL to acquire FC upon TGRL lipolysis by LPL was reduced, resulting in its negative correlation with the aortic accumulation of [ 3 H]-cholesterol. These data additionally attest FC transfer to HDL upon TGRL lipolysis by LPL as a negative metric of CVD and directly link HDL metabolism to postprandial atherosclerosis. Interestingly, FC transfer was not compromised in high HDL-C apoA-I transgenic mice, in contrast to our observations of reduced FC transfer in human subjects with extremely high HDL-C, potentially reflecting low CETP and LCAT activities in mice.

Together, our data provide evidence that HDL capacity to acquire FC upon TGRL lipolysis by LPL in vitro, measured using our novel assay, may represent a superior biomarker of cardiovascular risk as compared to plasma HDL-C levels presently employed in clinical practice. The assay is simple, rapid, does not require sophisticated equipment and can be easily run in clinical routine. Moreover, these data support our RRT hypothesis to explain the U-shape association between HDL-C levels and CVD by intravascular FC transfer from TGRL to HDL in a pathway that can be essential for triglyceride metabolism and energy production. We propose that plasma concentration of HDL-C is an imperfect static measure of cholesterol flux through this dynamic pathway, which is directly associated with the efficacy of cholesterol removal from the circulation only at low-to-normal HDL-C. Importantly, HDL exerts a plethora of other biological activities, including its capacity to efflux cholesterol from peripheral cells, to protect LDL from oxidative modification, to reduce production of pro-inflammatory phospholipids and to beneficially impact glucose metabolism. [START_REF] Allard-Ratick | HDL: Fact, fiction, or function? HDL cholesterol and cardiovascular risk[END_REF][START_REF] Richart | High-density lipoprotein and cardiac glucose metabolism: Implications for management of acute coronary syndromes[END_REF] None of these wellstudied activities is however known to be reduced at both low and extremely high concentrations of HDL and can thereby account for the complex non-linear relationship between HDL-C levels and CVD. We conclude that this relationship primarily reflects the lipid acceptor role of HDL in the lipolysis. Comorbidities in the patient group included hypertension (77%), Type 2 diabetes (18%) and heart failure (18%). Less than half of the patients were smokers (36%). Eight healthy nonsmoking normolipidemic male subjects were recruited as controls. Only male subjects were included in the study to minimise hormonal effects on lipoprotein metabolism. Written informed consent was obtained from all study subjects and the project was approved by the Ethics Committee of InCor in accordance with local institutional guidelines conformed to the Declaration of Helsinki.

Patients with Type 2 diabetes (T2D). Seventeen treatment-naïve adult patients with T2D diagnosed according to the American Diabetes Association (ADA) criteria 1 were recruited from the Ramon Carrillo Centre (La Matanza, Buenos Aires, Argentina). All diabetic patients were well-controlled and defined as those presenting HbA1c < 7% and fasting glucose < 7.22 mmol/L (130 mg/dL), following the ADA glycemic recommendations for patients with venous blood collected into sterile, evacuated tubes (Vacutainer). Plasma was immediately separated by low-speed centrifugation at 4 °C, aliquoted and frozen at -80 °C under nitrogen.

To evaluate free cholesterol (FC) transfer to serum-derived HDL, serum samples were also obtained from some control subjects.

Clinical and biological parameters

Circulating levels of total cholesterol (TC), low-density lipoprotein-cholesterol (LDL-C), HDL-C, triglyceride (TG), glucose and HbA1c were measured using commercially available kits. Plasma apolipoprotein (apo) A-I and apoB were quantitated by immunoturbidimetry (DiaSys, France). High-sensitivity C-reactive protein (hsCRP) was measured by immunoassay (Roche Diagnostics, Mannheim, Germany). Activities of CETP and LCAT were measured in plasma using fluorometric kits from Roar Biomedical (New York, NY, USA).

Isolation of lipoproteins

Single-step ultracentrifugation (UC). To isolate TGRL (d <1.019 g/mL), a single-step UC was employed. 2 Briefly, plasma was mixed with a required amount of dry solid KBr to increase its density to 1.019 g/mL, layered by a density solution of 1.019 g/mL and ultracentrifuged at 45000 rpm and 15℃ for 24h. Upper layer containing TGRL was removed and extensively dialysed against phosphate-buffered saline (PBS; pH 7.4) at 4°C in the dark.

