Olivier Dudas
email: olivier.dudas@univ-amu.fr

Gunter Malle
email: malle@mathematik.uni-kl.de

Gunter Malle Rationality

RATIONALITY OF EXTENDED UNIPOTENT CHARACTERS

Keywords: . 2010 Mathematics Subject Classification. 20C08, 20C15, 20C33 graph automorphism, rationality of extensions, unipotent characters

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The work of George Lusztig has shown the singular importance of unipotent characters in the representation theory of finite reductive groups and hence of finite simple groups of Lie type. Recent research has focussed on rationality properties of characters of almost simple groups, and this naturally leads to the problem of understanding fields of values, and fields of realisation, of extensions of unipotent characters to groups of Lie type extended by graph or graph-field automorphisms. The latter can be viewed as groups of rational points of suitable disconnected algebraic groups. While the rationality properties of unipotent characters themselves have long been known, due to the work of Lusztig [START_REF] Lusztig | Rationality properties of unipotent representations[END_REF] and Geck [START_REF] Geck | Character values, Schur indices and character sheaves[END_REF] (see e.g. [8, Cor. 4.5.6]), their extensions to disconnected groups have so far not been studied systematically; Digne-Michel [4, Thm II.3.3] considered characters in the principal series and [14, Prop. 2] deals with SU 3 (q).

The field of values of a unipotent character ρ is generated by its Frobenius eigenvalue (see [8,Prop. 4.5.5]). Here, we show that this Frobenius eigenvalue also governs the field of values of extensions of ρ. Our first result concerns cuspidal characters: Theorem 1. Let G be a simple algebraic group with a Frobenius map F and a commuting non-trivial graph automorphism σ. Then any cuspidal unipotent character ρ of G = G F has an extension ρ to G σ with Q( ρ) = Q(ρ), unless G = 2 A n-1 (q) with n = t 2 ≡ 2, 3 (mod 4) for some t ≥ 2, in which case

Q( ρ) = Q( √ -q).
In particular, Q( ρ) is generated by a δth root of the Frobenius eigenvalue of ρ, where δ ≥ 1 is minimal such that F δ acts trivially on the Weyl group of G.

Using earlier results on Frobenius-Schur indicators exhibits the following connection:

Corollary 2. In the situation of Theorem 1, ρ has a rational extension to G σ if and only if ρ has Frobenius-Schur indicator +1.

For arbitrary unipotent characters, we obtain: Theorem 3. Let G be a simple algebraic group with a Frobenius map F and a commuting non-trivial graph automorphism σ. Then any σ-invariant rational unipotent character ρ of G = G F has a rational extension to G σ , unless one of:

(1) G = A n-1 (q), q is not a square and ρ is labelled by a partition λ = (λ 1 ≥ . . . ≥ λ r ) of n with

i λ i 2 - i λ ′ i 2 + n 2 ≡ 1 (mod 2),
where

λ ′ = (λ ′ 1 ≥ . . . ≥ λ ′ s )
is the partition conjugate to λ;

(2) G = E 6 (q), q is not a square and ρ is one of φ 64,4 or φ 64,13 ; or (3) ρ lies in the Harish-Chandra series of a cuspidal unipotent character of a group of type 2 A n-1 (q) labelled by a 2-core of size n ≡ 2, 3 (mod 4). In cases ( 1) and ( 2), the extensions have character field

Q( √ q), in the third Q( √ -q).
The case of cuspidal characters is settled in Section 2, in Section 3 we reduce the general case to the former, thus proving Theorem 3. In Section 4 we discuss extensions by an exceptional graph automorphism for types B 2 , G 2 and F 4 .

Cuspidal unipotent characters

We consider the following setup. Let G be a connected reductive linear algebraic group with a Frobenius endomorphism F : G → G defining an F q -structure, and set G := G F , the finite group of F -fixed points. We further assume that G has a graph automorphism σ commuting with F . We set G = G σ the semidirect product of G with σ, and G := G F the corresponding extension of G with σ.

2.1. Deligne-Lusztig varieties and unipotent characters. Let B be the flag variety of G, that is, the variety of Borel subgroups of G. The actions of F and σ on G induce commuting endomorphisms of B. The group G acts by simultaneous conjugation on B×B and the orbits are parametrized by the elements of the Weyl group W of G. Given w ∈ W , we denote by O(w) the corresponding orbit and we define the Deligne-Lusztig variety as in [12, 3.3] by

X w = {B ∈ B | (B, F (B)) ∈ O(w)}.
The action of G on B × B restricts to an action of the finite group G on X w . Furthermore, F (resp. σ) induces a finite morphism (resp. an isomorphism) between X w and X F (w) (resp. X σ(w) ). Consequently, if δ is the smallest integer such that F δ acts trivially on W , then any Deligne-Lusztig variety has an action of F δ commuting with G.

For ℓ a prime not dividing q we denote by R w the corresponding Deligne-Lusztig character of G, given by

R w (g) = i∈Z (-1) i Tr g | H i c (X w , Q ℓ ) for g ∈ G.
This generalised character of G does not depend on ℓ (see e.g. [12, 1.2]). The complexvalued characters which appear as constituents of the various R w are called the unipotent characters of G; we denote them Uch(G). By a result of Lusztig [12, 3.9], for any unipotent character ρ of G, F δ acts by the same eigenvalue of Frobenius ω ρ on any ρ-isotypic component H i c (X w , Q ℓ ) ρ of any ℓ-adic cohomology group of any X w , up to multiplication by integral powers of q δ . 2.2. Reduction to simple groups. Here we follow the arguments in [8, Rem. 4.2.1] building upon [8, 1.5.9-1.5.13]. The centre Z(G) is characteristic in G, so σ induces a graph automorphism on (G/Z(G)) F . As unipotent characters have Z(G) F in their kernel, for the purpose of studying rationality of extensions of unipotent characters we may therefore assume G is semisimple. Let G sc and G ad be simply connected respectively adjoint groups of the same type as G. Then these possess corresponding graph automorphisms again denoted σ. Furthermore, there are natural F -and σ-equivariant epimorphisms G sc → G → G ad such that the images of the respective F -fixed points contain the derived subgroup of the F -fixed points of the target. Now unipotent characters have the centre in their kernel and restrict irreducibly to the derived subgroup. Since σ stabilises the centre and the derived subgroup, the rationality properties of extensions of unipotent characters of G F agree with those of any group isogenous to it. By passing to a group of adjoint type we see that we may hence assume for our purposes that G is simple (of a chosen isogeny type), which we do from now on.

In particular, we can always assume that δ ∈ {1, 2, 3} and F = σ r in its action on W , for some r ∈ {1, 2, 3} with one of r or δ being equal to 1.

2.3.

Eigenvalues of F and character extensions. We keep the above setting. We first look at the extensions over local fields given by the ℓ-adic cohomology of the Deligne-Lusztig varities.

Proposition 2.1. Let d be the order of σ (recall that d ∈ {2, 3}). Let ρ ∈ Uch(G) be rational valued and σ-invariant. Assume that there is w ∈ W such that (1) the F -orbit of w has length δ and is σ-stable; and

(2) the multiplicity of ρ in R w is not divisible by d.

Then for every extension

ρ of ρ to G σ , the field of values Q ℓ ( ρ) is contained in the splitting field of x d -ω d/δ ρ over Q ℓ . Furthermore, there is at least one extension ρ which is Q ℓ -valued if and only if there is a δth root of ω ρ in Q ℓ .
Remark 2.2. Since the Frobenius eigenvalue ω ρ is uniquely determined up to integral powers of q δ , the conclusion of Proposition 2.1 is well-defined.

Proof. We consider the subvariety 1) it has an action of both F and σ and for all i we have

X = X w ⊔ X F (w) ⊔ • • • ⊔ X F δ-1 (w) of B. By (
H i c (X) ρ ∼ = H i c (X w ) ρ ⊕ H i c (X F (w) ) ρ ⊕ • • • ⊕ H i c (X F δ-1 (w) ) ρ as Q ℓ G-
modules with F permuting cyclically the δ summands. By (2) there is some i for which the multiplicity of ρ in H i c (X w ) ρ is not divisible by d. Thus there also is a generalised eigenspace H i c (X w ) ρ,µ of F δ on H i c (X w ) ρ with the same property. Here, as cited above, the eigenvalue µ differs from ω ρ by an integral power of q δ . First assume δ = 1. Then H := H i c (X w ) ρ,µ is a Q ℓ G-module in which not all extensions of ρ can occur with the same multiplicity. Since d ∈ {2, 3}, at least one of the extensions must then be distinguished by its multiplicity and thus have values in Q ℓ . The others are obtained by tensoring with linear characters of G/G, which have values in the splitting field of

x d -1, so of x d -ω d ρ . Now assume δ = d. Then F has characteristic polynomial (x d -µ) m in its action on ⊕ δ-1 j=0 H i c (X F j (w) ) ρ,µ , with m = dim H i c (X w ) ρ,µ
and m/ρ(1) not divisible by d. Let K be the splitting field of x d -µ over Q ℓ . Then Gal(K/Q ℓ ) permutes the generalised F -eigenspaces as it permutes the eigenvalues, that is, as it acts on the roots of x d -µ. If K contains a zero of x d -µ, there is a Q ℓ -rational eigenspace and we can argue as before.

If Q ℓ does not contain a zero of x δ -ω ρ , then K/Q ℓ is an extension of degree δ (recall that δ ≤ 3) and the δ different generalised eigenspaces of F are Galois conjugate over Q ℓ . Thus, the same holds for the δ different extensions of ρ.

We now lift the rationality properties to Q.

Lemma 2.3. Let ρ be a rational valued unipotent character of G. Assume that for every ℓ = p, ρ has an extension ρ ℓ to G which takes values in Q ℓ . Then ρ has an extension to G which takes values in Q.

Proof. We argue by contradiction. Assume all extensions of ρ are defined over a proper extension K of Q, generated by a root of f ∈ Q[x], say. Since the sum of the extensions has values in Q and δ ∈ {2, 3} this means that K (and hence f ) has degree δ over Q. Note that K/Q is abelian. Thus by Dirichlet there are infinitely many primes ℓ such that f is also irreducible over Q ℓ , that is, its roots generate an extension of degree δ. Hence ρ does not have any Q ℓ -rational extension, contradicting our assumption. Proof. We consider the various cases according to Lusztig's classification of cuspidal unipotent characters. By [8, Thm 4.5.11] they are all σ-invariant. If G is of type D n with n = (2t) 2 for some t ≥ 1 and o(σ) = 2, then the class of W labelled by the bi-partition (-; 4t-1, 4t-3, . . . , 1) contains σ-stable elements (since the centraliser in W of σ, of type B n-1 , contains the class labelled (-; 4t -1, 4t -3, . . . , 3)), and by [13, Prop. 2.14], the unique cuspidal unipotent character ρ occurs exactly once in the corresponding Deligne-Lusztig character. Thus Proposition 2.1 applies to show that ρ has rational extensions. If G is of type D 4 with o(σ) = 3, then there exists a σ-stable element w ∈ W in the class labelled by the bi-partition (-; 31), and using Chevie [15] the unique cuspidal unipotent character ρ of G appears with multiplicity 1 in R w . Thus Proposition 2.1 applies again.

