Brownian paths in an alcove and the Littelmann path model Manon Defosseux To cite this version: Manon Defosseux. Brownian paths in an alcove and the Littelmann path model. 2021. hal- 03183785v3 HAL Id: hal-03183785 https://hal.archives-ouvertes.fr/hal-03183785v3 Preprint submitted on 24 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BROWNIAN PATHS IN AN ALCOVE AND THE LITTELMANN PATH MODEL WHAT WE KNOW WHAT WE DO NOT KNOW WHAT WE HOPE MANON DEFOSSEUX Abstract. We present some results connecting Littelmann paths and Brow- nian paths in the framework of affine Kac{Moody algebras. We prove in par- ticular that the string coordinates associated to a specific sequence of random Littelmann paths converge towards their analogs for Brownian paths. At the end we explain why we hope that our results will be the first steps on a way which could hopefully lead to a Pitman type theorem for a Brownian motion in an alcove associated to an affine Weyl group. 1. Introduction A Pitman's theorem states that if fbt; t ≥ 0g is a one dimensional Brownian motion, then fbt − 2 inf bs; t ≥ 0g 0≤s≤t is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob's sense to remain forever positive [10]. Philippe Biane, Philippe Bougerol and Neil O'Connell have proved in [2] that a similar theorem exists in which the real Brow- nian motion is replaced by a Brownian motion on a finite dimensional real vector space. A finite Coxeter group acts on this space and the positive Brownian motion is replaced by a conditioned Brownian motion with values in a fundamental domain for the action of this group. In that case, the second process is obtained by applying to the first one Pitman transformations in successive directions given by a reduced decomposition of the longest word in the Coxeter group. Paper [3] gives a similar representation theorem for a space-time real Brownian motion f(t; bt): t ≥ 0g conditioned to remain in the cone 0 C = f(t; x) 2 R × R : 0 ≤ x ≤ tg: 0 Actually C is a fundamental domain for the action on R+ × R of an affine Coxeter 1 group of type A1. This affine Coxeter group, which is not a finite group, is generated by two reflections and it could be natural to think that one could obtain a space-time Brownian motion conditioned to remain in C0 applying successively and infinitely to a space-time Brownian motion two Pitman transformations corresponding to these two reflections. We have proved with Philippe Bougerol in [3] that this is not the case. Actually a L´evytype transformation has to be added at the end of the successive Pitman transformations if we want to get a Pitman's representation theorem in this case. It is now natural to ask if such a theorem exists for the other affine Coxeter 1 groups. We will focus on Coxeter groups of type An, with n ≥ 1. Such a Coxeter group is the Weyl group of a type A extended affine Kac{Moody algebra. The 1 2 MANON DEFOSSEUX presence of the L´evytransformations in the case when n = 1 makes the higher rank statement quite open. When n = 1, the proof of the Pitman type theorem in [3] rests on an approxi- mation of the affine Coxeter group by a sequence of dihedral groups for which the results of [2] are applicable. Such an approximation does not exist for the higher ranks. Nevertheless, another approximation exists of the Brownian model that we are interested in. It involves the Littelmann path model. The Littelmann's model is a combinatorial model which allows to describe weight properties of some particular integrable representations of Kac{Moody algebras. Philippe Biane, Philippe Bougerol and Neil O'Connell pointed out in [1] the fun- damental fact that the Pitman transformations are intimately related to the Lit- telmann path model. In the case of an affine Lie algebra, this model allows to construct random paths which approximates the Brownian model that we are in- terested in. The knowledge of the Littelmann paths properties gives then a way to get a better understanding of those of the Brownian paths. Unfortunately this approach didn't lead for the moment to a Pitman type the- orem in an affine framework. Nevertheless, we have obtained several non