Conditioned random walks from Kac-Moody root systems
aa r X i v : . [ m a t h . C O ] D ec CONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS
C´EDRIC LECOUVEY, EMMANUEL LESIGNE AND MARC PEIGN´E
Abstract.
Random paths are time continuous interpolations of random walks. By using Lit-telmann path model, we associate to each irreducible highest weight module of a Kac Moodyalgebra g a random path W . Under suitable hypotheses, we make explicit the probability of theevent E : “ W never exits the Weyl chamber of g ”. We then give the law of the random walkdefined by W conditioned by the event E and prove this law can be recovered by applying to W a path transform of Pitman type. This generalizes the main results of [15] and [10] to KacMoody root systems and arbitrary highest weight modules. Our approach here is new and morealgebraic that in [15] and [10]. We indeed fully exploit the symmetry of our construction underthe action of the Weyl group of g which permits to avoid delicate generalizations of the resultsof [10] on renewal theory. Introduction
The purpose of the paper is to study conditionings of random walks using algebraic and com-binatorial tools coming from representation theory of Lie algebras and their infinite-dimensionalgeneralizations (Kac-Moody algebras). We extend in particular some results previously obtainedin [15], [16], [1], [10] and [11] to random paths in the weight lattice of any Kac-Moody algebra g . To do this, we consider a fixed g -module V in the category O int (a convenient generalizationof the category of Lie algebras finite dimensional representations). It decomposes as the directsum of its weight spaces, each such space being parametrized by a vector of the weight latticeof g . The transitions of the random walk associated to V are then the weights of V .The prototype of the results we obtain appears in the seminal paper [15] by O’Connell whereit is shown that the law of the one-way simple random walk W in Z n conditioned to stay inthe cone C = { ( x , . . . , x n ) ∈ Z n | x ≥ · · · ≥ x n ≥ } and with drift in the interior ˚ C of C ,is the same as the law of a Markov chain H obtained by applying to W a generalization ofthe Pitman transform. This transform is defined via an insertion procedure on semistandardtableaux classically used in representation theory of sl n ( C ). The transition matrix of H canthen be expressed in terms of the Weyl characters (Schur functions) of the irreducible sl n ( C )-modules. Here the transitions of the random walk W are the vectors of the standard basis of Z n which correspond to the weights of the defining representation C n of sl n ( C ). In addition to theinsertion procedure on tableaux and some classical facts about representation theory of sl n ( C ),the main ingredients of O’Connell’s result are a Theorem of Doob on Martin boundary togetherwith the asymptotic behavior of tensor product multiplicities associated to the decompositionsof V ⊗ ℓ in its irreducible components (which in this case are counted by standard skew tableaux).We consider in [10] more general random walks W with transitions the weights of a finite-dimensional irreducible g -module V where g is a Lie algebra. The law of W is constructed so thatthe probabilities of the paths only depend of their lengths and their ends. We then show that theprocess H obtained by applying to W a generalization of the Pitman transform introduced in [1]is a Markov chain. When V is a minuscule representation (i.e. when the weights of V belong tothe same orbit under the action of the Weyl group of g ) and W has drift in the interior ˚ C of the Date : December 20, 2013. cone C of dominant weights, we prove that H has the same law as W conditioned to never exit C . Similarly to the result of O’Connell, this common law can be expressed in terms of the Weylcharacters of the simple g -modules. Nevertheless the methods differ from [15] notably becausethere was no previously known asymptotic behavior for the relevant tensor multiplicities in themore general cases we study. In fact, we proceed by establishing a quotient renewal theorem forgeneral random walks conditioned to stay in a cone. When W is not defined from a minusculerepresentation, we also show that the law of W conditioned to never exit C cannot coincide withthat of H .In [11], we use the renewal theorem of [10] and insertion procedures on tableaux appearingin the representation theory of the Lie superalgebras gl ( m, n ) and q ( n ) to extend the results of[15] to one way simple random walks conditioned to never exit cones C ′ for examples of cones C ′ different from C .In view of the results of [10], it is natural to ask whether the Markov chain H is related toa suitable conditioning of W in the non minuscule case. Also what can be said about the lawof W conditioned to never exit C ? In the sequel, we will answer both questions (partially forthe second) not only for random walks defined from representations of Lie algebras but, moregenerally, for similar random walks with transitions the weights of a highest weight module V ( κ )associated to a Kac-Moody algebra g of rank n .By using Littelmann path model [13], one can associate to V ( κ ) a countable set of piecewisecontinuous linear paths B ( π κ ) in the weight lattice P ⊂ R n of g . These paths (called elementaryin the sequel) are regarded as functions π : [0 , → R n such that π (0) = 0 and π (1) ∈ P . Theweights of V ( κ ) are then the elements π (1) , π ∈ B ( π κ ) . The set B ( π κ ) has the structure of acolored and oriented graph isomorphic to the crystal graph of V ( κ ) as defined by Kashiwara.We use the crystal graph structure on B ( π κ ) to endow it as in [10] with a probability density p .This yields a random variable X defined on B ( π κ ) with probability distribution p . Let ( X ℓ ) ℓ ≥ be a sequence of i.i.d. random variables with the same law as X . We then define a continuousrandom path W such that for any t ≥ W ( t ) = X (1) + · · · + X ℓ − (1) + X ℓ ( ℓ − t ) for any t ∈ [ ℓ − , ℓ ]. The sequence W = ( W ℓ ) ℓ ≥ defined by W ℓ = W ( ℓ ) is then a random walk withtransitions the weights of V ( κ ) as considered in [10]. The main result of the paper is that, when W has drift in ˚ C (i.e. in the interior of the Weyl chamber of g ), the law of its conditioning bythe event E = ( W ( t ) ∈ C for any t ≥
0) can be simply expressed in terms of the Weyl-Kaccharacters. So the results of [10] remain true for a conditioning holding on the whole continuoustrajectory (not only on its discrete version at integer time). We also prove that the conditionedlaw so obtained coincides with the law of the image of W by the generalized Pitman transform.When g is finite-dimensional and κ is minuscule we recover in particular the main results of[15] and [10]. On the representation theory side, our results also lead to asymptotic behavior oftensor product multiplicities of Kac-Moody highest weight modules.Nevertheless our approach differ from that of [10] since we do not use any renewal theorem.Our strategy is more algebraic: we exploit the symmetry of the representations with respect tothe Weyl group W of g and study simultaneously a family of random paths W w indexed by theelements w ∈ W . In particular our proofs are independent of the results of [15] and [10].The paper is organized as follows. In Section 2, we introduce the notions of random walkand random path used in the paper. Section 3 recalls the necessary background on Kac-Moodyalgebras and their representations and summarize some important results on Littelmann’s pathmodel. The random path W and the random walk W associated to V ( κ ) are introduced inSection 4 together with the generalized Pitman transform and the Markov chain H . In Section 5, ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 3 we use a process of symmetrization to define the random paths W w , w ∈ W from W = W . Thisallows us to give an explicit expression of the harmonic function µ P µ ( W ( t ) ∈ C for any t ≥ W starting at µ remainsin C . We also extend it to the case of random walks defined from non irreducible representationsof simple Lie algebras. Finally Section 7 is devoted to additional results: we give asymptoticbehavior of tensor power multiplicities and also compare the probabilities P µ ( W ( t ) ∈ C for any t ≥
0) and P µ ( W ℓ ∈ C for any ℓ ≥ MSC classification:
Random paths
Background on Markov chains.
Consider a probability space (Ω , F , P ) and a countableset M . A sequence Y = ( Y ℓ ) ℓ ≥ of random variables defined on Ω with values in M is a Markovchain when P ( Y ℓ +1 = µ ℓ +1 | Y ℓ = µ ℓ , . . . , Y = µ ) = P ( Y ℓ +1 = µ ℓ +1 | Y ℓ = µ ℓ )for any any ℓ ≥ µ , . . . , µ ℓ , µ ℓ +1 ∈ M . The Markov chains considered in the sequelwill also be assumed time homogeneous, that is P ( Y ℓ +1 = λ | Y ℓ = µ ) = P ( Y ℓ = λ | Y ℓ − = µ )for any ℓ ≥ µ, λ ∈ M . For all µ, λ in M , the transition probability from µ to λ is thendefined by Π( µ, λ ) = P ( Y ℓ +1 = λ | Y ℓ = µ )and we refer to Π as the transition matrix of the Markov chain Y . The distribution of Y iscalled the initial distribution of the chain Y .In the following, we will assume that M is a subset of the euclidean space R n for some n ≥ Y = ( Y ℓ ) ℓ ≥ has full support, i.e. P ( Y = λ ) > λ ∈ M . In [10], we have considered a nonempty set C ⊂ M and anevent E ∈ T such that P ( E | Y = λ ) > λ ∈ C and P ( E | Y = λ ) = 0 for all λ / ∈ C ;this implied that P ( E ) >
0, we could thus define the conditional probability Q relative to thisevent: Q ( · ) := P ( ·| E ). For example, we considered the event E := ( Y ℓ ∈ C for any ℓ ≥ A continuous time Markov process Y = ( Y ( t )) t ≥ on (Ω , F , P ) with values in R n is a family ofrandom variables defined on (Ω , F , P ) such that, for any integer k ≥
1, any 0 ≤ t < · · · < t k +1 and any Borel subsets B , · · · , B k +1 of R n , one gets P ( Y ( t k +1 ) ∈ B k +1 | Y ( t ) ∈ B , Y ( t ) ∈ B , · · · , Y ( t k ) ∈ B k ) = P ( Y ( t k +1 ) ∈ B k +1 | Y ( t k ) ∈ B k ) . This is the Markov property, that we will use very often. In the following, we shall need a moregeneral version of this property which is a consequence of the above. One can indeed show thatfor any T ≥ A ⊂ ( R n ) ⊗ [0 ,T ] , B ⊂ R n and C ⊂ ( R n ) ⊗ [ T, + ∞ [ , one gets P (( Y ( t )) t ≥ T ∈ C | ( Y ( t )) ≤ t ≤ T ∈ A, Y ( T ) ∈ B ) = P (( Y ( t )) t ≥ T ∈ C | Y ( T ) ∈ B ) . In the sequel, we will assume the two following conditions.(1) For any integer ℓ ≥
0, one gets(1) Y ℓ := Y ( ℓ ) ∈ M P − almost surelyIt readily follows that the sequence Y = ( Y ℓ ) ℓ ≥ is a M -valued Markov chain. C´EDRIC LECOUVEY, EMMANUEL LESIGNE AND MARC PEIGN´E (2) For any 0 ≤ s ≤ t and any Borel subsets A, B ∈ R n (2) P ( Y ( t + 1) ∈ B | Y ( s + 1) ∈ A ) = P ( Y ( t ) ∈ B | Y ( s ) ∈ A ) . Combining this condition with the Markov property, one checks that for any T ≥ x ∈ R n , the conditional distribution of the process ( Y ( t + 1)) t ≥ T with respect to theevent ( Y ( T + 1) = x ) is equal to the one of ( Y ( t )) t ≥ T with respect to ( Y ( T ) = x ).In the following, we will assume that the initial distribution of the Markov process ( Y ( t )) t ≥ has full support, i.e. P ( Y (0) = λ ) > λ ∈ M . We will also consider a nonemptyset C ⊂ R n and will assume that the probability of the event E := ( Y ( t ) ∈ C for any t ≥ Q relative to E is thus well defined. The followingproposition can be deduced from our hypotheses and the Markov property of Y . We postponeits proof to the appendix. Proposition 2.1.