Finally, for G of type E 6 , there are two cuspidal characters E 6 [θ], E 6 [θ 2 ] with Frobenius eigenvalue a primitive third root of unity θ, respectively θ 2 . Let w ∈ W be in class E 6 . Then w can be chosen σ-stable. Again using [15], both E 6 [θ] and E 6 [θ 2 ] occur in R w with multiplicity 1. Let H be a cohomology group of X w containing ρ with odd multiplicity, for ℓ a prime with ℓ ≡ 1 (mod 3), so √ -3 ∈ Q ℓ . Since σ fixes w, it acts on H and so H ρ contains the two extensions ρ 1 , ρ 2 of ρ to G with different multiplicities. Thus, they must be Q ℓ -rational. Since this is true for all ℓ ≡ 1 (mod 3), the character field of ρ i is contained in

Q(θ) = Q(ρ), hence equal to Q(θ). Proposition 2.5. Let G = A n-1 (-q) = 2 A n-1 (q)
where n = t+1 2 with t ≥ 1, and let ρ be the cuspidal unipotent character of G. Let σ be the graph-field automorphism of G of order 2. Then the two extensions of ρ to G = G σ are rational-valued if n 2 is even, and are algebraically conjugate over Q( √ -q) if n 2 is odd. Proof. The cuspidal unipotent character of 2 A n-1 (q) is labelled by the staircase partition λ = (t, t -1, . . . , 1) (see e.g. [8, Prop. 4.3.6]). The Frobenius eigenvalue of ρ is ω ρ = (-q) ( n 2 ) (up to multiplication by powers of q 2 ) by [12, Rem. (a) after Thm 3.34] and [8, Prop. 4.3.7]). An application of the Murnaghan-Nakayama rule shows that the irreducible character of S n labelled by λ takes value ±1 on elements w ∈ W = S n of cycle type (2t -1, 2t -5, . . .). Since the multiplicities of unipotent characters in Deligne-Lusztig characters in type A n-1 are given by the character table of W (see [8, Cor. 2.4.19]), this means that ρ has multiplicity ±1 in the Deligne-Lusztig character R w . Also, no conjugate of w is centralised by σ, hence neither by F . Thus the assumptions of Proposition 2.1 are satisfied and the conclusion follows.

Remark 2.6. Alternatively, by the arguments given for the principal series case in the proof of Theorem 3 below, the conclusion of Proposition 2.5 would follow if the full endomorphism algebra of the cohomology of X w 0 , where w 0 ∈ W is the longest element, is indeed the Iwahori-Hecke algebra of type A n-1 at parameter -q as speculated in [12, 3.10(b)] (see also the general conjectures in [1, 1B]). Proof. First assume that G = 2 D n (q) where n = (2t + 1) 2 with t ≥ 1, and let ρ be the cuspidal unipotent character of G. Let w ∈ W be in the class labelled by (-; 4t + 1, 4t -1, . . . , 1). An application of Asai's formula [8, Thm 4.6.9] shows that ρ appears with multiplicity ±1 in R w (see also [13, 2.19]). By [7, Thm 4.11] the Frobenius eigenvalue of ρ is ω ρ = 1, up to multiplication by integral powers of q 2 , so an application of Proposition 2.1 allows us to conclude. Now assume G = 3 D 4 (q), with σ the graph-field automorphism of G of order 3. If ρ is the cuspidal unipotent character 3 D 4 [-1] of G then (using Chevie[15]), ρ occurs with multiplicity 1 in X w for w of type F 4 . The class of w is not F -stable, so Proposition 2.1 applies. In this case the eigenvalue of F 3 for ρ equals ω ρ = -1 (up to integral powers of q 3 ) by [11, (7.3)]. Next, let ρ be the cuspidal unipotent character 3 D 4 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF], with Frobenius eigenvalue ω ρ = 1, by [7, Rem. 4.9]. It occurs with multiplicity 1 in X w for w of type F 4 (a 1 ), which can be chosen not σ-invariant. Again Proposition 2.1 applies.

Finally, let G = 2 E 6 (q). The cuspidal unipotent character ρ = 2 E 6 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF] with ω ρ = 1 (by [7, Rem. 4.9]) appears with multiplicity 1 in the Deligne-Lusztig character R w for w in class 3A 2 . Choosing w not σ-stable, we conclude as in the previous case. Let now ρ be one of the cuspidal unipotent characters 2 E 6 [θ], 2 E 6 [θ 2 ] of G with ω ρ = θ, θ 2 respectively [11, (7.4)(e)]. For these the claim follows precisely as for the non-rational cuspidal characters of E 6 (q) in the proof of Proposition 2.4.

We are now ready to show our first main result:

Proof of Theorem 1. This follows from Propositions 2.4, 2.5 and 2.7 as those cover all relevant cases.

Proof of Corollary 2. If ρ is not real-valued the assertion holds trivially. We now discuss the real-valued cuspidal unipotent characters. All of them are rational by [8, Cor. 4.5.6]. The Frobenius-Schur indicators of all these characters of untwisted groups are +1 by Lusztig [13, Thm 0.2], and by Proposition 2.4 they possess rational extensions.

By Ohmori [START_REF] Ohmori | The Schur indices of the cuspidal unipotent characters of the finite unitary groups[END_REF] the cuspidal unipotent character ρ of the unitary group G = SU n (q) with n = t(t + 1)/2 has Frobenius-Schur indicator (-1) ⌊n/2⌋ . Now ⌊n/2⌋ is the F q -rank of G, which in turn is congruent modulo 2 to the exponent i in the Frobenius eigenvalue ω ρ = (-q) i of ρ by [12, Rem. (a) after Thm 3.34]. The claim in this case thus follows from Proposition 2.5, observing that ⌊n/2⌋ and n 2 have the same parity. For the orthogonal group G = SO - 2n (q) where n = (2t + 1) 2 , the Frobenius-Schur indicator of the cuspidal unipotent character equals +1 by [13, 1.13], the Frobenius-Schur indicator of the cuspidal unipotent character 2 E 6 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF] of 2 E 6 (q) is +1 by [5, 6.2], and similarly the indicators of the two cuspidal unipotent characters of 3 D 4 (q) are also +1 by [11, (7.6)] and [5, 6.2]. So for the latter groups we may conclude by Proposition 2.7.

Harish-Chandra theory

We now consider arbitrary unipotent characters.

Proof of Theorem 3. Let ρ ∈ Uch(G) be σ-invariant, so it has an extension ρ to G = G σ . First assume ρ lies in the principal series, so it occurs as constituent in the permuta-

tion module Q ℓ [G/B], whence ρ occurs in M := Q ℓ [ G/B]. Now as a Q ℓ G × End G (M)
bimodule, M decomposes as the direct sum of irreducible submodules M φ indexed by φ ∈ Irr(End G (M)) affording χ φ ⊗ φ for some χ φ ∈ Irr( G) in the principal series. Clearly, any Galois automorphism of Q ℓ permutes the M φ as it permutes the characters φ, and hence it permutes the χ φ in the same way. So the rationality statement for ρ follows from the corresponding one for the extended Hecke algebra in [4, Thm II.3.3]. (See also the proof of [START_REF] Geck | Character values, Schur indices and character sheaves[END_REF]Prop. 5.5].) Now assume we are not in that case. Let L ≤ G be a (split) Levi subgroup and λ a cuspidal unipotent character of L such that ρ lies in the Harish-Chandra series of λ, so ρ, R G L (λ) = 0. Thus, ρ corresponds to a character φ of the relative Weyl group

W ′ := W G (L, λ) (see [8, Thm 3.2.

5]). It is known that this relative Weyl group is of type

A 2 , B n , G 2 or F 4 in the cases we consider and that L can also be chosen to be σ-stable (see e.g. [8, Tab. 4.8]). Moreover, as λ is the unique cuspidal unipotent character of L, it is also σ-stable. Furthermore, σ acts trivially on W ′ except possibly if W ′ has type A 2 . By [9, Thm 3.3 and (3.6)] any such φ is of parabolic type, that is, there is a parabolic subgroup

W ′ J of W ′ such that 1 W ′ W ′ J , φ = 1. Note that we may assume W ′ J is proper in W ′ if W ′ = 1. This is clear if φ = 1 W ′ ,
and φ = 1 W ′ occurs with multiplicity 1 in the permutation character on any parabolic subgroup. Let M ≥ L be the Levi subgroup of G corresponding to W ′ J and χ be the unipotent character of M in the Harish-Chandra series (L, λ) corresponding to 1 W ′ J . Then by the comparison theorem [8, Thm 3.2.7] this means that R G M (χ), ρ = 1. First assume σ has order 2 and let ρ 1 , ρ 2 be the two extensions of ρ to G. We claim that ρ 1 has the same rationality property as some extension λ 1 of λ to L = L σ . If W ′ = 1 then L = G and the claim is obvious. If W ′ = 1 then W ′ J < W ′ as argued above. Exclude for the moment the case that W ′ has type A 2 . Then σ stabilises (W ′ J , χ), so χ has two extensions

χ 1 , χ 2 to M = M σ . Then R G M (χ 1 + χ 2 ), ρ 1 = R G M (Ind M M χ), ρ 1 = Ind G G R G M (χ), ρ 1 = R G M (
χ), ρ = 1 and thus (after possibly interchanging ρ 1 and ρ 2 ) R G M (χ 1 ), ρ 1 = 1 and R G M (χ 1 ), ρ 2 = 0. Consequently ρ 1 has the same rationality properties as χ 1 , which by induction has the same rationality properties as λ 1 , showing our claim. Now, the rationality properties of extensions of the cuspidal unipotent character λ were discussed in Section 2, which gives the desired conclusion. Now assume that W ′ has type A 2 . Then by [8, Tab. 4.8] we have G = E 6 (q) and λ is the cuspidal unipotent character of L of type D 4 . Of the three characters ρ of G in this Harish-Chandra series, two have multiplicity one in R G L (λ) and thus by our previous argument, possess rational extensions to G. The Deligne-Lusztig character R w for w in the class E 6 (with characteristic polynomial Φ 3 Φ 12 ) contains the third character D 4 ,r with multiplicity -1. Since the class E 6 contains elements centralised by σ, Proposition 2.1 shows that D 4 ,r has a rational extension to G.