Density gradient UC. HDL was isolated from plasma by a single step, isopycnic non-denaturing density gradient UC using a Beckman SW41 Ti rotor at 40,000 rpm for 44 hours in a Beckman XL70 ultracentrifuge at 15°C by a slight modification of the method of Chapman et al. 3 as previously described. 4 Briefly, plasma density was increased to 1.21 g/mL by adding dry solid KBr. A discontinuous density gradient was then constructed as follows: 2 mL of 1.24 g/mL NaCl/KBr solution, 3 mL plasma (d 1.21 g/mL), 2 mL of 1.063 g/mL NaCl/KBr solution, 2.5 mL of 1.019 g/mL NaCl/KBr solution, and 2.5 mL of NaCl solution (d 1.006 g/mL).

After centrifugation, gradients were collected from the top of the tube in fractions corresponding to VLDL and IDL (d <1.019 g/mL), LDL (d 1.019 to 1.063 g/mL), HDL2b (d 1.063 to 1.091 g/mL), HDL2a (d 1.091 to 1.110 g/mL), HDL3a (d 1.110 to 1.133 g/mL), HDL3b (d 1.133 to 1.156 g/mL), and HDL3c (d 1.156 to 1.179 g/mL). Total HDL was prepared by reconstituting the five HDL subpopulations at their plasma concentrations. If necessary, the bottom of the tube containing plasma from control subjects was collected as lipoprotein-deficient plasma (LPDP). Density limits were defined using a calibration curve of density versus volume derived from control gradients containing only salt solutions. All fractions were extensively dialysed against PBS at 4°C in the dark before use.

ApoB depletion. To separate HDL from apoB-containing lipoproteins, apoB precipitating reagent containing phosphotungstic acid (40 mg/mL) and MgCl 2 (100 mg/mL) was added to plasma at a ratio of 1:10 v/v. The samples were incubated at room temperature for 10 min and centrifuged at 13 000 rpm for 30 min. ApoB-depleted reference plasma was prepared by adding the apoB precipitant to a reference plasma sample obtained from a healthy normolipidemic male donor.

Chemical composition of lipoproteins

Total protein, TC, FC, phospholipid (PL) and TG contents of isolated lipoproteins were determined using commercially available assays. Cholesteryl ester (CE) was calculated by multiplying the difference between total and free cholesterol concentrations by 1.67. Total lipoprotein mass was calculated as the sum of total protein, CE, FC, PL and TG. Molecular masses of HDL2b and 3c of 450 and 160 kDa and their protein contents of 28 and 58%, respectively, were used to calculate molar concentrations of the particles. 5,6 Measurement of HDL capacity to acquire FC during TGRL lipolysis by LPL To characterise the transfer of TopF to HDL obtained from clinical plasma samples, TGRL and apoB-depleted plasma were used at the final concentration of 0.34 mmol/L (30 mg triglyceride/dL) and dilution of 30-fold, respectively. Lipid concentrations were evaluated in the samples using commercially available kits. Assay conditions (lipoprotein concentrations, incubation time, etc.) were chosen to model those of postprandial TGRL lipolysis in humans.

A reference apoB-depleted plasma sample obtained from one healthy normolipidemic control plasma aliquoted and stored at -80°C was included in each series of measurements and all values obtained in clinical samples were normalised to that observed in the reference plasma.

In some experiments, transfer of FC was evaluated fluorometrically using Amplex® Red kit (ThermoFisher Scientific, Montigny-le-Bretonneux, France).

Postprandial aortic cholesterol accumulation in mice

Postprandial aortic accumulation of cholesterol was evaluated in vivo after a gavage with 100 µCi (3.7 MBq) of [ 3 H]-cholesterol administrated with olive oil (100 µL) to three groups of mice greatly differing in HDL-C levels, including low-HDL-C apoA-I knock-out mice (SN 002055 B6.129P2-Apoa1tm1Unc/J), high-HDL-C human apoA-I transgenic mice (SN 001927 C57BL/6-Tg(APOA1)1Rub/J) and control wild-type mice (C57BL/6, all strains from Charles River Laboratories, L'Arbresle, France). The groups contained similar numbers of males and females and the sex ratio did not differ between the groups. The animals were euthanized 2h after the gavage, their aortas removed and weighted, and specific radioactivity measured using a MicroBeta 2450 Perkin-Elmer liquid scintillation counter.

The capacity of murine HDL to acquire TopF from TGRL during LPL-induced lipolysis was evaluated in apoB-depleted EDTA plasma obtained from the animals several weeks before the gavage experiment, following an overnight fast. All animal experiments were approved by the local ethics committee for animal care.

Statistical analysis

Incubations were performed in duplicates. As several individual samples were employed in every experiment, experimental errors reported primarily reflect biological variability across individuals rather than experimental reproducibility.