Let ( Y ( t )) t ≥ be a continuous time Markov process with values in R n satisfyingconditions (1) and (2) and C ⊂ R n such that the event E := ( Y ( t ) ∈ C for any t ≥ has positiveprobability measure. Then, under the probability Q ( · ) = P ( ·| E ) , the sequence ( Y ℓ ) ℓ ≥ is still aMarkov chain with values in C ∩ M and transition probabilities given by (3) ∀ µ, λ ∈ C ∩ M Q ( Y ℓ +1 = λ | Y ℓ = µ ) = Π E ( µ, λ ) P ( E | Y = λ ) P ( E | Y = µ ) where Π E ( µ, λ ) = P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1] | Y ℓ = µ ) . We will denote by Y E thisMarkov chain To simplify the notations we will denote by C the set C ∩ M as soon as we will consider theMarkov chain ( Y ℓ ) ℓ ≥ and Π E = (Π( µ, λ )) µ,λ ∈C the “restriction” of the transition matrix Π tothe event E whereΠ E ( µ, λ ) = P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1] | Y ℓ = µ ) . So Π E ( µ, λ ) gives the probability of the transition from µ to λ when Y ( t ) remains in C for t ∈ [ ℓ, ℓ + 1].A substochastic matrix on the countable set M is a map Π : M × M → [0 ,
1] such that P y ∈ M Π( x, y ) ≤ x ∈ M. If Π , Π ′ are substochastic matrices on M , we define theirproduct Π × Π ′ as the substochastic matrix given by the ordinary product of matrices:Π × Π ′ ( x, y ) = X z ∈ M Π( x, z )Π ′ ( z, y ) . A function h : M → R is harmonic for the substochastic transition matrix Π when we have P y ∈ M Π( x, y ) h ( y ) = h ( x ) for any x ∈ M . Consider a (strictly) positive harmonic function h .We can then define the Doob transform of Π by h (also called the h -transform of Π) settingΠ h ( x, y ) = h ( y ) h ( x ) Π( x, y ) . We then have P y ∈ M Π h ( x, y ) = 1 for any x ∈ M. Thus Π h is stochastic and can be interpretedas the transition matrix for a certain Markov chain.An example is given in formula (3): the state space is now C , the substochastic matrix isΠ E and the harmonic function is h E ( µ ) := P ( E | Y = µ ); the transition matrix Π Eh E is thetransition matrix of the Markov chain Y E . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 5
Elementary random paths.
Consider a Z -lattice P with finite rank d . Set P R = P ⊗ Z R so that P can be regarded as a Z -lattice of rank d in R d . An elementary path is a piecewisecontinuous linear map π : [0 , → P R such that π (0) = 0 and π (1) ∈ P . Two paths π and π are considered as identical if there exists a piecewise, surjective continuous and nondecreasingmap u : [0 , → [0 ,
1] such that π = π ◦ u .The set F of continuous functions from [0 ,
1] to P R is equipped with the norm k·k ∞ of uniformconvergence : for any π ∈ F , on has k π k := sup t ∈ [0 , k π ( t ) k where k·k denotes the euclideannorm on R d . Let B be a countable set of paths and fix a probability distribution p = ( p π ) π ∈ B on B such that p π > π ∈ B . Let X be a random variable defined on a probability space(Ω , F , P ) and with distribution p (in other words P ( X = π ) = p π for any π ∈ B ) . The variable X admits a moment of order 1 (namely E ( k X k ) < + ∞ ) when the series of functions P π p π k π k converges on [0 , m := E ( X ) = X π ∈ B p π π. The concatenation π ∗ π of two elementary paths π and π is defined by π ∗ π ( t ) = (cid:26) π (2 t ) for t ∈ [0 , ] ,π (1) + π (2 t −
1) for t ∈ [ , . In the sequel, C is a closed convex cone in P R with interior ˚ C and we set P + = C ∩ P .2.3. Random paths.
Let B be a set of elementary paths and ( X ℓ ) ℓ ≥ a sequence of i.i.d.random variables with law X where X is the random variable with values in B introduced in2.2. We define the random process W as follows: for any ℓ ∈ Z > and t ∈ [ ℓ, ℓ + 1] W ( t ) := X (1) + X (1) + · · · + X ℓ − (1) + X ℓ ( t − ℓ ) . The sequence of random variables W = ( W ℓ ) ℓ ≥ := ( W ( ℓ )) ℓ ≥ is a random walk with set ofincrements I := { π (1) | π ∈ B } .For any ℓ ≥
1, let ψ ℓ be the map defined by ∀ µ ∈ C ψ ℓ ( µ ) = P µ ( W ( t ) ∈ C for any t ∈ [0 , ℓ ])so that ψ ℓ ( µ ) is the probability that W starting at µ remains in C for any t ∈ [0 , ℓ ]. As ℓ → + ∞ ,the sequence of functions ( ψ ℓ ) ℓ ≥ converges to the function ψ defined by ∀ µ ∈ C ψ ( µ ) = P µ ( W ( t ) ∈ C for any t ≥ . Proposition 2.2.
Assume E ( k X k ) < + ∞ and m (1) / ∈ ˚ C . Then for any µ ∈ C , we have ψ ( µ ) = 0 .Proof. Observe that ψ ( µ ) = P µ ( W ( t ) ∈ C for any t ≥ ≤ P µ ( W ℓ ∈ C for any ℓ ≥ W (see [10]for more details), we have P µ ( W ℓ ∈ C for any ℓ ≥
0) = 0 when m (1) / ∈ ˚ C . Thus ψ ( µ ) = 0 when m (1) / ∈ ˚ C . (cid:3) Remark:
The hypothesis E ( k X k ) < + ∞ suffices in fact to prove also that ψ ( µ ) > m (1) ∈ ˚ C and there exists at least π ∈ B such that Im π ⊂ C . In the context of the paper, thiswill readily follows from Theorem 6.2 so we do not pursue in this direction. C´EDRIC LECOUVEY, EMMANUEL LESIGNE AND MARC PEIGN´E Representations of symmetrizable Kac-Moody algebras
Symmetrizable Kac-Moody algebras.
Let A = ( a i,j ) be a n × n generalized Cartanmatrix of rank r . This means that the entries a i,j ∈ Z satisfy the following conditions(1) a i,j ∈ Z for i, j ∈ { , . . . , n } , (2) a i,i = 2 for i ∈ { , . . . , n } , (3) a i,j = 0 if and only if a j,i = 0 for i, j ∈ { , . . . , n } .We will also assume that A is indecomposable : given subsets I and J of { , . . . , n } , thereexists ( i, j ) ∈ I × J such that a i,j = 0. We refer to [6] for the classification of indecomposablegeneralized Cartan matrices. Recall there exist only three kinds of such matrices: when all theprincipal minors of A are positive, A is of finite type and corresponds to the Cartan matrix of asimple Lie algebra over C ; when all the proper principal minors of A are positive and det( A ) = 0the matrix A is said of affine type ; otherwise A is of indefinite type . For technical reasons, fromnow on, we will restrict ourselves to symmetrizable generalized Cartan matrices i.e. we willassume there exists a diagonal matrix D with entries in Z > such that DA is symmetric.The root and weight lattices associated to a generalized symmetrizable Cartan matrix aredefined by mimic the construction for the Lie algebras. Let P ∨ be a free abelian group of rank2 n − r with Z -basis { h , . . . , h n } ∪ { d , . . . , d n − r } . Set h := P ∨ ⊗ Z C and h R := P ∨ ⊗ Z R . Theweight lattice P is then defined by P := { γ ∈ h ∗ | γ ( P ∨ ) ⊂ Z } .Set Π ∨ := { h , . . . , h n } . One can then choose a set Π := { α , . . . , α n } of linearly independentvectors in P ⊂ h ∗ such that α i ( h j ) = a i,j for i, j ∈ { , . . . , n } and α i ( d j ) ∈ { , } for i ∈{ , . . . , n − r } . The elements of Π are the simple roots . The free abelian group Q := L ni =1 Z α i isthe root lattice . The quintuple ( A, Π , Π ∨ , P, P ∨ ) is called a generalized Cartan datum associatedto the matrix A . For any i = 1 , . . . , n , we also define the fundamental weight ω i ∈ P by ω i ( h j ) = δ i,j for j ∈ { , . . . , n } and ω i ( d j ) = 0 for j ∈ { , . . . , n − r } .For any i = 1 , . . . , n , we define the simple reflection s i on h ∗ by(4) s i ( γ ) = γ − h i ( γ ) α i for any γ ∈ P .The Weyl group W is the subgroup of GL ( h ∗ ) generated by the reflections s i . Each element w ∈ W admits a reduced expression w = s i · · · s i r . One can prove that r is independent of thereduced expression considered so the signature ε ( w ) = ( − r is well-defined. Definition 3.1.
The
Kac-Moody algebra g associated to the quintuple ( A, Π , Π ∨ , P, P ∨ ) is the C -algebra generated by the elements e i , f i , i = 1 , . . . , n and h ∈ P together with the relations (1) [ h, h ′ ] = 0 for any h, h ′ ∈ P , (2) [ h, e i ] = α i ( h ) e i for any i = 1 , . . . , n and h ∈ P , (3) [ h, f i ] = − α i ( h ) f i for any i = 1 , . . . , n and h ∈ P , (4) [ e i , f j ] = δ i,j h i for any i, i = 1 , . . . , n , (5) ad ( e i ) − a i,j ( e j ) = 0 for any i, j = 1 , . . . , n such that i = j , (6) ad ( f i ) − a i,j ( f j ) = 0 for any i, j = 1 , . . . , n such that i = j ,where ad ( a ) ∈ End g is defined by ad ( a )( b ) = [ a, b ] := ab − ba for any a, b ∈ g . Denote by g + and g − the subalgebras of g generated by the e i ’s and the f i ’s, respectively. Wehave the triangular decomposition g = g + ⊕ h ⊕ g − and h is called the Cartan subalgebra of g .For any α ∈ Q , set g α := { x ∈ g | [ h, x ] = α ( h ) x for any h ∈ h } . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 7
The algebra g then decomposes on the form g = M α ∈ Q g α where dim g α is finite for any α ∈ Q . The roots of g are the nonzero elements α ∈ Q such that g α = { } . We denote by R the set of roots of g . Set Q + := L ni =1 Z ≥ α i , R + := R ∩ Q + and R − = R ∩ ( − Q + ). Then one can prove that R = R + ∪ R − and R − = − R + as for the finitedimensional Lie algebras. For any γ = P ni =1 a i α i ∈ Q + , we set ht ( γ ) := n X i =1 a i . We have the decomposition g = M α ∈ R + g α ⊕ h ⊕ M α ∈ R − g α . For any α ∈ R + , we set dim g α = m α the multiplicity of the root α in g . The set R + is infinite assoon as A is not of finite type; the multiplicity m α may be greater than 1 but is always boundedas follows (see [6] § m α ≤ n ht ( α ) for any α ∈ R + . When A is not of finite type, the Weyl group W is also infinite and there exist roots α ∈ R whichdo not belong to any orbit W α i , i = 1 , . . . , n of a simple root; these roots are called imaginaryroots in contrast to real roots which belong to the orbit of a simple root α i .The root system associated to a matrix A of finite type is well known (see for instance[2]) and are classified in four infinite series ( A n , B n , C n and D n ) and five exceptional systems( E , E , E , F , G ). In contrast, few is known on the root system associated to a matrix ofindefinite type. In the intermediate case of the affine matrices, there also exists a finite classifi-cation which makes appear seven infinite series and seven exceptional systems. The root systemcan be described as follows. First, the rows and columns of A can be ordered such that thesubmatrix A ◦ of size ( n − × ( n −
1) obtained by deleting the row and column indexed by n in A is the Cartan matrix of a finite root system R ◦ . The kernel of A has dimension 1; moreprecisely, there exists a unique n -tuple ( a , . . . , a n ) of positive relatively prime integers such that A t ( a , . . . , a n ) = 0 and the vector δ = P ni =1 a i α i then belongs to R . The sets of real roots, ofimaginary roots, of positive real roots and positive imaginary roots can be completely describedin terms of roots in R ◦ and δ . We refer to [6] p. 83 for a complete exposition and only recall thefollowing facts we need in the sequel. In particular, we do not need the complete description ofthe sets R re + which strongly depends on the affine root system considered. We have R re + ⊂ { α + kδ | α ∈ R ◦ , k ∈ Z > }∪ R ◦ + except for the affine root system A (2)2 n in which case R re + ⊂ { α + kδ | α ∈ R ◦ , k ∈ Z > } ∪ {
12 ( α + (2 k − δ | α ∈ R ◦ , k ∈ Z > }∪ R ◦ + . We also have in all affine cases(6) R im + = { kδ | k ∈ Z > } and R + = R re + ∪ R im + .The multiplicities of the positive roots verify (see [6] Corollary 8.3).(7) m α = 1 for α ∈ R re + and m α ≤ n for α ∈ R im + . C´EDRIC LECOUVEY, EMMANUEL LESIGNE AND MARC PEIGN´E
The category O int of g -modules. Let g be a symmetrizable Kac-Moody algebra. Wenow introduce a category of g -modules whose properties naturally extend those of the finite-dimensional representations of simple Lie algebras. Definition 3.2.