If σ has order 3, then necessarily G has type D 4 or 3 D 4 . Here, only the cuspidal characters do not lie in the principal series, and for those, the claim was shown above.

Example 3.1. Let ρ be any of the two unipotent characters of G = 2 E 6 (q) in the Harish-Chandra series of type 2 A 5 . Then the extensions of ρ to G have character field Q( √ -q), by Proposition 2.5 in conjunction with Theorem 3. These are Ennola-dual to the principal series characters φ 64,4 and φ 64,13 of E 6 (q), thus we see that the occurring irrationalities in Theorem 3 do obey the Ennola principle.

We can also understand completely the groups of type D 4 extended by its full group of grah-automorphisms: Corollary 3.2. Let G = D 4 (q) and Γ ∼ = S 3 its full group of graph automorphisms. Then all Γ-invariant unipotent characters of G have a rational extension to G = G.Γ.

Proof. Let ρ ∈ Uch(G) be Γ-invariant. Let σ ∈ Γ have order 3. By Proposition 2.4, ρ has one rational extension ρ 1 to G σ and two algebraically conjugate ones. Thus ρ 1 must be Γ-invariant as well and further extends to two characters ρ 1 , ρ 2 of G. The restrictions of these to G τ , where τ ∈ Γ has order 2, are the two extensions of ρ to G τ , so rational again by Proposition 2.4. But then ρ 1 , ρ 2 must also be rational valued.

Exceptional graph automorphisms

The groups B 2 (2 2f +1 ), G 2 (3 2f +1 ), F 4 (2 2f +1 ) with f ≥ 0 possess exceptional outer graph automorphisms of order 2 not induced by an automorphism of the ambient algebraic group (in particular one can not use the results in Section 2). In the smaller two cases, the rationality properties of extended unipotent characters were determined by Brunat [2,3]. More precisely we have the following: Proof. Let G be of type F 4 with a Steinberg endomorphism F 0 such that G = G F for F = F 2 0 . We may assume that F 0 induces σ on G. For the characters in the principal series the claim follows from the character table of the extended Hecke algebra in [6, Tab. VI]. For the characters B 2 ,1, B 2 ,ǫ we can argue as in the proof of Theorem 3 since they are parametrised by linear characters of the relative Weyl group, of type B 2 (see [8, Tab. 4.8]), which thus have multiplicity 1 in its regular character. Now let w ∈ W be a σ-invariant element, and µ be an eigenvalue of F on H = H i c (X w ) for some i. Let λ = √ µ be a root of µ. Then the action of G on H

:= (H ⊗ Q ℓ Q ℓ (λ)) µ
extends to an action of G where σ acts as λ -1 F 0 and the traces of all gσ, for g ∈ G, lie in λQ ℓ . Thus, any Galois automorphism sending λ to -λ will interchange the multiplicities in H of the two extensions of any irreducible character ρ of G. In particular, if ρ has odd multiplicity in H, then its two extensions are only defined over Q ℓ (λ). Now the eigenvalues of F can be computed by evaluating [4, III, Prop. 1.2 with Thm 1.3] with Chevie [15]. Specifically, if w ∈ W is a σ-invariant regular element of order 8, then F 4 [-1] occurs with odd multiplicity in some H i c (X w ) and with an eigenvalue of F equal to -q 3 .Since q is an odd power of 2 we have Q ℓ ( -q 3 ) = Q ℓ ( √ -2) and the previous argument shows that F 4 [-1] has an extension in H with character field Q( √ -2). For the two characters ρ = F 4 [±i], we use the same element w. In that case the eigenvalues of F are ±iq 3 . As before, this means that both characters possess extensions to G with values in Q( √ 2i). Now note that √ 2i = ±(1 + i) lies in Q(i), the character field of ρ.

Finally, the two cuspidal characters F I 4 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF], F II 4 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF] appear with odd mulitplicity in the q 6 -eigenspace of F on H i c (X w ) for w a σ-stable element in class D 4 (a 1 ). We can now argue as before to see that these characters possess rational extensions.

Remark 4.3. We expect (from Ennola duality and the known character table of F 4 (2).2), but have not been able to prove, that the unipotent character B 2 ,r possesses rational extensions, and the cuspidal characters ρ = F 4 [θ], F 4 [θ 2 ] have extensions with character field Q(ρ).

Remark 4.4. A way to see that the cuspidal unipotent character F 4 [-1] of G = F 4 (q) has non-real extensions to G is as follows: Let ℓ > 2 be a prime dividing q 4 + 1 and P a Sylow ℓ-subgroup of G. Then P has larger automiser in G than in G. Thus the Brauer tree of the principal ℓ-block of G is obtained by unfolding the Brauer tree of the principal ℓ-block of G around the exceptional vertex. Since F 4 [-1] and the trivial character lie on opposite sides of the exceptional vertex by [10, Thm 2.1(3)] and the trivial character has real extensions to G, the extensions of F 4 [-1] cannot lie on the real stem of the Brauer tree of the principal ℓ-block of G and thus can't be real-valued. This does, however, not exhibit the precise character field.

The argument also applies to the cuspidal unipotent characters of 2 A 2 (q) and 2 A 5 (q), and to G 2 [START_REF] Broué | Zyklotomische Heckealgebren[END_REF] of G 2 (q) (with suitable ℓ).

2. 4 .

 4 Extensions of cuspidal unipotent characters. We keep the above setting. For the classification and properties of unipotent characters we refer the reader to [8, § §4.3-4.5]. Recall that cuspidal unipotent characters have values in Q(ω ρ ). Proposition 2.4. Let G be simple of type D n or E 6 , σ a graph automorphism of G and F a commuting Frobenius map with δ = 1. Then any cuspidal unipotent character ρ of G = G F has an extension to G = G σ defined over Q(ρ).

Proposition 2. 7 .

 7 Let G be simple of type D n or E 6 , σ a graph automorphism of G and F a commuting Frobenius map with δ = o(σ). Then any cuspidal unipotent character ρ of G = G F has an extension to G = G σ with field of values Q(ρ).

Proposition 4. 1 (

 1 Brunat). (a) For G = B 2 (2 2f +1 ) all four invariant unipotent characters have rational extensions to the extension of G by the exceptional graph automorphisms. (b) For G = G 2 (3 2f +1 ), eight unipotent characters are invariant under the exceptional graph automorphism. Of these, the characters labelled φ 1,0 , φ 2,1 , φ

  Proof. The first statement follows by inspection of [2, Tab. 6], the second from [3, Tab. 11]. For the characters in the principal series this also already follows from [6, Tab. IV], using the arguments in the proof of Theorem 3. (Note that the entries in the first rows of both tables in loc. cit. should correctly read u l(w) , as confirmed by the authors.) Proposition 4.2. For G = F 4 (2 2f +1 ) with the exceptional graph automorphism σ of order 2, 21 unipotent characters are σ-invariant. Of these, the principal series characters φ 1,0 , φ 4,1 , φ 9,2 , φ 12,4 , φ 6,6 ′ , φ 6,6 ′′ , φ 4,8 , φ 9,10 , φ 4,13 , φ 1,24

	rational extensions, φ 2,2 has an extension with character field Q( and G 2 [θ 2 ] have extensions with character field Q( √ -3).	√	1,6 and G 2 [-1] have 3), while G 2 [1], G 2 [θ]
	The character φ 16,5 has extensions with character field Q(	√	2), the cuspidal characters
	F 4 [±i] have extensions with character field Q(i), and F 4 [-1] has extensions with character field Q( √ -2).

and the four characters

B 2 ,1, B 2 ,ǫ, F I 4 [1], F II 4 [1] possess rational extensions to G = G σ .

The second author gratefully acknowledges financial support by the DFG -Project-ID 286237555.


Kristijan Macek

Alejandro Vasquez Dizan

Thierry Govea

Roland Y Siegwart Fraichard

Siegwart

Towards Safe

Kristijan Maček

Dizan Vasquez

Thierry Fraichard

Roland Siegwart

Towards Safe Vehicle Navigation in Dynamic Urban Scenarios

Keywords: Motion planning, Dynamic environment, Motion safety, Urban navigation planiranje gibanja, dinamička okolina, sigurnost gibanja, urbana navigacija

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Background and Motivations

Autonomous mobile robots/vehicles navigation has a long history by now. Remember Shakey's pioneering efforts in the late sixties [START_REF] Nilsson | Shakey the robot[END_REF]. Today, the situation has dramatically changed as illustrated rather brilliantly by the 2007 DARPA Urban Challenge 1 . The challenge called for autonomous car-like vehicles to drive 96 kilometers through an urban environment amidst other vehicles (11 selfdriving and 50 human-driven). Six vehicles finished the race thus proving that autonomous urban driving could become a reality. Note however that, despite their strengths, the Urban Challenge vehicles have not yet met the challenge of fully autonomous urban driving (how about handling traffic lights or pedestrians for instance?).

Another point worth mentioning is that at least one collision took place between two competitors. It was nothing 1 http://www.darpa.mil/grandchallenge serious but it raises the important issue of motion safety, ie the ability for an autonomous robotic system to avoid collision with the objects of its environment. With robotic systems designed to operate in the real world among human beings in many cases, motion safety becomes critical. The size and the dynamics of the Urban Challenge vehicles make them potentially dangerous for themselves and their environment (especially when driving at high-speed). Therefore, before letting such autonomous systems move among people, it is vital to assert their operational motion safety.

In the last forty years, the number and variety of autonomous navigation schemes that have been proposed is huge (cf [START_REF] Siegwart | Introduction to Autonomous Mobile Robots[END_REF]). In general, these navigation schemes aims at fulfilling two key purposes: reaching a goal while avoiding collision with the objects of the environment. When it comes to collision avoidance, once again, many collision avoidance schemes have been proposed. Their aim of course is to ensure the robotic systems' safety. However the analysis carried out in [START_REF] Th | A short paper about motion safety[END_REF] of the most prominent navigation schemes (ie the ones currently used by robotics systems operating in real environments, eg [START_REF] Borenstein | The vector field histogramfast obstacle avoidance for mobile robots[END_REF][START_REF] Minguez | Nearness diagram (ND) navigation: collision avoidance in troublesome scenarios[END_REF][START_REF] Fox | The dynamic window approach to collision avoidance[END_REF][START_REF] Fiorini | Motion planning in dynamic environments using velocity obstacles[END_REF]) shows that, especially in dynamic environments, motion safety is not guaranteed (in the sense that it is easy to find situations where collisions will eventually occur). To some extent, this is due to the fact that safety is a concept that is taken for granted. In other words, the meaning of safety is never formally stated and, above all, the operational conditions of such collision avoidance schemes are seldom (if never) spelled out.