Distributions of all variables were analysed for normality using Kolmogorov-Smirnov test.

Normally distributed variables are expressed as means ± SD and non-Gaussian distributed variables are expressed as median (interquartile range

Figure 1 .

 1 Figure 1. Kinetics and dose-dependences of the transfer of fluorescent FC from TGRL to HDL

Figure 2 .

 2 Figure 2. Kinetic parameters of FC transfer from TGRL to HDL during LPL-induced lipolysis and

Figure 3 .

 3 Figure 3. Transfer of fluorescent FC from TGRL during LPL-induced lipolysis to HDL isolated

Figure 4 .

 4 Figure 4. Relationships of plasma HDL-C levels and HDL capacity to acquire fluorescent FC from

Fluorescent 23 -

 23 (dipyrrometheneboron difluoride)-24-norcholesterol (Avanti Polar Lipids, Alabaster, AL, USA), also known as TopFluor® cholesterol (TopF) and BODIPYcholesterol, was employed to trace the lipolytic transfer of FC to HDL during TGRL lipolysis.Human TGRLs were labelled with TopF and incubated in the presence of LPL and HDL at 37°C. At the end of the incubation, the lipoproteins were separated and HDL was analysed for the presence of TopF.To label human TGRL with TopF, TGRL was added to lipoprotein-deficient plasma (LPDP) at the LPDP:TGRL ratio of 1:100 by volume. The mixture was filtered with 0.8 µm filter (RC Membrane, Phenomenex, USA), the tube was covered with aluminium foil and TopF was added as a chloroformic solution followed by overnight incubation at 37°C under gentle stirring. Labelled TGRL was separated from unbound fluorescent lipid using filtration through a PD-10 Sephadex column (GE Healthcare, SephadexTM G-25M, UK). Absence in the eluate of other, than TGRL, material containing fluorescent lipid was confirmed by FPLC.Triglyceride concentration in purified labelled TGRL was measured by photometry and TopF fluorescence was registered at the excitation/emission wavelengths of 500/525 nm using Gemini Microplate Reader (Molecular Devices, USA) to verify the labelling. Briefly, a specific TopF fluorescence per unit TG concentration was calculated and TGRL samples revealing the specific fluorescence below a pre-defined threshold were discarded.To evaluate the transfer of TopF, Tris buffer (0.4 M, pH 8) was mixed on ice with required amounts of TopF-labelled TGRL and HDL, and LPL from Pseudomonas sp. or from bovine milk (both from Sigma, France) was added to start lipolysis at 37°C. At the end of the incubation, the reaction mixture was placed on ice and apoB precipitant containing phosphotungstic acid and MgCl 2 was added. The time-point corresponding to 0h was prepared by placing the mixture on ice immediately after adding LPL. Following incubation at room temperature and centrifugation, HDL-containing supernatant was aspirated, filtered through a 0.45 μm filter (RC Membrane, Phenomenex, USA) and transferred to a black microplate for fluorescence reading as described above. Fluorescence of the TGRL sample alone was also measured at a triglyceride concentration employed in the assay and fluorescent values measured in HDL were expressed as a percentage of fluorescence of such standard sample. To evaluate effects on the transfer of specific inhibitors for CETP and LCAT, the assay was performed in the absence or presence of torcetrapib (25 µM) or iodoacetamide (750 µM).

  high HDL-C (>2.59 mmol/L [>100 mg/dL]) and healthy normolipidemic controls HDL-C, high-density lipoprotein-cholesterol; LDL-C, low-density lipoproteincholesterol; TC, total cholesterol; TG, triglycerides; *** p<0.001, ** p<0.01, * p<0.05 vs. controls.

5

 5 ‡ ‡ ‡ † † † † § § Overall mortality in CANHEART Overall mortality in CCHS and CV mortality in CANHEART TopF transfer % Overall mortality in CANHEART Overall mortality in CCHS and CV mortality in CANHEART HDL-C, mg/dl Supplemental Figure 6

Table 2 .

 2 Clinical and biological characteristics of patients with T2D and healthy controls apoA-I, apolipoprotein A-I; apoB, apolipoprotein B; BMI, body mass index; HDL, high-density lipoprotein; HDL-C, high-density lipoprotein-cholesterol; hsCRP, highly sensitive C-reactive protein; LDL-C, low-density lipoprotein-

	Spearman's correlation coefficients were calculated to evaluate relationships between
	variables. The statistical software employed was Statistica (StatSoft, Tulsa, OK, USA).
	The ggplot2 package from R was employed to graphically represent relationships between
	means ± SDs of two variables across several groups of subjects.