The category O int is the category of g -modules M satisfying the followingproperties: (1) The module M decomposes in weight subspaces on the form M = M γ ∈ P M γ where M γ := { v ∈ M | h ( v ) = γ ( h ) v for any h ∈ h } . (2) For any i = 1 , . . . , n , the actions of e i and f i are locally nilpotent i.e. for any v ∈ M ,there exists integers p and q such that e pi · v = f qi · v = 0 . For any γ ∈ P , let e γ be the generator of the group algebra C [ P ] associated to γ . By definition,we have e γ e γ ′ = e γ + γ ′ for any γ, γ ′ ∈ P and the group W acts on C [ P ] as follows: w ( e γ ) = e w ( γ ) for any w ∈ W and any γ ∈ P .The irreducible modules in the category O int are the irreducible highest weight modules, theyare parametrized by the integral cone of dominant weights P + of g defined by P + := { λ ∈ P | λ ( h i ) ≥ i = 1 , . . . , n } . The irreducible highest weight module V ( λ ) of weight λ ∈ P + decomposes as V ( λ ) = L γ ∈ P V ( λ ) γ ;observe that dim V ( λ ) is infinite when g is not of finite type, nevertheless the weight space V ( λ ) γ is always finite-dimensional and we set K λ,γ := dim( V ( λ ) γ ). Furthermore, we havedim V ( λ ) λ = 1 and e i ( v ) = 0 for any i = 1 , . . . , n and v ∈ V ( λ ) λ ; the elements of V ( λ ) λ thuscoincide up to a multiplication by a scalar and are called the highest weight vectors .The character s λ of V ( λ ) is defined by s λ := P γ ∈ P K λ,γ e γ ; it is invariant under the action ofthe Weyl group W since K λ,γ = K λ,w ( γ ) for any w ∈ W . Observe that the orbit W · γ intersects P + exactly once when K λ,γ > ρ ∈ P such that ρ ( h i ) = 1 for any i = 1 , . . . , n ; we have theKac-Weyl character formula : Theorem 3.3.
For any λ ∈ P + , we have s λ = P w ∈ W ε ( w ) e w ( λ + ρ ) − ρ Q α ∈ R + (1 − e − α ) m α . The category O int is stable under the tensor product of g -modules. Moreover, every module M ∈ O int decomposes has a direct sum of irreducible modules. Given λ (1) , . . . , λ ( k ) a sequenceof dominant weights, consider the module M := V ( λ (1) ) ⊗ · · · ⊗ V ( λ ( r ) ). Then dim M γ is finitefor any γ ∈ P , the character of M can be defined by char( M ) := P γ ∈ P dim M γ e γ and we havechar( M ) = s λ (1) · · · s λ ( r ) . Each irreducible component of M appears finitely many times in this decomposition, in otherwords there exist nonnegative integers m M,λ such that M ≃ M λ ∈ P + V ( λ ) ⊕ m M,λ or equivalently char( M ) := X λ ∈ P + m M,λ s λ . Consider κ, µ ∈ P + and ℓ ∈ Z ≥ . We set(8) V ( µ ) ⊗ V ( κ ) ⊗ ℓ = X λ ∈ P + V ( λ ) ⊕ f κ,ℓλ/µ and m λµ,κ = f κ, λ/µ . In the sequel, we will fix κ ∈ P + and write f κ,ℓλ/µ = f ℓλ/µ for short. ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 9
Littelmann path model.
The aim of this paragraph is to give a brief overview of thepath model developed by Littelmann and its connections with Kashiwara crystal basis theory.We refer to [12], [13], [14] and [7] for examples and a detailed exposition. Let g be a sym-metrizable Kac-Moody algebra associated to the quintuple ( A, Π , Π ∨ , P, P ∨ ) where A is a n × n symmetrizable generalized Cartan matrix with rank r . In the following, it will be convenient tofix a nondegenerate symmetric bilinear form h· , ·i on h ∗ R invariant under W . For any root α , weset α ∨ = α h α,α i . We have seen that P is a Z -lattice with rank d = 2 n − r . We define the notion ofelementary piecewise linear paths in P R := P ⊗ Z R as we did in § P be the set of suchelementary paths having only rational turning points (i.e. whose inflexion points have rationalcoordinates) and ending in P i.e. such that π (1) ∈ P . The Weyl group W acts on P as follows:for any w ∈ W and η ∈ P , the path w [ η ] is defined by(9) ∀ t ∈ [0 , w [ η ]( t ) = w ( η ( t ))and the weight wt( η ) of η is defined by wt( η ) = η (1).We now define operators ˜ e i and ˜ f i , i = 1 , . . . , n, acting on P ∪ { } . If η = , we set ˜ e i ( η ) =˜ f i ( η ) = ; when η ∈ P , we need to decompose η into a union of finitely many subpaths andreflect some of these subpaths by s α i according to the behavior of the map h η : (cid:26) [0 , → R t
7→ h η ( t ) , α ∨ i i . Let m η for the minimum of the function h η . Since h η (0) = 0, we have m η ≤ m η > −
1, then ˜ e i ( η ) = . If m η ≤ −
1, set t := inf { t ∈ [0 , | h η ( t ) = m η } and let t ∈ [0 , t ] be maximal such that m η ≤ h η ( t ) ≤ m η + 1 for any t ∈ [ t , t ] (see figure 1). Choose r ≥ t = t (0) < t (1) < · · · < t ( r ) = t satisfying the following conditions: for 1 ≤ a ≤ r (1) either h η ( t ( a − ) = h η ( t ( a ) ) and h η ( t ) ≥ h η ( t ( a ) ) on [ t ( a − , t ( a ) ],(2) or h η is strictly decreasing on [ t ( a − , t ( a ) ] and h η ( t ) ≥ h η ( t ( a − ) on [0 , t ( a − ].We set t ( − = 0 and t ( r +1) = 1 and, for 0 ≤ a ≤ r + 1, we denote by η a the elementary pathdefined by ∀ u ∈ [0 , η a ( u ) = η ( t ( a − + u ( t ( a ) − t ( a − )) − η ( t ( a − ) . Observe that η a is the elementary path whose image translated by η ( t ( a − ) coincides with therestriction of η on [ t ( a − , t ( a ) ]; the path η decomposes as follows η = η ∗ η ∗ · · · ∗ η r ∗ η r +1 . For 1 ≤ a ≤ r + 1, we also set η ′ a = η a in case (1) and η ′ a = s α i ( η a ) in case (2). For i ∈ { , · · · , n } ,we set ˜ e i ( η ) = (cid:26) if h η (1) < m η + 1 ,η ∗ η ′ ∗ · · · ∗ η ′ r ∗ η r +1 otherwise.To define the ˜ f i , we first propose another decomposition of the path η . If h η (1) < m η + 1,then ˜ f i ( η ) = . Otherwise ( h η (1) ≥ m η + 1), set t ′ := sup { t ∈ [0 , | h η ( t ′ ) = m η } and let t ′ ∈ [ t ′ ,
1] be minimal such that h η ( t ) ≥ m η + 1 for t ∈ [ t ′ ,
1] (see figure 1). Choose r ≥ t ′ = t (0) < t (1) < · · · < t ( r ) = t ′ satisfying the following conditions: for 1 ≤ a ≤ r (3) either h η ( t ( a − ) = h η ( t ( a ) ) and h η ( t ) ≥ h η ( t ( a − ) on [ t ( a − , t ( a ) ] , (4) or h η is strictly increasing on [ t ( a − , t ( a ) ] and h η ( t ) ≥ h η ( t ( a ) ) on [ t ( a ) , . We set t ( − = 0 and t ( r +1) = 1 and, for 0 ≤ a ≤ r + 1, we denote by η a the elementary pathdefined by ∀ u ∈ [0 , η a ( u ) = η ( t ( a − + u ( t ( a ) − t ( a − )) − η ( t ( a − ) . mm+1 α η η η Figure 1.
Paths η , η = ˜ e i ( η ) and η = ˜ f i ( η )As above, the path η decomposes as η = η ∗ η ∗ · · · ∗ η r ∗ η r +1 ; for 1 ≤ a ≤ r + 1, we thus set η ′ a = η a in case (3) and η ′ a = s α i ( η a ) in case (4) and the operator ˜ f i , ≤ i ≤ n, is defined by˜ f i ( η ) = (cid:26) if h η (1) < m η + 1 ,η ∗ η ′ ∗ · · · ∗ η ′ r ∗ η r +1 otherwise . Remarks:
1. When g is finite-dimensional, the symmetric bilinear form h· , ·i can be assumedpositive so that elements of W are isometries. The paths η, ˜ e i ( η ) and ˜ f i ( η ) have the same length.This is no longer true when g is of affine or indefinite type.2. When ˜ e i ( η ) is computed, the segments of η which are replaced by their symmetric under s α i correspond to intervals where h η is strictly decreasing. This implies that h η ( t ) ≤ h ˜ e i ( η ) ( t )for any t ∈ [0 , h η ( t ) ≥ h ˜ f i ( η ) ( t ) for any t ∈ [0 , e i and ˜ f i satisfy the following properties : Proposition 3.4. (1)
Assume ˜ e i ( η ) = ; then ˜ e i ( η )(1) = η (1) + α i and ˜ f i (˜ e i ( η )) = η . (2) Assume ˜ f i ( η ) = ; then ˜ f i ( η )(1) = η (1) − α i and ˜ e i ( ˜ f i ( η )) = η . (3) A path η ∈ P satisfies ˜ e i ( η ) = for any i = 1 , . . . , n if and only if Im η + ρ is containedin ˚ C . Remark:
It also directly follows from the definition of ˜ f i ( η ) that there exists a piecewise linearincreasing map g defined on [0 ,
1] satisfying(10) η ( t ) − ˜ f i ( η )( t ) = g ( t ) α i for any t ∈ [0 , ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 11 and g (0) = 0 , g (1) = 1.We may endow P with the structure of a Kashiwara crystal: this means that P has thestructure of a colored oriented graph by drawing an arrow η i → η ′ between the two paths η, η ′ of P as soon as ˜ f i ( η ) = η ′ (or equivalently η = ˜ e i ( η ′ )). For any η ∈ P , we denote by B ( η ) theconnected component of η i.e. the subgraph of P obtained by applying operators ˜ e i and ˜ f i , i = 1 , . . . , n to η .For any path η ∈ P and i = 1 , . . . , n , set ε i ( η ) = max { k ∈ Z ≥ | ˜ e ki ( η ) = } and ϕ i ( η ) =max { k ∈ Z ≥ | ˜ f ki ( η ) = } ; one easily checks that ε i ( η ) and ϕ i ( η ) are finite.We now introduce the following notations • P min Z is the set of integral paths , that is paths η such that m η = min t ∈ [0 , {h η ( t ) , α ∨ i i} belongs to Z for any i = 1 , . . . , n . • C is the cone in h ∗ R defined by C = { x ∈ h ∗ R | x ( h i ) ≥ } . • ˚ C is the interior of C ; it is defined by ˚ C = { x ∈ h ∗ R | x ( h i ) > } .One gets the Proposition 3.5.
Let η and π two paths in P min Z . Then (1) the concatenation π ∗ η belongs to P min Z , (2) for any i = 1 , . . . , n we have (11) ˜ e i ( η ∗ π ) = (cid:26) η ∗ ˜ e i ( π ) if ε i ( π ) > ϕ i ( η )˜ e i ( η ) ∗ π otherwise, and ˜ f i ( η ∗ π ) = (cid:26) ˜ f i ( η ) ∗ π if ϕ i ( η ) > ε i ( π ) η ∗ ˜ f i ( π ) otherwise.In particular, ˜ e i ( η ∗ π ) = if and only if ˜ e i ( η ) = and ε i ( π ) ≤ ϕ i ( η ) for any i = 1 , . . . , n . (3) ˜ e i ( η ) = for any i = 1 , . . . , n if and only if Im η is contained in C . The following theorem summarizes crucial results of Littelmann (see [12], [13] and [14]).