Contributions and Paper Outline

In the past few years, the Autonomous Systems Lab. of the Swiss Federal Institute of Technology have been active with self-driving cars through the SmartTer initiative 2 . It has developed a navigation architecture that has evolved over the years. The primary purpose of this paper is to present the latest developments of the deliberative part of this navigation architecture. These developments are geared towards autonomous driving in dynamic urban environments with a particular focus on the motion safety issue.

The deliberative part of the architecture features two key modules working together in a hierarchical fashion: the Route Planner (high-level) and the Partial Motion Planner (low-level).

The purpose of the Route Planner is to provide the Partial Motion Planner with a valid route towards a given goal. A route in this case comprises a geometric path augmented with additional information based on the structure of the environment considered. Such a route should comply with the standard regulations for vehicles driving in a urban setting. This means that factors such as speed limits and stop signs should be taken into account.

It is up to the Partial Motion Planner to take care of all the gory details of the actual driving. It relies upon the route and a local model of the vehicle's environment (with up to date information about the fixed and the moving objects) in order to determine the next motion command to apply to the vehicle.

As the name suggests, a Partial Motion Planning scheme is used [START_REF] Petti | Partial motion planning framework for reactive planning within dynamic environments[END_REF], [START_REF] Maček | Motion planning for car-like vehicles in dynamic urban scenarios[END_REF] in order to (a) take into account the decision time constraint imposed by dynamic environments and (b) improve convergence towards the desired goal.

Motion safety is dealt with by the Partial Motion Planner. Two safety levels respectively called Passive and Passive Friendly are explored and their operational conditions are explicitly stated.

NAVIGATION ARCHITECTURE

Fig. 1 presents an overview of the navigation architecture of the SmartTer platform. It is structured in layers, where, in most cases, higher level components interact with all the lower level layers in the hierarchy. The only exception is the world model, which is directly accessed by all the levels of hierarchy. Given that this paper focuses on route planning and partial motion planning we will just briefly discuss the high-level components and their interaction here.

Mission Manager. It is responsible of translating

high-level tasks (eg pick-up a person at an address) into initial and a goal configurations that the vehicle should reach in order to accomplish those tasks.

2. Route planning. It is concerned with finding a global route for the vehicle between the initial and goal configurations given by the mission manager, using information about the static characteristics of the environment (e.g. lane geometry, speed limits). We assume that knowledge about these characteristics is available a priori so that it is possible to perform at least part of the computations off-line.

3. Partial Motion Planning. It is responsible of executing routes computed by the route planning module including collision avoidance and -for unstructured environments-motion planning. It integrates knowledge about the dynamic elements of the environment,and interacts tightly with the world model.

4. World model. The world model gathers all the available information about the environment, including the vehicle's localization, the environment structure and the current and predicted states of the other objects that are present in the environment. As mentioned above, it is different from other modules because it interacts with all the levels of the hierarchy.

The next two sections respectively describe Route Planning and Partial Motion Planning.

ROUTE PLANNING

The goal of route planning is to exploit prior knowledge about the environment in order to provide local navigation with a feasible and valid path between two points of the environment. In this context, valid means that the path should be collision free and comply with traffic regulations for vehicles driving in a urban setting, where factors such as speed limits and stop signs should be taken into account by the planner.

More specifically, prior knowledge comes in the form of a slightly augmented 3 Route Network Definition File (RNDF) [START_REF]Route network definition file (RNDF) and mission data file (MDF) formats[END_REF]. This information may be seen as a directed graph (see fig 2) where we distinguish between two types of nodes, waypoints, having a special meaning (eg an intersection or a parking entrance) and standard nodes which are used as a piecewise linear approximation to describe higher order curves. Additionally, unstructured, open areas of the environment are described as polygons having entry and exit points that are connected to other nodes of the graph.

From this graph, route planning is straightforward by applying the A* graph search algorithm using the length of the graph's edges as a cost function. Once a path has been found, it is smoothed out in order to guarantee its feasibility. The output of the module consists of a list of configurations

Q g = {q 1 g , • • • , q
Ng g } that the vehicle should 3 RNDF's have been augmented by including speed limits.

q i g = {x i g , y i g , θ i g , κ i g } (1)
Associated with every configuration, there is a constraint vector describing the bounds within which the vehicle's motion is considered as acceptable with respect to the task at hand and the traffic rules (eg speed limits), as well as a Boolean flag wp i , whose value is true when the configuration corresponds to a waypoint and false otherwise:

c i = {r i max , ∆θ i max , ∆κ i max , v i min , v i des , v i max , wp i } (2)
where r i max stands for the maximum distance between the object's actual position and the desired one; ∆θ i max and ∆κ i max are the maximum error tolerated for the heading angle and the curvature, respectively; and v i min , v i des and v i max are the minimum, desired and maximum velocities associated to the given configuration.

PARTIAL MOTION PLANNING

The Partial Motion Planner (PMP) is the core navigational module of the architecture.

Its primary purpose is to determine the motion command u that is sent to the vehicle controller at every time cycle.

The motion command u must meet the following requirements:

• Feasibility: it must take into account the dynamic constraints of the vehicle;

• Goal Convergence: it must eventually drive the vehicle towards the desired goal.

• Safety: it must ensure the safety of the vehicle, ie its ability to avoid collisions.

PMP operates iteratively with a time period T d which is determined by the environment (in an environment featuring moving objects, you have a limited time only to decide upon your future course of action otherwise you run the risk of being hit by a moving object). T d constitutes the decision time constraint.

To determine u, PMP (as the name suggests) applies the Partial Motion Planning principle [START_REF] Petti | Partial motion planning framework for reactive planning within dynamic environments[END_REF]: it tries to make the best possible use of the decision time T d available by computing a partial motion towards the goal. To that end, a diffusion technique is used to explore the state×time space of the vehicle and determine a partial motion π that is used during the next time cycle to drive the vehicle towards its goal.

Fig. 3 illustrates how PMP operates. Let t k denote the current time instant and the beginning of the k th PMP cycle. The previous PMP cycle has computed the partial motion π(t k-1 ) that starts at time t k . The k th PMP cycle then has to compute π(t k ) that will start at time t k+1 = t k + T d . The process is repeated until the goal is reached.

At every cycle, PMP takes as input an updated model of the environment that comprises:

• the route computed by the Route Planner, ie the list

Q g = {q 1 g , • • • , q Ng g } and the corresponding con- straints c i , i = 1 • • • N g (cf §3),
• a list O s of static objects: it is assumed that the static part of the environment is known from the road network structure and is described by a list of forbidden regions represented by closed polygons;

• a list O d of dynamic objects: one fundamental task of the World Modelling module is to provide PMP with an updated model of the environment of the vehicle at every time cycle. PMP must know what are the moving objects present in the environment and, most important, what their future behaviour will be. To that end, the World Modelling module features a Prediction module whose purpose is to estimate the future behaviour of the moving objects. It is assumed that the prediction is valid over a given prediction horizon of duration T p . The moving objects are described by a list of forbidden regions whose position varies over time. Their shape is modelled by rectangular bounding boxes which is suitable for vehicles and pedestrians alike (more general shapes could be used). The notation O d (t) is used to indicated the fact that their position varies over time.

Each partial motion π computed respects the dynamic constraints of the vehicle considered thus meeting the Feasibility requirement.

By nature, PMP aims at maximizing the lookahead of the navigation process (the exploration of the future is carried out as far as possible given the decision time T d available). In our opinion, this is one way to meet the Goal Convergence requirement.

Finally, each partial motion π computed will be safe in a predefined way (for instance, it will be guaranteed that the vehicle always have the possibility to brake down and stop before a collision occurs). This is the answer to the Safety requirement.

The next two sections respectively describe the diffusion technique used and how motion safety is handled.

DIFFUSION TECHNIQUE

The partial motion, ie the trajectory, that is to be computed for the vehicle can be described as a single parametric curve, eg polynomial or spline curve, or a concatenation of several geometrical primitives such as arcs or clothoids. The kinematic vehicle model is described by the Ackermann model:

ẋ = cos θ v l , ẏ = sin θv l , θ = v l L tan φ , (3)
with {x, y, θ} being the robot pose and {v l , φ} the longitudinal velocity and steering angle. Therefore the full vehicle state at the current navigation cycle t k can be described as:

s(t k ) = {x(t k ), y(t k ), θ(t k ), v l (t k ), φ(t k )} (4)
The dynamic update of the system can be described in the discrete general form:

ṡ(t k ) = f (s(t k ), u d (t k-1 )) (5)
where the dynamic level control input vector u d is the longitudinal acceleration vl (t k-1 ) and the steering rate θ(t k-1 ). The dynamic update function f encapsulates the physical dynamic model of the vehicle, including inertia and physical forces acting on the vehicle itself.

If a low-level control is implemented separately (cascade control) which handles directly the actuators of the vehicle, ie gas pedal (longitudinal acceleration vl ) and steering wheel torque (steering rate θ), the system function f represents the closed-loop response of the vehicle with low-level control. Therefore, the actual commands issued from the Partial Motion Planning level are the kinematic control reference values:

u = {v l,ref , φ ref } (6)
The control vector u is derived directly from the planned trajectory π at each navigation cycle. Moreover, if the latency of the low-level control is very small in comparison to the navigation cycle T d , the closed-loop vehicle response to a kinematic control reference value u can be described with circular arcs (ie if the steering angle φ ref is kept fixed during time T d . Using arc geometric primitives largely reduces the computational costs that would be incurred if the full numerical integration through the system function f would be performed for possible input u.

In order to explore different possible trajectories, the geometric arc primitives are concatenated in a randomized sampled diffusion scheme, where the trajectory structure grows in a RRT ("Rapidly-Exploring Random Tree") manner [START_REF] Lavalle | Randomized kinodynamic planning[END_REF]. The motion exploration phase starts with the current vehicle state s(t k )) and the control input (trajectory) from the previous cycle u(t k-1 ), the goal configuration(s) Q g , the set of dynamic obstacles O d (t) and static obstacles O s . The new exploration states are inserted into a search tree T as can be seen in Table 1. The available decision time for exploration is of duration T d , after which a valid trajectory with associated control inputs

u(t k ) = {v l,ref (t k ), φ ref (t k )} is returned or failure is sig- nalled.
Firstly, the newly formed tree T is initialized by its root node τ root , where the information contained in each node is of the form:

τ = {s, u, w τ ,t τ , d τ } (7)
The s represents the state of the vehicle, u the reference input that induced the state s, w τ the overall cost to reach the node τ, t τ the cumulative time with respect to the root of the tree T and d τ the node depth within the tree. Therefore, τ root which contains the predicted state s(t k+1 ) (PREDICT_STATE) for the given control input u(t k-1 ) from the previous navigation cycle and zeroed cost, cumulative time and tree depth. The state prediction for the root node is necessary due to the fact that the currently computed trajectory π(k) will be available only at the end of the current navigation cycle (see Fig. 3). The L 1 contains all the nodes that are appended to the current tree depth d T .