). Between-groups differences in normally distributed variables were analysed using Student's t-test. For non-Gaussian distributed variables, the non-parametric Mann-Whitney U test was employed. Differences in dichotomous variables were analyzed by Fisher's exact test. C, high-density lipoprotein-cholesterol; hsCRP, highly sensitive C-reactive protein; LDL-C, low-density lipoprotein-cholesterol; TC, total cholesterol; TG, triglycerides; VLDL-C, very low-density lipoprotein-cholesterol; *** p<0.001, ** p<0.01, * p<0.05 vs. controls. cholesterol; T2D, Type 2 diabetes; TC, total cholesterol; TG, triglycerides; *** p<0.001, * p<0.05 vs. controls.

Table 3 .

 3 Clinical and biological characteristics of subjects with high HDL-C (1.81 to 2.59 mmol/L [70 to 100 mg/dL]) and healthy normolipidemic controls HDL-C, high-density lipoprotein-cholesterol; LDL-C, low-density lipoprotein-cholesterol; TC, total cholesterol; TG, triglycerides; *** p<0.001, * p<0.05 vs. controls.

	Supplemental
		Controls (n=10)	Extremely high HDL-C (n=20)
	Age (years)	44±13	66±9**
	Sex (M/F)	4/6	3/17
	TC (mg/dL)	195±30	268±55*
	(mmol/L)	5.05±0.78	6.94±1.42*
	TG (mg/dL)	73±13	74±32
	(mmol/L)	0.82±0.15	0.84±0.36
	LDL-C (mg/dL)	118±35	136±53
	(mmol/L)	3.06±0.91	3.52±1.37
	HDL-C (mg/dL)	61±15	117±19***
	(mmol/L)	1.58±0.39	3.03±0.49***

Table 4 .

 4 Clinical and biological characteristics of subjects with extremely

Acknowledgements

Excellent technical assistance from Ms Julie Tomczyk, Mr Henri Voedts, Mr. Narcisse Medadje and Mr Jean Brerault is deeply appreciated.

Funding

These studies were supported by National Institute for Health and Medical Research (INSERM; Paris, France). We gratefully acknowledge further support from the University of Pierre and Marie Curie (UPMC; Paris, France), ICAN (Paris, France), China Scholarship Council (CSC, China), CAPES (Brazil), FAPESP (Brazil), and a collaborative grant from CONICET (Buenos Aires, Argentina), University of Buenos Aires (UBACyT CB23), Craveri Foundation and CONICET (PIP 516).

Conflict of Interest

Dr. Kontush reports grants from CSL and Pfizer, personal fees from American Heart Association, National Lipid Association, Sanofi and European Society of Cardiology, outside the submitted work. In addition, Dr. Kontush has a patent EP17306042 entitled "Novel assay of HDL function".

Authors' Contributions

Designing research studies: MF, MD, ML, LM, EF, FR, CVS, RDS, FB, GB, EG, TH, AC, EB, MG, PC, with LPL (190 U/mL) and either TGRL labelled with TopF (A) or non-labelled TGRL (B; both at 15 mg TG/dL), at final dilutions of apoB-depleted plasma of 2.5, 5, 10 and 20%. At the end of the incubation, HDL was separated from TGRL by apoB depletion, and TopF fluorescence or FC were measured in HDL at 500/525 nm or by the Amplex Red kit, respectively. Data from three to four independent experiments performed in duplicate with three individual HDL samples are shown; **p < 0.01, *p < 0.05 vs. 0%; † † p < 0.01, † p < 0.05 vs. 2.5%; ‡ ‡ p < 0.01, ‡ p < 0.05 vs. 5%.

Supplemental Figure 3. Transfer of FC from TGRL during LPL-induced lipolysis to

HDL isolated from patients with AMI. HDL was isolated by UC from patients with AMI (n=7) and healthy normolipidemic controls (n=6) and TopF transfer to HDL (4 mg protein/dL) from TGRL (30 mg TG/dL) was evaluated (A). In a separate experiment, HDL was isolated by apoB depletion from patients with AMI (n=7) and healthy normolipidemic controls (n=7) and FC transfer from TGRL (30 mg TG/dL) was evaluated by the Amplex Red kit at the concentration of apoB-depleted plasma of 3.3% (B); ** p < 0.01 vs. HDL from controls.

Supplemental Figure 4. Activities of CETP (A) and LCAT (B) in plasma of patients with

AMI, T2D, extremely high HDL-C and healthy normolipidemic controls. The measurements were performed using kits from Roar Biomedical in ten patients randomly selected from each group; * p < 0.05 vs. controls; a. u., arbitrary units.

Supplemental