Theorem 3.6.
Consider λ, µ and κ dominant weights and choose arbitrarily elementary paths η λ , η µ and η κ in P such that Im η λ ⊂ C , Im η µ ⊂ C and Im η κ ⊂ C and joining respectively to λ , to µ and to κ . (1) We have B ( η λ ) := { ˜ f i · · · ˜ f i k η λ | k ∈ N and ≤ i , · · · , i k ≤ n } \ { } . In particular wt( η ) − wt( η λ ) ∈ Q + for any η ∈ B ( η λ ) . (2) The graph B ( η λ ) is contained in P min Z . (3) If η ′ λ is another elementary path from to λ such that Im η ′ λ is contained in C , then B ( η λ ) and B ( η ′ λ ) are isomorphic as oriented graphs i.e. there exists a bijection θ : B ( η λ ) → B ( η ′ λ ) which commutes with the action of the operators ˜ e i and ˜ f i , i = 1 , . . . , n . (4) The crystal B ( η λ ) is isomorphic to the Kashiwara crystal graph B ( λ ) associated to the U q ( g ) -module of highest weight λ . (5) We have (12) s λ = X η ∈ B ( η λ ) e η (1) . (6) For any i = 1 , . . . , n and any b ∈ B ( η λ ) , let s i ( b ) be the unique path in B ( η λ ) such that ϕ i ( s i ( b )) = ε i ( b ) and ε i ( s i ( b )) = ϕ i ( b ) (in other words, s i acts on each i -chain C i as the symmetry with respect to the center of C i ). The actions of the s i ’s extend to an action of W on P which stabilizes B ( η λ ) . In particular, for any w ∈ W and any b ∈ B ( η λ ) , we have w ( b ) ∈ B ( η λ ) and wt( w ( b )) = w (wt( b )) . (7) For any b ∈ B ( η λ ) we have wt( b ) = P ni =1 ( ϕ i ( b ) − ε i ( b )) ω i . (8) Given any integer ℓ ≥ , set (13) B ( η µ ) ∗ B ( η κ ) ∗ ℓ = { π = η ∗ η ∗ · · · ∗ η ℓ ∈ P | η ∈ B ( η µ ) and η k ∈ B ( η κ ) for any k = 1 , . . . , ℓ } . The graph B ( η µ ) ∗ B ( η κ ) ∗ ℓ is contained in P min Z . (9) The multiplicity m λµ,κ defined in (8) is equal to the number of paths of the form µ ∗ η with η ∈ B ( η κ ) contained in C . (10) The multiplicity f ℓλ/µ defined in (8) is equal to cardinality of the set H ℓλ/µ := { π ∈ B ( η µ ) ∗ B ( η κ ) ∗ ℓ | ˜ e i ( π ) = 0 for any i = 1 , . . . , n and π (1) = λ } . Each path π = η ∗ η ∗ · · · ∗ η ℓ ∈ H ℓλ/µ verifies Im π ⊂ C and η = η µ . Remarks:
1. Combining assertion (2) of Proposition 3.4 together with assertions (1) and (5) ofthe Theorem 3.6, one may check that the function e − λ s λ is in fact a polynomial in the variables T i = e − α i , namely(14) s λ = e λ S λ ( T , . . . , T n )where S λ ∈ C [ X , . . . , X n ]. Observe also that the quantity S ∞ := Q α ∈ R + − e − α ) mα is a formalpower series in the variables T , . . . , T n . M. Kashiwara proved (see for instance [5] § B ( λ ) admits a projective limit B ( ∞ ) when λ tends to infinity and thatchar( B ( ∞ )) = X b ∈ B ( ∞ ) e wt( b ) = S ∞ . Now, since B ( λ ) can be embedded in B ( ∞ ) up to a translation by the weights by λ , we have(15) S λ ( T , . . . , T n ) ≤ S ∞ ( T , . . . , T n );in other words S ∞ ( T , . . . , T n ) = S λ ( T , . . . , T n ) + P µ ∈ Q + a µ T µ where the coefficients a µ arenonnegative integers.2. Using assertion (1) of Theorem 3.6, we obtain m λµ,δ = 0 only if µ + δ − λ ∈ Q + . Similarly,when f δ,ℓλ/µ = 0 one necessarily has µ + ℓδ − λ ∈ Q + .3. A minuscule weight is a dominant weight κ ∈ P + such that the weights of V ( κ ) are exactlythose of the orbit W · κ . In this case, if we take η κ : t tκ , the crystal B ( η κ ) contains only thepaths η : t tw ( κ ). In particular, these paths are lines.4: Given any path η λ such that Im η λ ⊂ C , the set of paths B ( η λ ) is in general very difficultto describe (even in the finite type cases). Nevertheless, for the classical types or type G and a particular choice of η λ , the sets B ( η λ ) can be made explicit by using generalizations ofsemistandard tableaux (see for example [9] and the references therein).The height ht ( η ) of a path η ∈ B ( η λ ) is the length of any path in B ( η λ ) from η λ to η . Forany a ≥
0, we denote by B ( η λ ) a the set of paths in B ( η λ ) at height a. Each subset B ( η λ ) a isfinite and we have(16) B ( η λ ) = G a ≥ B ( η λ ) a . This action should not be confused with that defined in (9) which does not stabilize B ( η λ ) in general. ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 13
By Proposition 3.4, ht ( η ) is equal to the number of simple roots appearing in the decompositionof wt( η λ ) − wt( η ) on the basis { α , . . . , α n } .4. Random paths and symmetrizable Kac-Moody algebras
Probability distribution on elementary paths.
Consider κ ∈ P + and a path π κ ∈P from 0 to κ such that Im π κ is contained in C . Let B ( π κ ) be the connected componentof P containing π κ . We now endow B ( π κ ) with a probability distribution p κ , which will becharacterized by the datum of a n -tuple τ = ( τ , . . . , τ n ) ∈ R n> (each τ i can be regardedas attached to the positive simple root α i ). For any u = u α + · · · + u n α n ∈ Q , we set τ u = τ u · · · τ u n n . Let π ∈ B ( π κ ): by assertion (1) of Theorem 3.6, one gets π (1) = wt( π ) = κ − n X i =1 u i ( π ) α i where u i ( π ) ∈ N for any i = 1 , . . . , n . We have S κ ( τ ) := S κ ( τ , . . . , τ n ) = P π ∈ B ( π κ ) τ κ − wt( π ) . Proposition 4.1.
For any κ ∈ P + , (1) if A is of finite type then < S κ ( τ ) < ∞ for any τ ∈ R n> , (2) if A is of affine type then < S κ ( τ ) < ∞ for any τ ∈ ]0 , n , (3) if A is of indefinite type then < S κ ( τ ) < ∞ for any τ ∈ ]0 , n [ n .Proof. The inequality S κ ( τ ) > τ i > i = 1 , . . . , n . When A is offinite type, the crystal B ( π κ ) is finite, so that S κ ( τ ) < ∞ . When A is not of finite type, let¯ τ = max( τ i , i = 1 , . . . , n ). We have by (15) S κ ( τ ) ≤ S ∞ ( τ ) = Y α ∈ R + − τ α ) m α ≤ Y α ∈ R + − ¯ τ ht ( α ) ) m α and it suffices to prove that(17) S ∗∞ (¯ τ ) = S ∞ (¯ τ , . . . , ¯ τ ) = Y α ∈ R + − ¯ τ ht ( α ) ) m α < + ∞ . • Assume first that A is of affine type different from A (2)2 n . By (6) and (7), we have Y α ∈ R + − ¯ τ ht ( α ) ) m α ≤ Y α ∈ R ◦ + − ¯ τ ht ( α ) + ∞ Y k =1 − ¯ τ kht ( δ ) ) n ! Y α ∈ R ◦ + ∞ Y k =1 − ¯ τ ht ( α + krδ ) ! since 0 < ¯ τ < α ∈ R + and R ◦ is finite. We have to prove that the infinite productsin the above expression are finite. Since ht ( δ ) ≥ n , we have ¯ τ h ( δ ) ≤ ¯ τ n ; moreover α + krδ ∈ Q + for any k ≥ α ∈ R ◦ . We therefore get + ∞ Y k =1 − ¯ τ kht ( δ ) ) n ≤ + ∞ Y k =1 − ¯ τ kn ! n < + ∞ since the series P + ∞ k =1 ln(1 − ¯ τ kn ) converges for ¯ τ n ∈ ]0 , τ rn ∈ ]0 ,
1[ one gets + ∞ Y k =1 − ¯ τ ht ( α + krδ ) ≤ + ∞ Y k =1 − ¯ τ ht ( α ) ¯ τ krn < + ∞ . The case A (2)2 n is obtained by the same arguments. • Secondly, assume that A is of indefinite type. By (5), we have S ∞ (¯ τ ) ≤ Y α ∈ R + (cid:18) − ¯ τ ht ( α ) (cid:19) n ht ( α ) .Moreover, since 0 < ¯ τ < β ∈ Q + and R + ⊂ Q + , we have also S ∗∞ (¯ τ ) ≤ Y β ∈ Q + β =0 − ¯ τ ht ( β ) ) n ht ( β ) = + ∞ Y k =1 Y β ∈ Q + ht ( β )= k − ¯ τ k ) n k with Y β ∈ Q + ht ( β )= k − ¯ τ k ) n k ≤ (cid:18) − ¯ τ k (cid:19) ( k +1) n n k since card( { β ∈ Q + | ht ( β ) = k } ) ≤ ( k + 1) n . We thus get S ∗∞ (¯ τ ) ≤ + ∞ Y k =1 − ¯ τ k ) n k ( k +1) n < + ∞ using the fact that the series P + ∞ k =1 n k ( k + 1) n ln(1 − ¯ τ k ) converges for ¯ τ ∈ ]0 , n [. (cid:3) The previous proposition has three important corollaries. First set T κ ( τ ) := T κ ( τ , . . . , τ n ) = P π ∈ B ( π κ ) ht ( π ) τ κ − wt( π ) . Corollary 4.2.
For any κ ∈ P + , (1) if A is of finite type then < T κ ( τ ) < ∞ for any τ ∈ R n> , (2) if A is of affine type then < T κ ( τ ) < ∞ for any τ ∈ ]0 , n , (3) if A is of indefinite type then < T κ ( τ ) < ∞ for any τ ∈ ]0 , n [ n .Proof. This is clear when A is of finite type. For assertion 2 and 3, let ¯ τ = max( τ i , i = 1 , . . . , n ).In the previous proof, we have established that S ∗∞ (¯ τ ) is finite. Set S ∗ κ (¯ τ ) = S κ (¯ τ , . . . , ¯ τ ). Since S ∗ κ (¯ τ ) ≤ S ∗∞ (¯ τ ), the series S ∗ κ (¯ τ ) is also finite. This means that for any ¯ τ ∈ ]0 ,
1[ we have S ∗ κ (¯ τ ) = X π ∈ B ( π κ ) ¯ τ ht ( π ) = X a ≥ m ( a )¯ τ a < + ∞ where m ( a ) is the number of paths in B ( π κ ) a (see (16)). It follows that T ∗ κ (¯ τ ) = P a ≥ am ( a )¯ τ a is also finite for any ¯ τ ∈ ]0 , T κ ( τ ) ≤ T κ (¯ τ , . . . , ¯ τ ) = X π ∈ B ( π κ ) ht ( π )¯ τ ht ( π ) = T ∗ κ (¯ τ ) < + ∞ . (cid:3) From now on , we write T for the set of n -tuples τ = ( τ , · · · , τ n ) ∈ R n> such that • τ i ∈ ]0 ,
1[ for 1 ≤ i ≤ n when A is of finite or affine type, • τ i ∈ ]0 , n [ for 1 ≤ i ≤ n when A is of indefinite type. Corollary 4.3.