PMP_SEARCH(s(t

k ), π(t k-1 ), Qg, O d (t), Os,T d ) 1 ts=0.0, d T =0; 2 s(t k+1 ) ← PREDICT_STATE(s(t k ), π(t k-1 )); 3 τroot.init(s(t k+1 ), u(t k-1 ), 0.0, 0.0, d T ); 4 T .init(τroot), L 1 .init(τroot), L 2 =∅

. Node expansion

The overall computation time t s can only be within the T d span (Table 1

.L5).

There are three tree expansion methods implemented in the current PMP scheme:

1. exploration: expansion towards a random configuration in the environment -q rand (Table 1.L9-L13);

2. goal search: expansion towards a goal configurationq goal (Table 1.L14-L18);

3. exhaustive search: extension from a given tree node (state) using a randomized control input -u rand (Table 1.L19-L23).

These cases are depicted further in Fig. 4. In case 1, the predicted state s(t k+1 ) is expanded towards a random configuration q rand in the workspace (s 1 (t k+2 )), whereas in case 2 the tree T is expanded towards the goal configuration q goal (s 2 (t k+2 )). In both cases the function EXTEND_STATE_SPACE first finds the node τ near in the tree T which is the closest to the chosen configuration according to a predefined distance metrics and than chooses the nearest feasible command with NEAR-EST_COMMAND function. The case 3 represents the direct search in the control space of currently kinodynamically feasible commands with RAND_COM_SPACE (s 3 (t k+2 )).

For all the three possible expansion methods, the newly obtained nodes/states and the arcs a that connect them to their predecessor nodes in the tree have to be checked for collision against all the dynamic and static obstacles. This is done by discretizing the connecting arc trajectories a in time and verifying possible polygonal intersections resulting from the vehicle and obstacles intermediate configurations in the EX-TEND_WITH_COLLISION_CHECK function. Moreover, if the predecessor of an expanded node is at the depth d τ = 0, additional safety check has to be performed in the function EXTEND_WITH_SAFETY_CHECK (Table 1.L32-L40) -the vehicle must be able to come to a full-stop without collision (the state is therefore ICS safe, further details in §6). In Fig. 4 this is depicted by the example of the break-state s 3 (t k+2 + T b ) originating from state s 3 (t k+2 ), where the collision-check test is performed for the duration of the breaking manoeuvre. The brake time T b is related to the dynamic capabilities of the vehicle, ie max linear deceleration vl,decc :

T b = v l vl,decc (8)
where the longitudinal velocity depends on each given state, in this case s 3 (t k+2 ).

As already mentioned, the diffusion process computation is limited to t s ≤ T d , where at each new computational increment another layer of nodes is added to the tree T . The final depth of T is not determined a-priori, nor how close the best trajectory will be with respect to the goal configuration. In Fig. 4 the final best trajectory is depicted with π(t k ), referring to the navigation cycle when it was computed, whereas the trajectory's final state s(t k+n ) can be arbitrarily far in the future (n is a free parameter).

Regarding the goal search, ie finding a trajectory towards a goal configuration, there are two approaches distinguished here:

1. waypoint following: the current set of waypoints Q g is the next topological node to reach in the environment (see Sec.3), such as an intersection, route crossing, lane changing waypoint, etc.;

2. route following: the current set of waypoints Q g is a collection of intermediate configurations, which together describe a route between topological nodes.

In case 1 the cost function of a node to determine the best trajectory is based on a distance metric s -{q g , c q } between the goal waypoint with its constraints and the last state s on a trajectory:

w τ (s) = α g • s -{q g , c q } + α t • t τ (s) (9)
α g and α t are the weighting factors between minimizing the distance to the waypoint and minimizing the cumulative time t τ (s) to reach it, respectively.

In case 2 there is more information available based on the environment structure, in the form of a route. These geometrical configurations can be followed with a type of path following technique, with the possibility of deviating from the route based on the tree structure, if the dynamic obstacles trajectories require such evasive manoeuvres. However, in the absence of dynamic obstacles O d and proper route definition according to the a-priori knowledge of static obstacles O s , the vehicle should follow the route as close as possible. Assuming that a path following controller (implementation details on the controller can be found in [START_REF] Solea | Trajectory planning with velocity planner for fully-automated passenger vehicles[END_REF][START_REF] Maček | Path following for autonomous vehicle navigation with inherent safety and dynamics margin[END_REF]) computes an error function E which describes the discrepancy between a particular vehicle state s and the route Q g , then the cost of a node can be computed in the error terms as:

w τ (s) = Ns j=1 E(s j , Q g ) (10)
where the set of discretized states {s j=1 = s(k + 1) . . . s j . . . s Ns = s} with the associated arcs a forms a trajectory π. The cost calculation step is performed in the INSERT_WITH_COST function (Tab. 1.L44-L47). The best trajectory π⋆ = π(k) which will be applied in the next navigation cycle can be therefore determined from node τ ⋆ with minimum overall cost w τ , iff τ ⋆ = ∅.

SAFETY ISSUES

The purpose of this section is to explore the safety issues related to the navigation scheme proposed.

The diffusion technique presented in §5 aims at building a tree embedded in the state×time space of the system R and to extract from this tree a partial motion π that is used during the next time cycle to drive the system towards its goal.

The concept of Inevitable Collision State (ICS) 4 [START_REF] Th | Inevitable collision states. a step towards safer robots?[END_REF] and the motion safety criteria introduced in [START_REF] Th | A short paper about motion safety[END_REF] show that it does not suffice that each partial motion π be collision-free to ensure the safety of R.

From a theoretical point of view, the safety of R is guaranteed if and only each π is ICS-free up to the time T d (that corresponds to the initial state of the partial motion that is to be computed at the next navigation cycle), because then, at the next navigation cycle, the navigation module always has a safe evasive manoeuvre available. Now, checking whether a given state of π is an ICS or not requires in theory the full knowledge of the environment of R and its future evolution, ie the knowledge of the space-time W × [0, ∞). In practice however, one has to deal with the sensors' limited field of views and the elusive nature of the future. Knowledge about the environment of R is thus limited both spatially and temporally: it is limited to W p × [0, T p ] where W p ⊂ W denotes the subset of the environment which is perceived and T p the prediction horizon.

To further ensure safety with respect to the objects that lie outside of W p , its boundary is treated in a manner similar to [START_REF] Th | Inevitable collision states. a step towards safer robots?[END_REF] or [START_REF] Alami | On the influence of sensor capacities and environment dynamics onto collision-free motion plans[END_REF] as a potentially moving object whose motion direction is unknown but whose velocity is upperbounded.

W p × [0, T p ] and the objects within, fixed and moving, yields in the state×time space of R a set of ICS which is only an approximation of the true set of ICS generated by

W × [0, T p ].
This is the very reason why it is impossible to guarantee an absolute level of safety (absolute in the sense that it can be guaranteed that R will never end up in an ICS and therefore crash eventually).

This intrinsic impossibility compels us to settle for weaker levels of safety. Although weaker, the important thing is that such levels of safety will be guaranteed given the information that R knows about its environment, ie given W p × [0, T p ]. We have explored two different levels of safety, they are detailed in the next two sections.

Safety Level #1

The first safety level we have seeked to enforce is the simplest one maybe. It guarantees that, should a collision ever occur, R will be at rest. In other words, if a collision is inevitable, it can be guaranteed that R always have the possibility to brake down and stop before the collision occurs. Such a safety level is a form of passive safety in the sense that R will never actively collide with an object. It is henceforth called Passive Safety and denoted by PS.

Under PS, a state s is considered as being safe iff there exists at least one braking manoeuvre starting at s which is collision-free until the time where R has stopped. PS yields the following definition for a safe state:

Def. 1 (Passive Safety) a state s is safe under PS (or p-safe) iff there exists at least one braking manoeuvre starting at s and collision-free until T b , with T b the time where R is at rest (the braking time).

In practice, the function EXTEND.WITH.SAFETY.-CHECK (cf Table 1) samples a finite and discrete set of braking manoeuvres and checks them for collision against W p × [0, T p ]. If one collision-free manoeuvre exists the state considered is labeled as p-safe and unsafe otherwise.

Safety Level #2

In a way, PS leaves most of the collision-avoidance burden to the other objects. In certain situations however, this is unsatisfactory: R may for instance decide to move on a railway track to reach its goal because, under PS, it is safe to do so (indeed R would have the time to stop before being hit by the train). Unfortunately, the train in spite of its best efforts may not be able to avoid crashing into R because of its own dynamics.

In an environment where the moving objects are assumed to be friendly, ie seeking to avoid collisions, and for which a certain knowledge about their dynamic properties is available, it can be desirable to enforce a stronger level of safety. This second safety level guarantees that, should a collision ever occur, R will be at rest and the colliding object would have had the time to slow down and stop before the collision had it wanted to. This safety level is henceforth called Passive Friendly Safety and is denoted PFS. It yields the following definition for a safe state:

Def. 2 (Passive Friendly Safety) a state s is safe under PFS (or pf-safe) iff there exists at least one braking manoeuvre starting at s and collision-free until T b + T ob , with T b the braking time of R, and T ob the maximum braking time of the moving objects present in the environment.

The conservative nature of Def. 2 should be noted. It is possible in practice to refine it in order for example to take into account the dynamics of the particular moving objects that would collide with R when it follows a particular braking manoeuvre. For the time being, Def. 2 is left as is.

Other safety levels could be proposed. The ultimate one of course is to determine safety with respect to the set of ICS which is defined by W p × [0, T p ]. Given the complexity of characterizing this ICS set, Passive Safety and Passive Friendly Safety constitutes interesting alternatives in the sense that they can be computed efficiently and provide an adequate level of safety.

EXPERIMENTAL PLATFORM

SmartTer

The SmartTer is a passenger car equipped for autonomous driving. The system has a localization module based on sensor fusion, in the line of that described in [START_REF] Dissanayake | The aiding of a low-cost strapdown inertial measurement unit using vehicle model constraints for land vehicle applications[END_REF]. To accurately localize the vehicle, four different sensors are used: DGPS, IMU, optical gyro and vehicle sensors (wheel encoders and steering angle sensor). The combination of their measurements allows the estimation of the vehicle's 6 degrees of freedom eg the 3D position (x, y, z) and the attitude (roll, pitch, heading). The details of the localization system used are described in [START_REF] Lamon | The SmartTer -a vehicle for fully autonomous navigation and mapping in outdoor environments[END_REF].

For environment detection we use one IBEO Alasca XT laser scanner mounted at the front of the vehicle and two Sick LMS 291 mounted on the roof to protect the vehicle corners as well as a Sony Camera for vision.