trivial results that we present here. In particular, we prove the convergence of the string coordinates arising in the framework of the Littelmann path model towards their analogs defined for the Brownian paths. Besides we use the Littelmann path ap- proach to try to guess which correction could be needed if a Pitman type theorem existed in this case. We propose here a conjectural correction, with encouraging simulations. Finally, notice that the space component of a space-time Brownian motion con- ditioned to remain in an affine Weyl chamber is equal, up to a time inversion, to a Brownian motion conditioned to remain in an alcove, so that our suggestion provides also a suggestion for a Pitman's theorem for this last conditioned process. These notes are organized as follows. In section 2 we recall the necessary back- ground about affine Lie algebras and their representations. The Littelmann path model in this context is explained briefly in section 3 where we recall in particular the definition of the string coordinates. In section 4 we define two sequences of random Littelmann paths. The first one converges towards a space-time Brownian motion in the dual of a Cartan subalgebra of an affine Lie algebra. The second one converges towards a space-time Brownian motion in an affine Weyl chamber. This last process is defined in section 5. The statements of the two convergences are given in section 6, where we also prove the convergence of the string coordinates associated to a sequence of random Littelmann paths towards their analogs for Brownian paths. Section 7 is devoted to explain what is missing in the perspective of a Pitman type theorem. Finally in section 8 we use the description of the highest weight Littelmann modules given in [9] to suggest transformations which could play the role of L´evytransformations in the case when n is greater than one. 2. Basic definitions This section is based on [7]. In order to make the paper as easy as possible to read, we consider only the case of an extended affine Lie algebra of type A. For _ 1 this we consider a realization (bh; Πb; Πb ) of a Cartan matrix of type An for n ≥ 1. That is to say ∗ _ _ _ Πb = fα0; : : : ; αng ⊂ bh and Πb = fα0 ; : : : ; αn g ⊂ bh BROWNIAN PATHS IN AN ALCOVE 3 with 8 2 if i = j _ < hαi; αj i = −1 if ji − jj 2 f1; ng when n ≥ 2 : −2 if ji − jj = 1 when n = 1, and dim bh = n + 2, where h·; ·i is the canonical pairing. We consider an element d 2 bh such that hαi; di = δi0; ∗ for i 2 f0; : : : ; ng and define Λ0 2 bh by _ hΛ0; di = 0 and hΛ0; αi i = δi0; for i 2 f0; : : : ; ng. We consider the Weyl group Wc which is the subgroup of GL(bh∗) generated by the simple reflexions sαi , i 2 f0; : : : ; ng, defined by _ ∗ sαi (x) = x − hx; αi iαi; x 2 bh ; i 2 f0; : : : ; ng, and equip bh∗ with a non degenerate Wc-invariant bilinear form (·|·) defined by 8 (αijαj) = 2 if i = j > < (αijαj) = −1 if ji − jj 2 f1; ng when n ≥ 2 (αijαj) = −2 if ji − jj = 1 when n = 1, > : (αijαj) = 0 otherwise, (Λ0jΛ0) = 0 and (αijΛ0) = δi0, i 2 f0; : : : ; ng. We consider as usual the set of integral weights ∗ _ Pb = fλ 2 bh : hλ, αi i 2 Z; i 2 f0; : : : ; ngg; and the set of dominant integral weights + ∗ _ Pb = fλ 2 bh : hλ, αi i 2 N; i 2 f0; : : : ; ngg: For λ 2 Pb+ we denote by V (λ) the irreducible highest weight module of highest 1 weight λ of an affine Lie algebra of type An with bh as a Cartan subalgebra and Πb as a set of simple roots. We consider the formal character X λ β chλ = mβe ; β2Pb λ ∗ where mβ is the multiplicity of the weight β in V (λ). If νb 2 bh satisfies (νbjα0) > 0 then the series X λ (βjνb) mβe ; β2Pb converges and we denote by chλ(νb) its limit. 3. Littelmann path model From now on we work on the real vector space n M h∗ = Λ ⊕ α : bR R 0 R i i=0 4 MANON DEFOSSEUX In this section, we recall what we need about the Littelmann path model (see mainly [9] for more details, and also [8]). Fix T ≥ 0. A path π is a piecewise linear function π : [0;T ] ! h∗ such that π(0) = 0. We consider the cone generated by P bR b+ C = fλ 2 h∗ : hλ, α_i ≥ 0; i 2 f0; : : : ; ngg: bR i A path π is called dominant if π(t) 2 C for all t 2 [0;T ].