For any µ ∈ P + and w ∈ W , the weight µ + ρ − w ( µ + ρ ) belongs to Q + ; moreover, for τ ∈ T , one gets (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X w ∈ W ε ( w ) τ µ + ρ − w ( µ + ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X w ∈ W τ µ + ρ − w ( µ + ρ ) < + ∞ . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 15
Proof.
By the Weyl-Kac character formula, one gets e − µ s µ = P w ∈ W ε ( w ) e w ( µ + ρ ) − ρ − µ Q α ∈ R + (1 − e − α ) m α . Since e − µ s µ and Q α ∈ R + (1 − e − α ) m α are polynomial in e − β with β ∈ Q + , we have µ + ρ − w ( µ + ρ ) ∈ Q + for any w ∈ W . Now observe that µ + ρ belongs to P + and is the dominant weight of V ( µ + ρ ), each w ( µ + ρ ) is thus also a weight of V ( µ + ρ ). Therefore, the coefficients of thedecomposition of s µ + ρ − P w ∈ W e w ( µ + ρ ) on the basis { e β | β ∈ P } are nonnegative; in other words X w ∈ W e w ( µ + ρ ) ≤ s µ + ρ which readily implies that P w ∈ W e w ( µ + ρ ) − µ − ρ ≤ e − µ − ρ s µ + ρ . By specializing e − α i = τ i , one gets P w ∈ W τ µ + ρ − w ( µ + ρ ) ≤ S µ + ρ ( τ ) < + ∞ . (cid:3) Definition 4.4.
We define the probability distribution p on B ( π κ ) setting p π = τ κ − wt( π ) S κ ( τ ) . Remark:
By Assertion 3 of Theorem 3.6, for π ′ κ another elementary path from 0 to κ such thatIm π ′ κ is contained in C , there exists an isomorphism Θ between the crystals B ( π κ ) and B ( π ′ κ )and one gets p π = p Θ( π ) for any π ∈ B ( π κ ). Therefore, the probability distributions we use onthe graph B ( π κ ) are invariant by crystal isomorphisms.Let X a random variable with values in B ( π κ ) and probability distribution p ; as a directconsequence of Proposition 4.1, we get the Corollary 4.5.
The variable X admits a moment of order . Moreover the series of functions m = X π ∈ B ( π κ ) p π π converges uniformly on [0 , .Proof. We can decompose B ( π κ ) = F a ≥ B ( π κ ) a as in (16). Then, we get for any t ∈ [0 , m ( t ) = X a ≥ X π ∈ B ( π κ ) a p π π ( t ) . Consider π ∈ B ( π κ ) a and set π = ˜ f i · · · ˜ f i a ( π κ ). By (10) and an immediate induction, thereexist increasing piecewise linear maps g , . . . , g a from [0 ,
1] to itself with g k (0) = 0 and g k (1) = 1for any k = 1 , . . . , a such that π ( t ) = π κ ( t ) − ( g ( t ) α i + · · · + g a ( t ) α i a ) . In particular k π ( t ) − π κ ( t ) k ≤ k α k + · · · + k α a k ≤ Ca where C = max α ∈ π k α k is the norm ofthe longest simple root. We thus get k π ( t ) k ≤ k π κ ( t ) k + k π ( t ) − π κ ( t ) k ≤ M + Cht ( π )where M = max t ∈ [0 , k π κ ( t ) k . We obtain max t ∈ [0 , k p π π ( t ) k ≤ ( M + Cht ( π ) τ κ − wt( π ) S κ ( τ ) . But theseries S κ ( τ ) = X π ∈ B ( π κ ) τ κ − wt( π ) and T κ ( τ ) = X π ∈ B ( π κ ) ht ( π ) τ κ − wt( π ) converge by Proposition 4.1 and Corollary 4.2. This means that the series of functions m converges uniformly on [0 , (cid:3) Random paths of arbitrary length.
We now extend the notion of elementary randompaths. Assume that π , . . . , π ℓ a family of elementary paths; the path π ⊗ · · · ⊗ π ℓ of length ℓ is defined by: for all k ∈ { , . . . , ℓ − } and t ∈ [ k, k + 1](18) π ⊗ · · · ⊗ π ℓ ( t ) = π (1) + · · · + π k (1) + π k +1 ( t − k ) . Let B ⊗ ℓ ( π κ ) be the set of paths of the form b = π ⊗ · · · ⊗ π ℓ where π , . . . , π ℓ are elementarypaths in B ( π κ ); there exists a bijection ∆ between B ⊗ ℓ ( π κ ) and the set B ∗ ℓ ( π κ ) of paths in P obtained by concatenations of ℓ paths of B ( π κ ):(19) ∆ : (cid:26) B ⊗ ℓ ( π κ ) −→ B ∗ ℓ ( π κ ) π ⊗ · · · ⊗ π ℓ π ∗ · · · ∗ π ℓ . In fact π ⊗ · · · ⊗ π ℓ and π ∗ · · · ∗ π ℓ coincide up to a reparametrization and we define the weightof b = π ⊗ · · · ⊗ π ℓ settingwt( b ) := wt( π ) + · · · + wt( π ℓ ) = π (1) + · · · + π ℓ (1) . We now endow B ⊗ ℓ ( π κ ) with the product probability measure p ⊗ ℓ defined by(20) p ⊗ ℓ ( π ⊗ · · · ⊗ π ℓ ) = p ( π ) · · · p ( π ℓ ) = τ ℓκ − ( π (1)+ ··· π ℓ (1)) S κ ( τ ) ℓ = τ ℓκ − wt( b ) S κ ( τ ) ℓ . In particular, for any b, b ′ in B ⊗ ℓ ( π κ ) such that wt( b ) = wt( b ′ ), one gets p ⊗ l ( b ) = p ⊗ l ( b ′ ) . Write Π ℓ : B ⊗ ℓ ( π κ ) → B ⊗ ℓ − ( π κ ) the projection defined by Π ℓ ( π ⊗ · · · ⊗ π ℓ − ⊗ π ℓ ) = π ⊗ · · · ⊗ π ℓ − ; the sequence ( B ⊗ ℓ ( π κ ) , Π ℓ , p ⊗ ℓ ) ℓ ≥ is a projective system of probability spaces. We denoteby ( B ⊗ N ( π κ ) , p ⊗ N ) its injective limit; the elements of B ⊗ N ( π κ ) are infinite sequences b = ( π ℓ ) ℓ ≥ and by a slight abuse of notation, we will also write Π ℓ ( b ) = π ⊗ · · · ⊗ π ℓ .Now let X = ( X ℓ ) ℓ ≥ a sequence of i.i.d. random variables with values in B ( π κ ) and proba-bility distribution p ; the random path W on ( B ⊗ N ( π κ ) , p ⊗ N ) are thus defined by W ( t ) := Π ℓ ( X )( t ) = X ⊗ X ⊗ · · · ⊗ X ℓ − ⊗ X ℓ ( t ) for t ∈ [ ℓ − , ℓ ] . By (18), the path W coincides with the one defined in § Proposition 4.6. (1)
For any β, η ∈ P , one gets P ( W ℓ +1 = β | W ℓ = η ) = K κ,β − η, τ κ + η − β S κ ( τ ) . (2) Consider λ, µ ∈ P + we have P ( W ℓ = λ, W = µ, W ( t ) ∈ C for any t ∈ [0 , ℓ ]) = f ℓλ/µ τ ℓκ + µ − λ S κ ( τ ) ℓ . In particular P ( W ℓ +1 = λ, W ℓ = µ, W ( t ) ∈ C for any t ∈ [ ℓ, ℓ + 1]) = m λµ,κ τ κ + µ − λ S κ ( τ ) . Proof.
1. We have P ( W ℓ +1 = β | W ℓ = η ) = X π ∈ B ( b π ) β − η p π where B ( b π ) β − η is the set of paths in B ( b π ) of weight β − η . We conclude noticing that all thepaths in B ( b π ) β − η have the same probability τ κ + η − β S κ ( τ ) and card( B ( b π ) β − η ) = K κ,β − η . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 17
2. By Assertion 7 of Theorem 3.6, we know that the number of paths in B ( π µ ) ∗ B ∗ ℓ ( π κ )starting at µ , ending at λ and remaining in C is equal to f ℓλ/µ . Since the map ∆ defined in (19) isa bijection, the integer f ℓλ/µ is also equal to the number of paths in B ( π µ ) ⊗ B ⊗ ℓ ( π κ ) starting at µ , ending at λ and remaining in C . Moreover, each such path has the form b = b µ ⊗ b ⊗ · · · ⊗ b ℓ where b ⊗ · · · ⊗ b ℓ ∈ B ⊗ ℓ ( π κ ) has weight λ − µ . Therefore we have p b = τ ℓκ + µ − λ S κ ( τ ) ℓ . (cid:3) The generalized Pitman transform.
By Assertion 8 of Theorem 3.6, we know that B ⊗ ℓ ( π κ ) is contained in P min Z . Therefore, if we consider a path b ∈ B ⊗ ℓ ( π κ ) , its connectedcomponent B ( b ) is contained in P min Z . Now, if η ∈ B ( b ) is such that ˜ e i ( η ) = 0 for any i =1 , . . . , n , we should have Im η ⊂ C by Assertion 3 of Proposition 3.5; Assertion 1 of Theorem 3.6thus implies that η is the unique path in B ( b ) = B ( η ) such that ˜ e i ( η ) = 0 for any i = 1 , . . . , n .This permits to define the generalized Pitman transform on B ⊗ ℓ ( π κ ) as the map P whichassociates to any b ∈ B ⊗ ℓ ( π κ ) the unique path P ( b ) ∈ B ( b ) such that ˜ e i ( η ) = 0 for any i = 1 , . . . , n . By definition, we have Im P ( b ) ⊂ C and P ( b )( ℓ ) ∈ P + .Let W be the random path of § H setting(21) H ( t ) = P (Π ℓ ( W ))( t ) for any t ∈ [ ℓ − , ℓ ] . For any ℓ ≥
1, we set H ℓ := H ( ℓ ); one gets the Theorem 4.7.
The random sequence H := ( H ℓ ) ℓ ≥ is a Markov chain with transition matrix (22) Π( µ, λ ) = S λ ( τ ) S κ ( τ ) S µ ( τ ) τ κ + µ − λ m λµ,κ where λ, µ ∈ P + .Proof. Consider µ = µ ( ℓ ) , µ ( ℓ − , . . . , µ (1) a sequence of elements in P + . Let S ( µ (1) , . . . µ ( ℓ ) , λ )be the set of paths b h ∈ B ⊗ ℓ ( π κ ) remaining in C and such that b h ( k ) = µ ( k ) , k = 1 , . . . , ℓ and b ( ℓ +1) = λ . Consider b = b ⊗ · · · ⊗ b ℓ ⊗ b ℓ +1 ∈ B ⊗ ℓ +1 ( π κ ). We have P ( b ⊗ · · · ⊗ b k )( k ) = µ ( k ) for any k = 1 , . . . , ℓ and P ( b )( ℓ + 1) = λ if and only if P ( b ) ∈ S ( µ (1) , . . . µ ( ℓ ) , λ ). Moreover, by(20), for any b h ∈ S ( µ (1) , . . . µ ( ℓ ) , λ ), we have P ( b ∈ B ( b h )) = X b ∈ B ( b h ) p b = X b ∈ B ( b h ) τ ( ℓ +1) κ − wt( b ) S κ ( τ ) ℓ +1 ;combining (12) and (14), one obtains P ( b ∈ B ( b h )) = τ ( ℓ +1) κ − λ S λ ( τ ) S κ ( τ ) ℓ +1 , which only depends on λ .This gives P ( H ℓ +1 = λ, H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = X b h ∈S ( µ (1) ,...µ ( ℓ ) ,λ ) X b ∈ B ( b h ) p b = card( S ( µ (1) , . . . µ ( ℓ ) , λ )) τ ( ℓ +1) κ − λ S λ ( τ ) S κ ( τ ) ℓ +1 . By assertion 9 of Theorem 3.6 and an easy induction, we have alsocard( S ( µ (1) , . . . µ ( ℓ ) , λ )) = ℓ − Y k =1 m µ ( k +1) µ ( k ) ,κ × m λµ,κ . We thus get P ( H ℓ +1 = λ, H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = ℓ − Y k =1 m µ ( k +1) µ ( k ) ,κ × m λµ,κ τ ( ℓ +1) κ − λ S λ ( τ ) S κ ( τ ) ℓ +1 . Similarly P ( H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = ℓ − Y k =1 m µ ( k +1) µ ( k ) ,κ τ ℓκ − µ S λ ( τ ) S κ ( τ ) ℓ , this readily implies P ( H ℓ +1 = λ | H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = P ( H ℓ +1 = λ, H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) P ( H k = µ ( k ) , ∀ k = 1 , . . . , ℓ )= S λ ( τ ) S κ ( τ ) S µ ( τ ) τ κ + µ − λ m λµ,κ . (cid:3) Symmetrization In § p on a crystal B ( π κ ) where κ ∈ P + and π κ is an elementary path from 0 to κ remaining in the cone C . This distribution depends on τ ∈ R n> and Proposition 4.1 gives a sufficient condition to ensure that S κ ( τ ) is finite. Sincethe characters of the highest weight representations are symmetric under the action of the Weylgroup, it is possible to define, starting from the distribution p and for each w in the Weyl group W of g , a probability distribution p w which reflects this symmetry.5.1. Twisted probability distribution.