Simulator

In order to facilitate system integration and to perform tests where the ground truth is available, we have performed our experiments using the Morsel simulator. Morsel has been developed in our laboratory and is based on the Panda3d engine [START_REF] Goslin | The panda3d graphics engine[END_REF].

Morsel features emulation for the two lower level layers of fig. 1. This includes cameras, range sensors, and standard cinematic models such as Ackermann, differential, etc.

The simulator can be easily extended using Python or C++. Another interesting feature is its ability run in "real" or "virtual" time modes. In the former mode, the simulator will try to produce sensor output at the real frequency; in the second mode time is emulated by putting world events in a priority queue according to their frequency.

Since our system uses the Carmen [START_REF] Montemerlo | Carmen: Carnegie mellon robot navigation toolkit[END_REF] library for communications, low-level sensor and actuator driver modules are effectively decoupled from high-level algorithm. This makes it possible to plug the simulator to those algorithms in a transparent fashion.

SIMULATION RESULTS

Simulation results are presented for different structured and unstructured urban environment scenarios including intersection handling (Fig. 6 to Fig. 9), in-lane driving (Fig. 10 and Fig. 11) and parking lot negotiation (Fig. 12 and Fig. 13). The dynamic obstacles presented in different scene views are pedestrians and static and moving vehicles, whose shapes are described as polygons, more specifically rectangular shapes. Their estimated positions and future trajectories are depicted in navigation views which also includes the static part of the environment that is described via polylines (for instance lane and parking lot boundaries). Collision checking between the ego-vehicle and environemental obstacles has to be performed for the whole given time horizon. The navigation scheme presented in the results is based on waypoint following (see §5, case 1) for goal search where the currently active goal waypoint is drawn in the navigation views as a circular region with a specified orientation (according to the given constraints). The trajectory diffusion starting from the ego-vehicle contains tree nodes that are free of obstacles (green) and prohibited nodes (marked red), whereas the currently best trajectory towards the waypoint is marked in magenta. Note that the given navigation snapshots are based on a certain time instant t k and that the prohibited regions depend on all the future motions of the obstacles. It can be seen from the results that the ego-vehicle is both able to reach the waypoint within given constraints while negotiating moving obstacles and preventing head-on collisions. Furthermore, no adaptation of the navigation scheme is needed for a particular environment, proving its generic applicability.

CONCLUSION AND OUTLOOK

This paper has presented the deliberative part of the navigation architecture for the SmartTer platform, comprising two main component: (a) route planning, which finds a set of configurations between two given points of the environment while taking into account given traffic rules; and (b) partial motion planning, which handles the actual execution of the plan while taking into account the dynamic elements of the world.

A key aspect of the navigation architecture proposed is that a special attention is paid to the motion safety issue, ie the ability to avoid collisions. Different safety levels are explored and their operational conditions are explicitly spelled out (something which is usually not done).

The results depict safe navigation in dynamic scenarios which include both moving vehicles and pedestrians, where the architecture has been implemented in both real and simulated platforms, although only simulation results are provided at the time of paper writing.

On the theoretical side, an interesting research direction is the exploration of more advanced motion prediction techniques in order to improve both the accuracy and the time horizon of our safety checks.

  2 http://www.smart-team.ch

Fig. 1 .

 1 Fig. 1. Overview of the proposed navigation architecture

Fig. 2 .Fig. 3 .

 23 Fig. 2. Detail of final DARPA's RNDF file showing waypoints (red), standard nodes (blue) and unstructured areas (green polygons)

Fig. 4

 4 Fig. 4. Node expansion

Fig. 5 .

 5 Fig. 5. The SmartTer passenger car equipped for autonomous driving. Behind the windscreen we have the camera system, on the sides of the roof we have the tilted laser scanners for corner protection. Currently, the front Sick has been replaced by an Alasca Scanner

Fig. 6 .Fig. 7 .

 67 Fig. 6. Intersection handling I (scene view)

Fig. 8 .Fig. 9 .

 89 Fig. 8. Intersection handling II (scene view)

Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 10111213 Fig. 10. In lane (scene view)

Table 1 .

 1 Diffusion process of the Partial Motion Planner

	, τ⋆=∅;

rand ← RAND_COM_SPACE(u(t k-1 )); 21 τnew ← EXPAND_COM_SPACE(τext, u rand ,. . . ); 22 if not (τnew = ∅) then 23 INSERT_WITH_COST(τnew, Qg, T , L 2 , d T , τ⋆);

A state is an ICS iff a collision eventually occurs no matter how R moves.

ACKNOWLEDGMENTS

This work has partly been supported by the EC under contract number FP6-IST-027140-BACS, as well as by the European project URUS (Ubiquitous Networking Robots in Urban Settings).


Kelig Mahé
email: kelig.mahe@ifremer.fr

Lea Joly

Solene Telliez

Jose-Luis Zambonino Infante

Léo Meunier Cédric

Kirsteen M Mackenzie

Carolina Giraldo

Kélig Mahé

Léa Justine Joly

Solène Telliez

José Luis Zambonino-Infante

Cédric Léo Meunier

K Mahé

Effect of temperature and CO2 concentration on the morphogenesis of sagittal otoliths in Atlantic herring (Clupea harengus) larvae

Keywords: otolith shape, otolith area, temperature effect, ocean acidification, Elliptic Fourier descriptors, directional asymmetry

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Otoliths are calcified biomineralised structures overlying the sensory epithelia in the inner ear of fish.

They are formed by calcium carbonate crystals embedded in a non-collagenous organic matrix composed of acidic proteins and polysaccharides [START_REF] Degens | Molecular structure and composition of fish otoliths[END_REF][START_REF] Popper | Why otoliths? Insights from inner ear physiology and fisheries biology[END_REF]. They are metabolically inert and do not resorb in periods of stress, but grow throughout an individual's life, in correlation with its ontogenic growth [START_REF] Casselman | Determination of age and growth[END_REF]. Because of the positive correlation between otolith and fish growths, otoliths have been widely used in fisheries science to understand the fish growth and the potential effects which controlled it (i.e. environment, fishing pressure; [START_REF] Enberg | Fishing-induced evolution of growth: concepts, mechanismsand the empirical evidence[END_REF][START_REF] Marty | Temporal trends in age and size at maturation of four North Sea gadid species: cod, haddock, whiting and Norway pout[END_REF][START_REF] Carbonara | Does fish growth respond to fishing restrictions within Marine Protected Areas? A case study of the striped red mullet in the south-west Adriatic Sea (central Mediterranean)[END_REF]. In Atlantic herring (Clupea harengus), otolith microstructure has been used to study environmental effects (temperature, salinity or feeding activity) on the growth of the larvae [START_REF] Folkvord | Early life history of herring larvae in contrasting feeding environments determined by otolith microstructure analysis[END_REF]2004 ;[START_REF] Johannessen | Differences in growth pattern between spring and autumn spawned herring (Clupea harengus L.) larvae[END_REF]Berg et al., 2017;[START_REF] Denis | Ontogenetic changes in the larval condition of Downs herring: use of a multi-index approach at an individual scale[END_REF][START_REF] Tonheim | Comparison of Otolith Microstructure of Herring Larvae and Sibling Adults Reared Under Identical Early Life Conditions[END_REF], and otolith shape used as a tool for stock identification [START_REF] Turan | Otolith shape and meristic analysis of herring (Clupea harengus) in the North-East Atlantic[END_REF][START_REF] Burke | Otolith shape analysis: its application for discriminating between stocks of Irish Sea and Celtic Sea herring (Clupea harengus) in the Irish Sea[END_REF][START_REF] Libungan | Otolith shape: a population marker for Atlantic herring Clupea harengus[END_REF]. Atlantic herring are of high commercial importance, with around 1,640,000 tons caught in 2016 (FAO FishStat data), but are vulnerable to the effects of climate change, particularly at the larval stage [START_REF] Hufnagl | Physiological individual-based modelling of larval Atlantic herring (Clupea harengus) foraging and growth: insights on climate-driven life-history scheduling[END_REF].

Otolith growth, and ultimately global otolith shape are well-known to result from the combination of genetic heterogeneity, ontogeny (physiological processes) and the influence of environmental (biotic and abiotic) factors [START_REF] Vignon | Environmental and genetic determinant of otolith shape revealed by a non-indigenous tropical fish[END_REF][START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF][START_REF] Hüssy | Trace Element Patterns in Otoliths: The Role of Biomineralization[END_REF]. Global climate change is projected to cause warming of the ocean surface by 3 to 5°C by 2100, and acidification of waters with a drop of 0.4 pH units in the worst-case scenario (IPCC 2019). The speed at which global change is occurring will not allow some species to adapt; in addition, early life history stages which

do not yet have fully functional physiological homeostatic mechanisms could be particularly affected [START_REF] Melzner | Physiological basis for high CO2 tolerance in marine ectothermic animals: pre-adaptation through lifestyle and ontogeny?[END_REF]. Some ontogenic processes, such as otolith morphogenesis, could thus be particularly impacted by the alteration of environmental conditions.

A previous study on herring larvae testing four temperature conditions showed that the size and the width of the otolith daily increment were both temperature-dependent [START_REF] Folkvord | Temperature-dependent otolith growth in Norwegian spring-spawning herring (Clupea harengus L.) larvae[END_REF]. Other studies on the effects of ocean acidification on fish otoliths have been mostly conducted on larval developmental stages. Several studies have shown that elevated levels of CO2 caused an increase in otolith size or modification of the otolith shape for multiple species [START_REF] Munday | Ocean acidification does not affect the early life history development of a tropical marine fish[END_REF]Bignami et al., 2013;[START_REF] Maneja | Effects of ocean acidification on the calcification of otoliths of larval Atlantic cod Gadus morhua[END_REF]Reveillac et al., 2015;Coll-Llado et al., 2021). Consequently, acidification can lead to decoupling of otolith and body growth for some species (Reveillac et al., 2015;[START_REF] Franke | Effect of ocean acidification on early life stages of Atlantic herring (Clupea harengus L.)[END_REF][START_REF] Frommel | Egg and early larval stages of Baltic cod, Gadus morhua, are robust to high levels of ocean acidification[END_REF][START_REF] Perry | Effect of ocean acidification on growth and otolith condition of juvenile scup, Stenotomus chrysops[END_REF].

Here, we aimed to investigate in Atlantic herring how otolith shape and size variability is affected by the warming and acidification conditions projected by 2100 under the worst-case IPCC scenario during the first early life stages. This otolith shape was investigated within individual (i.e. directional bilateral asymmetry) and within the population (i.e. between individuals).