Recall τ = ( τ , . . . , τ n ) ∈ T is fixed. Given any w ∈ W , we want to define a probability distribution on B ( κ ) for each w ∈ W . Recall that w ( α i ) is a (real) root of g for any w ∈ W and any simple root α i ; this root is neither simple oreven positive in general. By general properties of the root systems, we know that w ( α i ) can bedecomposed as follows w ( α i ) = α k + · · · + α k r or − ( α k + · · · + α k r )where α k , . . . , α k r are simple roots depending on w . Let us define the n -tuple τ w = ( τ w , . . . , τ wn ) ∈ R n> setting τ wi = (cid:26) Q rs =1 τ k s if w ( α i ) = α k + · · · + α k r , Q rs =1 τ − k s if w ( α i ) = − ( α k + · · · + α k r ) , that is(23) τ wi = τ w ( α i ) . More generally for any ¯ u = u α + · · · + u n α n ∈ Q, we have( τ w ) ¯ u = ( τ w ) u · · · ( τ wn ) u n = τ w (¯ u ) . Observe also that τ w / ∈ T in general; indeed we have the following Lemma 5.1. τ ( w ) ∈ T if and only if w = 1 .Proof. It suffices to show that for any w ∈ W \ { Id } distinct from the identity, there is at leasta simple root α i such that w ( α i ) = − ( α k + · · · + α k r ) ∈ − Q + . Indeed, we will have in thatcase τ wi = τ k ··· τ kr > τ ( w ) ∈ T . Consider w ∈ W \ { Id } such that w ( α i ) ∈ Q + for any i = 1 , . . . , ℓ . Let us decompose w as a reduced word w = s i · · · s i t ; by lemma 3.11 in [6], wemust have w ( α i t ) ∈ − Q + , hence a contradiction. This means that w = 1. (cid:3) ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 19
Consider κ ∈ P + . Recall that we have by definition s κ = e κ S κ ( T , . . . , T n ) where T i = e − α i .Since s κ is symmetric under W , we have s κ = e w ( κ ) S κ ( T w , . . . , T wn ) with T wi = e − w ( α i ) for any i = 1 , . . . , n . Therefore S κ ( T w , . . . , T wn ) = e κ − w ( κ ) S κ ( T , . . . , T n ) for any w ∈ W . Since κ − w ( κ ) belongs to Q + , we can specialize each T i in τ i . Then T wi is specialized in τ wi andwe get(24) S κ ( τ w ) = τ w ( κ ) − κ S κ ( τ ) , in particular, it is finite. Definition 5.2.
For any w ∈ W and any integer ℓ ≥ , let p w be the probability distribution on B ( π κ ) ⊗ ℓ defined by: for any b ∈ B ( π κ ) ⊗ ℓ p wb := ( τ w ) ℓκ − wt( b ) S κ ( τ w ) ℓ = τ ℓw ( κ ) − wt( w ( b )) S κ ( τ w ) ℓ where w ( b ) is the image of b under the action of W (see Assertion 6 of Theorem 3.6). Inparticular, p = p coincides with the probability distribution (20). The following lemma states that the probabilities p w and p coincide up to the permutation ofthe elements in B ( π κ ) ⊗ ℓ given by the action of w described in Assertion 6 of Theorem 3.6. Lemma 5.3.
For any w ∈ W and any b ∈ B ( π κ ) ⊗ ℓ , we have p wb = p w ( b ) , where w ( b ) is theimage of b under the action of W (see Assertion 6 of Theorem 3.6).Proof. Recall that wt( w ( b )) = w (wt( b )); therefore p w ( b ) = τ ℓκ − wt( w ( b )) S κ ( τ ) ℓ . On the other hand, by(24) we have p wb := τ ℓw ( κ ) − wt( w ( b )) S κ ( τ w ) ℓ = τ ℓw ( κ ) − wt( w ( b )) τ ℓw ( κ ) − ℓκ S κ ( τ ) ℓ and the equality p wb = p w ( b ) follows. (cid:3) Twisted random paths.
Let w ∈ W and denote by X w the random variable defined on( B ( π κ ) , p w ) with law given by: P ( X w = π ) = p wπ = p w ( π ) for all π ∈ B ( π κ ) . Set m w := E ( X w ) and m := m . Proposition 5.4.
Assume τ ∈ T . One gets (1) m (1) ∈ ˚ C , (2) m w = w − ( m ) , (3) m w (1) ∈ ˚ C if and only if w is equal to the identity.Proof.
1. By definition of ˚ C , we have to prove that h i ( m (1)) > i = 1 , . . . , n . Recallthat m = P π ∈ B ( π κ ) p π π ; observe that the quantity c i = h i ( m (1)) = X π ∈ B ( π κ ) p π h i ( π (1))is well defined by Corollary 4.5. We can decompose the crystal B ( π κ ) in its i -chains, that isthe sub-crystal obtained by deleting all the arrows j = i . When g is not of finite type, thelengths of these i -chains are all finite but not bounded. The contribution to c i of any i -chain C : a i → a i → · · · i → a k of length k is equal to c i ( C ) = k X j =0 p a i h i (wt( a j )) . Since ˜ e i ( a ) = 0 and ˜ f k +1 i ( a ) = 0, we obtain h i (wt( a )) = k . By definition of the distribution p and Proposition 3.4,we have the relation p a j = τ ji p a . Finally, we get c i ( C ) = p a k X j =0 τ ji ( k − j ) = p a ⌊ k/ ⌋ X j =0 ( k − j )( τ ji − τ k − ji ) . In particular the hypothesis τ i ∈ ]0 ,
1[ for any i = 1 , . . . , n implies that c i ( C ) > i -chain of length k >
0; one thus gets c i > B ( π k ) contains at least an i -chain oflength k >
0, otherwise the action of the Chevalley generators e i , f i on the irreducible module V ( π λ ) would be trivial.2. By Lemma 5.3, we can write m w = X π ∈ B ( π κ ) p w ( π ) π = X π ′ ∈ B ( π κ ) p π ′ w − ( π ′ ) = w − X π ′ ∈ B ( π κ ) p π ′ π ′ = w − ( m )where we use assertion 7 of Theorem 3.6 in the third equality.3. Since m w = w − ( m ) and m (1) ∈ ˚ C , one gets m w (1) / ∈ C because C is a fundamentaldomain for the action of the Weyl group W on the tits cone X = ∪ w ∈ W w ( C ) (see [6] Proposition3.12). (cid:3) Now let X w = ( X wℓ ) ℓ ≥ be a sequence of i.i.d. random variables defined on B ( π κ ) withprobability distribution p w . The random process W w = ( W wt ) t> is defined by: for all ℓ ≥ t ∈ [ ℓ − , ℓ ] W w ( t ) := Π ℓ ( X w )( t ) = X w ⊗ X w ⊗ · · · ⊗ X wℓ − ⊗ X wℓ ( t ) . By (18), the random walk W w is defined as in § W w . For any ℓ ∈ Z ≥ , we also definethe function ψ wℓ on P + setting ψ wℓ ( µ ) := P µ ( W w ( t ) ∈ C for any t ∈ [0 , ℓ ]) . The quantity ψ wℓ ( µ ) is equal to the probability of the event “ W w starting at µ remains in thecone C until the instant ℓ ”. We also introduce the function ψ w ( µ ) := P µ ( W w ( t ) ∈ C , t ≥ . For w = 1, we simply write ψ and ψ ℓ instead of ψ and ψ ℓ .The following proposition is a consequence of the previous lemma, Proposition 2.2 and Corol-lary 4.5. Proposition 5.5. (1)
We have lim ℓ → + ∞ ψ wℓ ( µ ) = ψ w ( µ ) for any µ ∈ P + . (2) If w = 1 , then ψ w ( µ ) = 0 for any µ ∈ P + . (3) If w = 1 , then ψ ( µ ) ≥ for any µ ∈ P + . Similarly to Proposition 4.6 and using (24), we obtain the
Proposition 5.6. (1)
For any weights β and η , one gets P ( W wℓ +1 = β | W wℓ = η ) = K κ,β − η, τ w ( κ + η − β ) S κ ( τ w ) = K κ,β − η, τ κ + w ( η ) − w ( β ) S κ ( τ ) . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 21 (2)
For any dominant weights λ and µ , one gets P ( W wℓ = λ, W w = µ, W w ( t ) ∈ C for any t ∈ [0 , ℓ ]) = f ℓλ/µ τ w ( ℓκ + µ − λ ) S κ ( τ w ) ℓ = f ℓλ/µ τ ℓκ + w ( µ ) − w ( λ ) S κ ( τ ) ℓ . In particular P ( W wℓ +1 = λ, W wℓ = µ, W w ( t ) ∈ C for any t ∈ [ ℓ, ℓ + 1]) = m λµ,κ τ w ( κ + µ − λ ) S κ ( τ w ) = m λµ,κ τ κ + w ( µ ) − w ( λ ) S κ ( τ ) . Law of the conditioned random path
The harmonic function ψ . By Assertion 2 of the previous proposition, we can write(25) ψ wℓ ( µ ) = P µ ( W w ( t ) ∈ C for any t ∈ [0 , ℓ ]) = X λ ∈ P + f ℓλ/µ τ ℓκ + w ( µ ) − w ( λ ) S κ ( τ ) ℓ where f ℓλ/µ is the number of highest weight vertices in the crystal(26) B ( µ ) ⊗ B ( κ ) ⊗ ℓ ≃ M λ ∈ P + B ( λ ) ⊕ f ℓλ/µ . By interpreting (26) in terms of characters, we get(27) s µ × s ℓκ = X λ ∈ P + f ℓλ/µ s λ . The Weyl character formula s λ = P w ∈ W ε ( w ) e w ( λ + ρ ) − ρ Q α ∈ R + (1 − e − α ) m α yields Y α ∈ R + (1 − e − α ) m α s µ × s ℓκ = X λ ∈ P + f ℓλ/µ X w ∈ W ε ( w ) e w ( λ + ρ ) − ρ . In the previous formal series in C [[ P ]], all the monomials e w ( λ + ρ ) − ρ with w ∈ W and λ ∈ P + are distinct (see [6] Proposition 3.12). We thus also have(28) Y α ∈ R + (1 − e − α ) m α s µ × s ℓκ = X w ∈ W ε ( w ) X λ ∈ P + f ℓλ/µ e w ( λ + ρ ) − ρ or equivalently Y α ∈ R + (1 − e − α ) m α S µ × S ℓκ = X w ∈ W ε ( w ) X λ ∈ P + f ℓλ/µ e w ( λ + ρ ) − ρ − ℓκ − µ . We now need the following lemma.
Lemma 6.1.
For any w ∈ W and µ ∈ P + , set Π wℓ ( µ ) := X λ ∈ P + f ℓλ/µ τ ℓκ + ρ + µ − w ( λ + ρ ) S κ ( τ ) ℓ . We then have lim ℓ → + ∞ Π wℓ ( µ ) = 0 when w = 1 and the series P w ∈ W ε ( w )Π wℓ ( µ ) convergesuniformly in ℓ . Proof.
Using (25), one gets(29) Π wℓ ( µ ) = τ ρ − w ( ρ )+ µ − w ( µ ) X λ ∈ P + f ℓλ/µ τ ℓκ + w ( µ ) − w ( λ ) S κ ( τ ) ℓ = τ ρ − w ( ρ )+ µ − w ( µ ) ψ wℓ ( µ ) . Fix w = 1. Since τ ρ − w ( ρ )+ µ − w ( µ ) does not depend on ℓ and lim ℓ → + ∞ ψ wℓ ( µ ) = 0 by Proposition5.5, we derive lim ℓ → + ∞ Π wℓ ( µ ) = 0 as desired.Now, we have obviously 0 ≤ ψ wℓ ( µ ) ≤ P w ∈ W τ ρ − w ( ρ )+ µ − w ( µ ) converges byCorollary 4.3. The uniform convergence in ℓ of the series P w ∈ W ε ( w )Π wℓ ( µ ) thus follows fromthe inequality | ε ( w )Π wℓ ( µ ) | ≤ τ ρ − w ( ρ )+ µ − w ( µ ) , which is a direct consequence of (29). (cid:3) We can now set τ i = e − α i in (28) and get(30) Y α ∈ R + (1 − τ α ) m α S µ ( τ ) = X w ∈ W ε ( w ) X λ ∈ P + f ℓλ/µ τ ℓκ + ρ + µ − w ( λ + ρ ) S κ ( τ ) ℓ .Consequently, we have Y α ∈ R + (1 − τ α ) m α S µ ( τ ) = X w ∈ W ε ( w )Π wℓ ( µ ) = Π ℓ ( µ ) + X w =1 ε ( w )Π wℓ ( µ )with Π ℓ ( µ ) = ψ ℓ ( µ ) , by (29). Letting ℓ → + ∞ , the previous lemma finally gives ψ ( µ ) = Y α ∈ R + (1 − τ α ) m α S µ ( τ ) . We have established the following theorem, which is the analogue in our context of Corollary7.4.3 in [10]:
Theorem 6.2.
For any µ ∈ P + , we have ψ ( µ ) = P µ ( W ( t ) ∈ C for any t ≥
0) = Y α ∈ R + (1 − τ α ) m α S µ ( τ ) > In particular, the harmonic function ψ is positive and does not depend on the dominant weight κ considered. Corollary 6.3.
The law of the random walk W conditioned by the event E := ( W ( t ) ∈ C for any t ≥ is the same as the law of the Markov chain H defined as the generalized Pitman transform of W (see Theorem 4.7). In particular, this law only depends on κ and not on the choice of thepath π κ such that Im π κ ⊂ C .Proof. Let Π be the transition matrix of W and Π E its restriction to the event E . We haveseen in § W conditioned by E is the h -transform of Π E by theharmonic function h E ( µ ) := P µ ( W ( t ) ∈ C for any t ≥ . By the previous theorem, we have h E = ψ . It also easily follows from Theorem 4.7 that thetransition matrix of H is the ψ -transform of Π E . Therefore both H and the conditioning of W by E have the same law. (cid:3) ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 23
Random walks defined from non irreducible representations.
For simplicity werestrict ourselves in this paragraph to the case where g is a (finite-dimensional) Lie algebra with(invertible) Cartan matrix A . In particular, m α = 1 for any α ∈ R + . Consider τ = ( τ , . . . , τ n ) ∈T . Then both root and weight lattices have the same rank n . Moreover, the Cartan matrix A is the transition matrix between the weight and root lattices. In particular, each weight β ∈ P decomposes on the basis of simple roots as β = β ′ α + · · · β ′ n α n where ( β ′ , . . . , β ′ n ) ∈ A Z n and we can set τ β = τ β ′ · · · τ β ′ n n .Let M be a finite dimensional g -module with decomposition in irreducible components M ≃ M κ ∈ κ V ( κ ) ⊕ a κ where κ is a finite subset of P + and a κ > κ ∈ κ . For each κ ∈ κ choose a path η κ in P from 0 to κ contained in C . Let B ( κ ) be the set of paths obtained by applying the operators˜ e i , ˜ f i , i = 1 , . . . , n to the paths η κ , κ ∈ κ . This set is a realization of the crystal of the g -module ⊕ κ ∈ κ V ( κ ) (without multiplicities) and we have B ( κ ) = G κ ∈ κ B ( η κ ) . Given π = π ⊗ · · · ⊗ π ℓ in B ⊗ ℓ ( κ ) such that π a ∈ B ( κ a ) for any a = 1 , . . . , ℓ , we set a π = a κ · · · a κ ℓ . By formulas (11), the function a is constant on the connected components of B ⊗ ℓ ( κ ).We are going to define a probability distribution on B ( κ ) compatible with its weight grad-uation and taking into account the multiplicities a κ . We cannot proceed as in (20) by workingonly with the root lattice of g since B ( κ ) contains fewer highest weight paths. So the underlyinglattice to consider is the weight lattice. We first setΣ M ( τ ) = X κ ∈ κ X π ∈ B ( η κ ) a κ τ − wt( π ) = X π ∈ B ( κ ) a π τ − wt( π ) = X κ ∈ κ a κ s κ ( τ ) = X κ ∈ κ a κ τ − κ S κ ( τ ) . We define the probability distribution p on B ( κ ) by setting p π = a κ τ − wt( π ) Σ M ( τ ) for any π ∈ B ( η κ ).When card( κ ) = 1, we recover the probability distribution of § M ( τ ) ℓ = X π ∈ B ⊗ ℓ ( κ ) a π τ − wt( π ) for any ℓ ≥ p ⊗ ℓ on B ⊗ ℓ ( κ ) such that p π = a π τ − wt( π ) Σ M ( τ ) ℓ for any π = π ⊗ · · · ⊗ π ℓ ∈ B ⊗ ℓ ( κ ) . Let X = ( X ℓ ) ℓ ≥ be a sequence of i.i.d. random variables defined on B ( κ ) with probabilitydistribution p . The random process W and the random walk W are then defined from X and p ⊗ N as in § W and its corresponding randomwalk W obtained from the set of elementary paths B ( κ ). We have then P ( W ℓ +1 = β | W ℓ = γ ) = K M,β − γ Σ M ( τ ) τ γ − β for any weights β and γ where K M,β − γ is the dimension of the space of weight β − γ in M . Weindeed have K M,β − γ = P κ ∈ κ a κ K κ,β − γ where K κ,β − γ is the number of paths η ∈ B ( κ ) such that η (1) = β − γ . Given λ and µ two dominant weights, we also get(31) P ( W ℓ +1 = λ | W ℓ = µ, W ( t ) ∈ C for any t ∈ [ ℓ, ℓ + 1]) = m λM,µ Σ M ( τ ) τ µ − λ where m λM,µ is the multiplicity of V ( λ ) in M ⊗ V ( µ ). We indeed have m λM,µ = P κ ∈ κ a κ m λκ,µ where m λκ,µ is the number of paths η ∈ B ( κ ) such that η (1) = λ − µ which remains in C .We define the generalized Pitman transform P and the Markov chain H as in § ℓ ≥
1, we yet write ψ ℓ ( µ ) = P µ ( W ( t ) ∈ C for any t ∈ [1 , ℓ ]). We then have ψ ℓ ( µ ) = X π ∈ B ⊗ ℓ ( κ ) ,µ + π ( t ) ∈C for t ∈ [0 ,ℓ ] p π = X λ ∈ P + X π ∈ B ⊗ ℓ ( κ ) , µ + π ( t ) ∈C for t ∈ [0 ,ℓ ] ,π ( ℓ )= λ a π τ µ − λ Σ M ( τ ) ℓ = X λ ∈ P + f ℓλ/µ τ µ − λ Σ M ( τ ) ℓ where f ℓλ/µ is the multiplicity of V ( λ ) in V ( µ ) ⊗ M ⊗ ℓ . We indeed have the equality f ℓλ/µ = P π ∈ B ⊗ ℓ ( κ ) , µ + π ( t ) ∈C for t ∈ [0 ,ℓ ] ,π ( ℓ )= λ a π by an easy extension of Assertion 10 in Theorem 3.6. Wecan now establish the following theorem. Theorem 6.4.
The law of the random walk W conditioned by the event E := ( W ( t ) ∈ C for any t ≥ is the same as the law of the Markov chain H defined as the generalized Pitman transform of W (see Theorem 4.7). The associated transition matrix Π E verifies (32) Π E ( µ, λ ) = S λ ( τ ) S µ ( τ )Σ M ( τ ) m λM,µ τ µ − λ and we have yet P µ ( W ( t ) ∈ C for any t ≥
0) = Y α ∈ R + (1 − τ α ) S µ ( τ ) . Proof.
The computation of the harmonic function ψ ( µ ) = P µ ( W ( t ) ∈ C for any t ≥
0) is similarto § Y α ∈ R + (1 − e − α ) e − µ s µ = X λ ∈ P + f ℓλ/µ X w ∈ W ε ( w ) e w ( λ + ρ ) − ρ − µ s ℓM where s M = char( M ). When we specialize τ i = e − α i in s M , we obtain Σ M ( τ ). Hence Y α ∈ R + (1 − τ α ) S µ ( τ ) = X λ ∈ P + f ℓλ/µ X w ∈ W ε ( w ) τ µ + ρ − w ( λ + ρ ) Σ M ( τ ) ℓ . If we set Π wℓ ( µ ) := X λ ∈ P + f ℓλ/µ τ ρ + µ − w ( λ + ρ ) Σ M ( τ ) ℓ , we yet obtain lim ℓ → + ∞ Π wℓ ( µ ) = 0 when w = 1 andΠ ℓ ( µ ) = ψ ℓ ( µ ). Moreover Y α ∈ R + (1 − τ α ) S µ ( τ ) = X w ∈ W ε ( w )Π wℓ ( µ ) = Π ℓ ( µ ) + X w =1 ε ( w )Π wℓ ( µ ) ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 25 so the harmonic function ψ = lim ℓ → + ∞ ψ ℓ is also given by ψ ( µ ) = Q α ∈ R + (1 − τ α ) S µ ( τ ). Since Π E is the Doob ψ -transform of the the restriction (31) of W to C , we obtain the desired expression(32) for Π E ( µ, λ ).To see that Π E coincides with the law of the image of W under the generalized Pitmantransform, we proceed as in Proof of Theorem 4.7. Consider µ = µ ( ℓ ) , µ ( ℓ − , . . . , µ (1) a sequenceof elements in P + . Let S ( µ (1) , . . . µ ( ℓ ) , λ ) be the set of paths b h ∈ B ⊗ ℓ ( κ ) remaining in C andsuch that b h ( k ) = µ ( k ) , k = 1 , . . . , ℓ and b ( ℓ +1) = λ . Consider b = b ⊗ · · · ⊗ b ℓ ⊗ b ℓ +1 ∈ B ⊗ ℓ +1 ( κ ).We have P ( b ⊗ · · · ⊗ b k )( k ) = µ ( k ) for any k = 1 , . . . , ℓ and P ( b )( ℓ + 1) = λ if and only if P ( b ) ∈ S ( µ (1) , . . . µ ( ℓ ) , λ ). Moreover, by (20), for any b h ∈ S ( µ (1) , . . . µ ( ℓ ) , λ ), we have P ( b ∈ B ( b h )) = X b ∈ B ( b h ) p b = X b ∈ B ( b h ) a b τ − wt( b ) Σ M ( τ ) ℓ +1 = a b h τ − λ S λ ( τ )Σ M ( τ ) ℓ +1 since a b = a b h for any b ∈ B ( b h ) . This gives P ( H ℓ +1 = λ, H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = τ − λ S λ ( τ )Σ M ( τ ) ℓ +1 X b h ∈S ( µ (1) ,...µ ( ℓ ) ,λ ) a b h = τ − λ S λ ( τ )Σ M ( τ ) ℓ +1 ℓ − Y k =1 m µ ( k +1) µ ( k ) ,M × m λµ,M also using extension of Assertion 10 in Theorem 3.6. Similarly P ( H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = τ − µ S µ ( τ )Σ M ( τ ) ℓ ℓ − Y k =1 m µ ( k +1) µ ( k ) ,M which implies P ( H ℓ +1 = λ | H k = µ ( k ) , ∀ k = 1 , . . . , ℓ ) = S λ ( τ ) S µ ( τ )Σ M ( τ ) m λM,µ τ µ − λ . (cid:3) Example: random walk to the height closest neighbors.