Materials and methods

Experimental design

A experiment with temperature and CO2 concentration (through the measure of pCO2 as partial pressure of CO2) as combined factors was conducted, one close to present day conditions of temperature and pH in the winter in the English Channel (Amb; 11°C and pH 8.0, pCO2 ~560 µatm), and the other a global change scenario of ocean warming and acidification (IPCC 2019, "Changing ocean, marine ecosystems and dependent communities") (OWA; 14°C and pH 7.6, pCO2 ~1660 µatm) (Supplementary Table 1). There were 3 replicates (tanks) for each experimental condition (i.e. temperature-CO2). Most of the eggs hatched on the 9th day of incubation (starting point of the experiment: 1 dph) and larvae were kept until 69 days post hatch (dph). Three days after hatching (3 dph), all larvae were counted and distributed equally in six 38 L conical black tanks (1500 larvae by tank), constituting the replicates of the experiment. A continuous flow through system of 20 L h -1 was used (i.e. open circulation system). To allow the acclimation of the larvae, the temperature in the OWA tanks was progressively increased from 11 to 14°C over 48 h. Each OWA tank was supplied by a 200 L header tank, where the heated water was enriched with CO2 to achieve the target value of pH 7.6.

Temperature and CO2 were checked twice per day (pH meter 330i, WTW, Germany; Table S1).

Oxygen saturation (oximeter: WTW Oxi 340, Bioblock scientific) and salinity were measured once per week, along with total alkalinity (TA) of each tank following the protocol of Anderson and Robinson (1946) and Strickland and Parsons (1972). Oxygen concentration was always above 88%. pCO2 was calculated using the excel macro CO2sys (Lewis and Wallace 1998) with the constant from Mehrbach et al., (1973) refit by Dickson and Millero (1987). To prevent any food limitation, the daily food quantity was distributed four times during the day to maintain an ad libitum level (Strain 2002), ensuring that there were always prey in the tank during the day. To be sure to allow proper feeding through time we used an increasing range of living prey sizes from phytoplankton to 24 h old nauplii, before weaning with feed granules (as described in [START_REF] Joly | Maturation of the digestive system of Downs herring larvae (Clupea harengus, Linnaeus, 1758): identification of critical periods through ontogeny[END_REF]. Throughout the study, mortality was monitored daily, and was comparable between all ponds with no influence of environmental conditions (for more details, [START_REF] Joly | Maturation of the digestive system of Downs herring larvae (Clupea harengus, Linnaeus, 1758): identification of critical periods through ontogeny[END_REF].

Experimental animal came from wild Atlantic herring from the Downs stock of the southern North Sea and English Channel [START_REF] Joly | Maturation of the digestive system of Downs herring larvae (Clupea harengus, Linnaeus, 1758): identification of critical periods through ontogeny[END_REF]. Herring larvae were reared until they reached the last larval development stage respectively from hatching to 47 days post-hatching in the OWA treatment (14°C, 1660 µatm) and 60 days post-hatching in the Amb treatment (11°C, 560 µatm) to obtain the same Growing Degree Days (GDD) for two temperature conditions. To follow the otolith growth and shape variation through time, 351 Atlantic herring from 32 to ~662°C.day (GDD, Figure 1) were sampled five times over the course of the experiment. At 32°C.day, 15 individuals were sampled, and then 42 fish for each experimental condition were sampled at each of the next four samplings. To assess the potential effect of temperature on the growth and shape of otoliths, we used the GDD approach which quantifies the thermal opportunity for growth by aggregating temperatures relevant to growth [START_REF] Mcmaster | Growing degree-days: one equation, two interpretations[END_REF], and is thus more precise than the calendar time approach when describing growth (Neuheimer and Taggart, 2007;[START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF]. The first sampling occurred at three days post-hatching (GDD=32°C.day). The four other samplings were then carried out after different time periods depending on the rearing temperature (Figure 1).

Otolith shape analysis

After measuring the total length (TL± 0.1 cm) of fish, their sagittal otoliths (left and right) were extracted from the cranial cavity and cleaned. The outline of each otolith was digitized using an image analysis system. To compare the shapes of the left and right otoliths, mirror images of the right otoliths were used. Otolith shape was assessed by analyzing first the otolith area (Oarea, µm²) and Elliptic Fourier Descriptors (EFDs; e.g. [START_REF] Lestrel | Fourier Descriptors and their Applications in Biology[END_REF]. Oarea seems to be a better univariate descriptor than longest length. For each otolith, the first 99 elliptical Fourier harmonics were extracted and normalized with respect to the first harmonic so as to be invariant to otolith size, rotation and starting point of contour description [START_REF] Kuhl | Elliptic Fourier features of a closed contour[END_REF]. To determine the number of harmonics required to reconstruct the otolith outline, the cumulated Fourier power (F) was used. Only the first 6 harmonics were necessary to ensure reconstruction of each otolith shape with a precision of F=99.99% [START_REF] Lestrel | Fourier Descriptors and their Applications in Biology[END_REF] and were thus used for further analyses.

The resulting matrix containing EFDs (as columns) for each otolith (as rows) was subjected to Principal Components Analysis (PCA) [START_REF] Rohlf | A Comparison of Fourier Methods for the Description of Wing Shape in Mosquitoes (Diptera: Culicidae)[END_REF]. and the 3 first principal components (PCs) were selected as otolith shape descriptors or shape matrix according to the broken stick model [START_REF] Legendre | Numerical Ecology[END_REF], which, in this case, corresponded to a threshold of 3.4% of the total variance explained [START_REF] Borcard | Otolith shape analysis and daily increment validation during ontogeny of larval and juvenile European chub Squalius cephalus[END_REF]. In total, these 3 PCs explained 82.4% of the total variance in the EFDs. This procedure allowed us to decrease the number of variables used to describe otolith shape variability through EFDs while ensuring that the main sources of shape variation were kept, and to avoid co-linearity between shape descriptors [START_REF] Rohlf | A Comparison of Fourier Methods for the Description of Wing Shape in Mosquitoes (Diptera: Culicidae)[END_REF].

Statistical analyses

Each environmental condition was the combination of temperature and CO2 concentration values.

Otolith area and larvae size differences were analysed using a post-hoc Tukey-HSD test among several values of GDD. The relationship between larval total length (TL) and otolith area (Oarea) in response to the environmental conditions (temperature and CO2 concentration) was tested using Analysis of Covariance after verifying the normality of residuals. Using the Fourier harmonics to describe the otolith shape, pRDA were used to test the explanatory variables of interest (i.e., side, combination of temperature and CO2 concentration, GDD value) using total length to correct for fish size. This pRDA was combined with permutation tests on the selected PC matrix and the explanatory matrix consisted. To analyse the potential anatomical differences described by the directional bilateral asymmetry between left and right otolith shape, partial redundancy analysis (pRDA) was modelled on the selected principal components (PCs) matrix using otolith side (left/right) as the potentially influential variable and the individual as the conditioned variable. To test the potential effect of climate change, the pRDA was applied with the environmental explanatory matrix (Temperature and CO2 concentration). To visualise shape differences, average shapes were rebuilt based on EFDs averaged for each group of individuals. Directional asymmetry and environmental effect amplitude were then computed as the percentage of non-overlapping surface between the reconstructed otolith average shapes relative to the total area they covered after superposition for 2 groups left side/right side or 2 environmental conditions as used for other otolith shape studies [START_REF] Mahé | Directional bilateral asymmetry in otolith morphology may affect fish stock discrimination based on otolith shape analysis[END_REF][START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF][START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF].

Statistical analyses were performed using the following packages in the statistical environment R: 'Vegan' (Oksanen et al., 2013), 'sp' (Bivand et al., 2013) and 'rgeos' (Bivand et al., 2013).

Results

Otolith growth

There is a positive relationship between otolith and fish larvae sizes. Otolith and larvae sizes increased stepwise from 32 to 670°C.day (Figure 2). Over the course of the experiment, the shape was modified leading to the loss of the initial circular shape of the otolith. Otolith morphogenesis was positively correlated with GDD.. Herring larvae were long and thin at hatching, then progressively developed dorsal and caudal fins. Body height increased around 436°C.day, and the pelvic fins were the last fins to differentiate at around 662°C.day. Otolith area and larvae size differences were analysed among several values of GDD. Larvae size as otolith area grow during the larvae life (i.e. with GDD value; Figure 3A). Larvae size increased significantly between 32°C.day and 87°C.day, while the first changes in otolith area appeared only at 436°C.day. However, after this GDD level, otolith area (i.e. otolith 2-dimensional measure) increased faster than larval size measure (i.e. fish 1-dimensional measure). Neither larvae size nor otolith area were significantly different between the two environmental conditions (temperature/CO2) at any value of GDD, with the exception of larvae size at 436°C.day (P<0.05). The relationship between Oarea and TL was always significant (P<0.05), but the slope was significantly higher in the OWA than in the Amb treatment (slopes difference; P<0.05; Figure 3B). For the same fish length, the area of otolith was bigger for the 2100 scenario (14°C/CO2 1660 µatm) than under present day conditions (11°C/CO2 560 µatm) (Figure 3B); this difference increased with fish size. Our experimental study showed that this significant correlation between otolith and fish growth was still observed in present day environmental conditions as well as for the 2100 scenario.

Otolith shape

No significant difference in shape was observed between left and right otoliths (Supplementary Table 2). The average percentage of non-overlapping surface between the two sides never exceeded 1% (Figure 4). Similarly, the results showed no significant environmental effect on otolith shape in herring larvae, with the average percentage of non-overlapping surface ranging from 0.73% to 1.48% (Supplementary Table 2; Figure 4).

Discussion

The otolith shape is regulated by a complex combination of endogenous and exogenous factors, including both abiotic environmental parameters (such as temperature and CO2), and biotic parameters (such as food availability). The influence of these factors is dependent on the ontogenetic stage (i.e. the development stage of the individual). In addition, the otoliths can be different between right and left inner ears as a consequence of potential developmental lateralization (e.g. side effect).

Environmental factors, especially temperature, have a greater influence than genetic differences for Atlantic cod [START_REF] Cardinale | Effects of sex, stock, and environment on the shape of known-age Atlantic cod (Gadus morhua) otoliths[END_REF][START_REF] Hüssy | Otolith shape in juvenile cod (Gadus morhua): Ontogenetic and environmental effects[END_REF][START_REF] Irgens | Otolith structure as indicator of key life history events in Atlantic cod[END_REF]. For seabass (Dicentrarchus labrax), increased temperature speeds up otolith morphogenesis and modifies the developmental pattern of the otolith shape [START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF]. Ocean acidification can also alter otolith shape [START_REF] Holmberg | Ocean acidification alters morphology of all otolith types in Clark's anemonefish (Amphiprion larkii)[END_REF]. For several species, individuals exposed to high CO2 had a larger otolith area and maximum length compared with controls; the increases were larger than could be explained by an increase in CaCO3 precipitation in the otoliths driven by the modification of the pH regulation in the endolymph [START_REF] Checkley | Elevated CO2 Enhances Otolith Growth in Young Fish[END_REF][START_REF] Munday | Ocean acidification does not affect the early life history development of a tropical marine fish[END_REF][START_REF] Réveillac | Ocean acidification reshapes the otolith-body allometry of growth in juvenile sea bream[END_REF]Coll-Llado et al., 2018). Changes in extracellular concentrations of carbonate and bicarbonate caused by acid-base regulation in a high CO2 environment could increase the precipitation of CaCO3 in the otolith [START_REF] Payan | Relationship between otolith and somatic growth: consequence of starvation on acid-base balance in plasma and endolymph in the rainbow trout Oncorhynchus mykiss[END_REF]. In studies on other species including herring, high CO2 had no effect on the larval sagittal otolith [START_REF] Franke | Effect of ocean acidification on early life stages of Atlantic herring (Clupea harengus L.)[END_REF][START_REF] Munday | Ocean acidification does not affect the early life history development of a tropical marine fish[END_REF][START_REF] Frommel | Egg and early larval stages of Baltic cod, Gadus morhua, are robust to high levels of ocean acidification[END_REF]Perry et al., 2015). Our experimental study showed that there is the relationship between otolith area and fish length according to the environmental conditions and validated the results observed in situ [START_REF] Folkvord | Early life history of herring larvae in contrasting feeding environments determined by otolith microstructure analysis[END_REF][START_REF] Johannessen | Differences in growth pattern between spring and autumn spawned herring (Clupea harengus L.) larvae[END_REF]Berg et al., 2017;[START_REF] Denis | Ontogenetic changes in the larval condition of Downs herring: use of a multi-index approach at an individual scale[END_REF][START_REF] Tonheim | Comparison of Otolith Microstructure of Herring Larvae and Sibling Adults Reared Under Identical Early Life Conditions[END_REF]. A previous study on herring larvae testing four temperature conditions (4°C, 12°C and two others shifted twice with 4/8/4°C and 12/8/12°C) showed that the size and the width of the otolith daily increment were both temperature-dependent [START_REF] Folkvord | Temperature-dependent otolith growth in Norwegian spring-spawning herring (Clupea harengus L.) larvae[END_REF]. The otolith area per fish size in our study increased with higher temperature and CO2 (i.e. lower pH). It should be noted that the feeding protocol chosen for this study provides ad-libitum feed, in order to avoid under-feeding situations that could have interfered with the otolith study. Nevertheless, it cannot be excluded that larvae under the 2100 scenario ingest more food than control larvae, which could also accelerate the increase of the otolith area in the 2100 scenario group. While growth mechanisms of otoliths and their morphogenesis during the early life stages of fish are poorly understood, shifts in otolith shape are linked to physiological modifications due to environmental disturbance [START_REF] Geffen | Methods of validating daily increment deposition in otoliths of larval fish, in Age and growth of fish[END_REF][START_REF] Vignon | Short-term stress for long-lasting otolith morphology -Brief embryological stress disturbance can reorient otolith ontogenetic trajectory[END_REF]. Early life stages are more vulnerable to environmental challenges because they have higher surface area to volume ratios and have not yet fully developed the homeostatic regulatory mechanisms which are found in adult fish [START_REF] Hurst | Effects of ocean acidification on hatch size and larval growth of walleye pollock (Theragra chalcogramma)[END_REF]. For Downs herring, a shift affecting larval condition has previously been observed in situ for individuals at 13 mm, and was driven by environmental conditions (temperature and prey concentration) (Dennis et al., 2017). Our experimental study on Downs herring covered this length range but the otoliths did not show any ontogenetic differences. Several previous studies showed that the increase in otolith area by fish size could mainly result from water acidification [START_REF] Checkley | Elevated CO2 Enhances Otolith Growth in Young Fish[END_REF][START_REF] Munday | Ocean acidification does not affect the early life history development of a tropical marine fish[END_REF][START_REF] Réveillac | Ocean acidification reshapes the otolith-body allometry of growth in juvenile sea bream[END_REF]Coll-Llado et al., 2018). Moreover, the elevated seawater CO2 can cause directional asymmetry [START_REF] Holmberg | Ocean acidification alters morphology of all otolith types in Clark's anemonefish (Amphiprion larkii)[END_REF]. For herring, the environmental factors we studied (temperature/CO2) did not affect the otolith shape between sides at the observed larval stage.

Consequently, it is likely that the level of response to CO2 increase might be a species-specific phenomenon. This capacity to maintain otolith shape in response to environmental changes is probably due to efficient intracellular ionic-regulation mechanisms in this species [START_REF] Ishimatsu | Fishes in high-CO2, acidified oceans[END_REF][START_REF] Melzner | Physiological basis for high CO2 tolerance in marine ectothermic animals: pre-adaptation through lifestyle and ontogeny?[END_REF]. This result may suggest that herring larvae are well equipped to cope with the environmental changes projected for 2100, as long as their energy and nutritional needs are well covered, which would not necessarily be the case in the natural environment. The faster growth of the larvae under the 2100 scenario may indeed lead to a greater prey requirement, and it cannot be ruled out that the increase in the area of the otoliths may also have affected their hearing ability and behaviour. Nevertheless, it should be noted that this species certainly has a good intracellular ionic regulation because it is confronted with different environments [START_REF] Ishimatsu | Fishes in high-CO2, acidified oceans[END_REF][START_REF] Melzner | Physiological basis for high CO2 tolerance in marine ectothermic animals: pre-adaptation through lifestyle and ontogeny?[END_REF]. Indeed, herring larvae typically hatch in littoral regions where pH and temperature can fluctuate because of freshwater inflow, so they may often experience such fluctuations and therefore be quite resistant to them. For more extreme values, however, the homeostatic mechanisms might not be sufficient to compensate for environmental stress and this could induce changes in otolith morphological development (Coll-Llado et al., 2018).

Ontogeny and environment are the factors that control most of the otolith development during the early stages of life. The otolith shape evolves from the basic round form to elongated shape due to faster growth along the anterior-posterior axis than along the dorsal-ventral axis [START_REF] Galley | Combined methods of otolith shape analysis improve identification of spawning areas of Atlantic cod[END_REF][START_REF] Mille | Sagittal otolith morphogenesis asymmetry in marine fishes[END_REF]Bounket et al., 2019;[START_REF] Mahé | Do environmental conditions (temperature and food composition) affect otolith shape during fish early-juvenile phase? An experimental approach applied to European Seabass (Dicentrarchus labrax)[END_REF]. Our study showed that the otolith shape of herring grew in this way during the larval period. At the adult stage, significant directional asymmetry has been observed in herring (Bird et al., 1986), while no lateralization has been observed at the larval stage. This bilateral effect increases over the life of the fish when considering a different trajectory of otolith morphogenesis between left and right sides. However, the asymmetry between left and right otoliths especially the fluctuating asymmetry may be underestimated in experiments due to the small number of tested individuals which are experiencing stressful conditions and present experimental conditions that do not require significant development of acoustic functions as in the natural environment [START_REF] Grønkjaer | Fluctuating asymmetry and nutritional condition of Baltic cod (Gadus morhua) larvae[END_REF]Diaz-Gil et al., 2015). Consequently, no lateralization has been observed in the experimental condition for the herring larval stage does not mean that this result will be the same in the natural environment. Although otolith shape is influenced by both abiotic and biotic environmental parameters and depends on individuals' genotype, environmental effects can also be perceived more with ontogeny, consequently, this developmental lateralization could be a phenotypically plastic response to environmental drivers rather than the consequence of the individual genotype, as previously suggested in Boops boops [START_REF] Mahé | Directional bilateral asymmetry in otolith morphology may affect fish stock discrimination based on otolith shape analysis[END_REF]. Sagittal otolith size and shape for each species could therefore be adaptive traits to different habitats and ecological niches [START_REF] Lychakov | Otolith regularities[END_REF][START_REF] Lombarte | Ecomorphological trends and phylogenetic inertia of otolith sagittae in Nototheniidae[END_REF].

Conclusions

The growth and shape of the otoliths studied here did not appear to be affected by the environmental conditions predicted for 2100. However, this experimental approach should be confirmed in situ.

Only the relationship between otolith and fish sizes seems to be environment-dependent. This information is important to understand the otolith morphogenesis and consequently when the otolith shape was used as a tool in fisheries science. In the future, to complete this approach, a crossed-effect experiment (four different settings: control, temperature increase, acidification, temperature increase and acidification) could be applied to quantify the effect of each environmental factor. In the same way, it could be important to understand the link between the environment modifications and the food consumption by fish to separate the direct and indirect effects of the climate change on the otolith and fish growth.

Figure 1 :

 1 Figure 1: Main steps of the experimental design for herring with five samplings by the level of GDD (Growing Degree Days) and by the number of dph (Days Post Hatching; grey value for 11°C; black value for 14°C). Each condition of temperature and CO2 concentration had three replicates. To test the effect of global change, present day conditions (11°C and pH 8.0, pCO2 560 µatm) versus the ocean warming and acidification conditions predicted for 2100 (14°C and pH 7.6, pCO2 1660 µatm) were used.

Figure 2 :

 2 Figure 2: Growth of otolith shape and larval fish by GDD value (°C.day).

Figure 3 :

 3 Figure 3: (A) Box-plot of otolith area (Oarea, µm²) and total length of fish (TL, cm) values by GDD value (°C.day) and environmental conditions (red for 14°C and 1660 µatm, and blue for 11°C and

Figure 4 :

 4 Figure 4: Percentage of non-overlapping surface between left and right otolith shape and for two environmental conditions by GDD value (°C.day) (arrows identify the main areas of difference between both otoliths).

Acknowledgments

This work was supported by the Institut Français de Recherche et d'Exploitation de la Mer and the Alfred Wegener Institute (AWI-MARUM-IFREMER partnership program, "CoCktAIL" Climate ChAnge effect on fIsh Larvae, doctoral support to L. Joly, 2018Joly, -2022)).

The authors thank technicians, researchers and students who helped to carry this experiment, Valérie

Berg, F., Husebø, A., Aanestad Godiksen, J., Slotte, A., Folkvord. A., 2017. Spawning time of Atlantic herring (Clupea harengus) populations within a restricted area reflects their otolith growth at the larval stage, Fish. Res. 194: 68-75, doi:10.1016/j.fishres.2017.05.009. Bignami, S., Sponaugle, S., Cowen, R., 2013. Response to ocean acidification in larvae of a large tropical marine fish, Rachycentron canadum. Glob. Chang. Biol. 19: 996-1006, doi:10.1111/gcb.12133. Bivand, R.S., Pebesma, E., Gomez-Rubio., V., 2013. Applied spatial data analysis with R, Second edition. New York: Springer.