We now study in detailthe case of the random walk in the plane with transitions 0 and the height closest neighbors.The underlying representation is not irreducible and does not decompose as a sum of minusculerepresentations. So the conditioning of this walk cannot be obtained by the methods of [10]. The root system of type C is realized in R = R ε ⊕ R ε . The Cartan matrix is A = (cid:18) − − (cid:19) The simple roots are then α = ε − ε and α = 2 ε . We have P = Z . The fundamentalweights are ω = ε and ω = ε + ε . We have C = { ( x, y ) ∈ R | x ≥ y ≥ } and P + = { λ =( λ , λ ) | λ ≥ λ ≥ } , the set of partitions with two parts . Choose τ ∈ ]0 , , τ ∈ ]0 , . For λ = ( λ , λ ) ∈ P + , we have λ = λ α + λ + λ α . Thus τ λ = τ λ ( √ t ) λ + λ .Consider the sp ( C )-module M = V (1) ⊕ a ⊕ V (1 , ⊕ a . The elementary paths in B ( κ ) canbe easily described from the highest weight paths π : t tε and γ : (cid:26) tε , t ∈ [0 , ] ε + 2( t − ) ε , t ∈ ] ,
1] in C . The results of [10] permit nevertheless to study the random walk in the space R with transitions ± ε ± ε ± ε corresponding to the weights of the spin representation of g = so . We obtain B ( κ ) = B ( π ) ⊕ B ( γ ) where(1) B ( π ) : π : t tε , π : t tε , π : t
7→ − tε and π : t
7→ − tε with t ∈ [0 , B ( γ ) : γ : (cid:26) tε , t ∈ [0 , ] ε + 2( t − ) ε , t ∈ ] , γ : (cid:26) tε , t ∈ [0 , ] ε − t − ) ε , t ∈ ] , ,γ : (cid:26) tε , t ∈ [0 , ] ε − t − ) ε , t ∈ ] , γ : (cid:26) tε , t ∈ [0 , ] ε − t − ) ε , t ∈ ] ,
1] and γ : (cid:26) − tε , t ∈ [0 , ] − ε − t − ) ε , t ∈ ] , . The crystal B ( κ ) is the union of the two following crystals π → π → π → π γ
12 2 → γ
12 1 → γ
22 1 → γ
21 2 → γ Observe that for the path γ , we have γ (0) = γ (1) = 0. The other transitions correspondto the 8 closest neighbors in the lattice Z .We now define the probability distribution p on the set B ( π ) ⊕ m ⊕ B ( γ ) ⊕ m . We haveΣ M ( τ ) = a τ + τ τ + τ τ τ √ τ + a τ + τ τ + τ τ + τ τ τ τ . The probability p is defined by p = a Σ M ( τ ) τ √ τ , p = a Σ M ( τ ) √ τ , p = a √ τ Σ M ( τ ) , p = a τ √ τ Σ M ( τ ) p = a Σ M ( τ ) τ τ , p = a Σ M ( τ ) τ , p = a Σ M ( τ ) , p = a τ Σ M ( τ ) , p = a τ τ Σ M ( τ ) . The set of positive roots is R + = { ε ± ε , ε , ε } and ρ = (2 , . The action of the Weyl group on Z yields the 8 transformations which preserves the square ofvertices ( ± , ± µ = ( µ , µ ) ∈ P + , we obtain by the Weyl character formulaand Theorem 6.2 ψ ( µ ) = P µ ( W ( t ) ∈ C , t ≥
0) = (1 − τ )(1 − τ )(1 − τ τ )(1 − τ τ ) S µ ( τ , τ ) = X w ∈ W ε ( w ) τ w ( µ + ρ ) − ( µ + ρ ) =1 + τ µ − µ +11 τ µ +22 + τ µ +41 τ µ + µ +32 + τ µ + µ +31 τ µ +12 − τ µ − µ +11 − τ µ +12 − τ µ +41 τ µ +22 − τ µ + µ +31 τ µ + µ +32 Moreover, the law of the random walk W conditioned by the event E := ( W ( t ) ∈ C for any t ≥ H defined as the generalized Pitman transform of W (see Theorem 4.7). To compute the associated transition matrix M, we need the tensor product ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 27 multiplicities m λµ,M = a m λ (1 , ,µ + a m λ (1 , ,µ . We have for any partitions λ and µ with two parts m λ (1 , ,µ = (cid:26) λ and µ are equal or differ by only one box0 otherwiseand m λ (1 , ,µ = (cid:26) λ and µ are equal or differ by two boxes in different rows0 otherwise.We thus have for any λ, µ ∈ P + Π E ( µ, λ ) = ψ ( λ ) ψ ( µ )Σ M ( τ ) (cid:16) a m λ (1 , ,µ + a m λ (1 , ,µ (cid:17) τ µ − λ √ τ µ + µ − λ − λ ) . Some consequences
In the remaining of the paper, we assume that g is of finite type and W is constructed froman irreducible g -module V ( κ ) in the category O int . Then the crystal B ( π κ ) has a finite numberof paths which all have the same length as π κ since W contains only isometries.7.1. Asymptotics for the multiplicities f ℓλ/µ . We will use later a quotient version of a locallimit theorem for these random paths; following [10], we may state the
Proposition 7.1.
Let ( g ℓ ) , ( h ℓ ) be two sequences in P such that the events ( W ℓ = g ℓ ) and ( W ℓ = g ℓ + h ℓ ) have non zero probability for ℓ > large enough. Assume there exists α < / such that lim ℓ − α k g ℓ − ℓm k = 0 and lim ℓ − / k h ℓ k = 0 . Then, when ℓ tends to infinity, one gets P ( W ℓ = g ℓ + h ℓ , W ( t ) ∈ C for any t ∈ [0 , ℓ ]) ∼ P ( W ℓ = g ℓ , W ( t ) ∈ C for any t ∈ [0 , ℓ ]) . Proof.
The proof of this statement follows line by line the one of Theorem 4.3 in [10]. Withoutloss of generality, we may assume that the law of the X ℓ is aperiodic in P , which means thatits support generates P and is not included in a coset of a proper subgroup of P : this readilyimplies that P ( W ℓ = g ℓ ) > P ( W ℓ = g ℓ + h ℓ ) > g ℓ ) ℓ and ( h ℓ ) ℓ satisfy theconditions of the proposition and ℓ large enough. When the law of the X ℓ is not aperiodic, therandom walk ( W ℓ ) ℓ has a finite number p of periodic classes and the condition P ( W ℓ = g ℓ ) > P ( W ℓ = g ℓ + h ℓ ) > g ℓ and g ℓ + h ℓ belong to the same periodicclass indexed by the value of ℓ modulo p ; the statement in this case follows from the one in theaperiodic one, by induction of the random walk on each periodic class.We fix a real number β such that < α < β < , set b ℓ = [ ℓ β ] and choose δ > B ℓ =:= B ( m, δ ) ⊂ C .As in [10], we first check that P ( W ℓ = g ℓ , W ( t ) ∈ C for any t ∈ [0 , ℓ ]) P ( W ℓ = g ℓ , W b ℓ ∈ B b ℓ , W ( t ) ∈ C for any t ∈ [0 , b ℓ ]) →
1; inothers words, one may “forget” the conditioning ( W ( t ) ∈ C for any t ∈ [ b ℓ , l ]) in the event( W ℓ = g ℓ , W ( t ) ∈ C for any t ∈ [0 , ℓ ]). The same result holds if one replaces g ℓ by g ℓ + h ℓ forlim ℓ − α k g ℓ + h ℓ − ℓm k = 0.To achieve the proof of the proposition, it now suffices to establish that P ( W ℓ = g ℓ + h ℓ , W b ℓ ∈ B b ℓ , W ( t ) ∈ C for any t ∈ [0 , b ℓ ]) P ( W ℓ = g ℓ , W b ℓ ∈ B b ℓ , W ( t ) ∈ C for any t ∈ [0 , b ℓ ]) → . Since the increments of the random walk ( W ℓ ) ℓ are independent with the same law, we maywrite P ( W ℓ = g ℓ , W b ℓ ∈ B b ℓ , W ( t ) ∈ C for any t ∈ [0 , b ℓ ])= X x ∈ B bℓ ∩ P + P ( W ℓ − b ℓ = g ℓ − x ) × P ( W ℓ = x, W ( t ) ∈ C for any t ∈ [0 , b ℓ ]) . This leads to the proposition since P ( W ℓ − b ℓ = g ℓ − x ) ∼ P ( W ℓ − b ℓ = g ℓ + h ℓ − x ) uniformly in x ∈ B b ℓ . (cid:3) Consider λ, µ ∈ P + and ℓ ≥ f ℓλ/µ > f ℓλ >
0. Then, we must have ℓκ + µ − λ ∈ Q + and ℓκ − λ ∈ Q + . Therefore µ ∈ Q and it decomposes as a sum of simple roots.In the sequel, we will assume the condition µ ∈ Q ∩ P + is satisfied.We assume the notation and hypotheses of Theorem 6.2. Consider a sequence λ ( ℓ ) of dominantweights such that λ ( ℓ ) = ℓm (1) + o ( ℓ ). Following Proposition 5.3 in [10], one gets the followingdecomposition(33) f ℓλ ( ℓ ) /µ f ℓλ ( ℓ ) = X γ ∈ P K µ,γ f ℓλ ( ℓ ) − γ f ℓλ ( ℓ ) = τ − µ X γ ∈ P K µ,γ f ℓλ ( ℓ ) − γ τ − λ ( ℓ ) + γ f ℓλ ( ℓ ) τ − λ ( ℓ ) τ µ − γ where the sums are finite since the set of weights in V ( µ ) is finite. By Assertion 2 of Proposition4.6, we have, for any γ ∈ P and ℓ large enough f ℓλ ( ℓ ) − γ τ − λ ( ℓ ) + γ f ℓλ ( ℓ ) τ − λ ( ℓ ) = f ℓλ ( ℓ ) − γ τ ℓκ − λ ( ℓ ) + γ f ℓλ ( ℓ ) τ ℓκ − λ ( ℓ ) = P ( W ℓ = λ ( ℓ ) − γ, W t ∈ C for any t ∈ [0 , ℓ ]) P ( W ℓ = λ ( ℓ ) , W t ∈ C for any t ∈ [0 , ℓ ])this last quotient tending to 1 when ℓ tends to infinity, by Proposition 7.1. This implieslim ℓ → + ∞ f ℓλ ( ℓ ) /µ f ℓλ ( ℓ ) = τ − µ X γ ∈ P K µ,γ τ µ − γ = τ − µ S µ ( τ ) . We have thus proved the following consequence of Theorem 6.2
Corollary 7.2.
For any µ ∈ Q ∩ P + , and any sequence of dominant weights of the form λ ( ℓ ) = ℓm (1) + o ( ℓ ) , we have lim ℓ → + ∞ f ℓλ ( ℓ ) /µ f ℓλ = τ − µ S µ ( τ ) . Remark:
One can regard this corollary as an analogue of the asymptotic behavior of thenumber of paths in the Young lattice obtained by Kerov and Vershik (see [8] and the referencestherein).7.2.
Probability that W stay in C . By Theorem 6.2, we can compute P µ ( W ( t ) ∈ C for any t ∈ [0 , ℓ ]). Unfortunately, this does not permit do make explicit P µ ( W ℓ ∈ C ∀ ℓ ≥ P µ ( W ( t ) ∈ C for any t ≥ ≤ P µ ( W ℓ ∈ C for any ℓ ≥ . Since we have assumed that g is of finite type, each crystal B ( π κ ) is finite. For any i =1 , . . . , n , write m ( i ) ≥ i -chains appearing in B ( π κ ). Set κ = P ni =1 ( m ( i ) − ω i . Observe that κ = 0 if and only if κ is a minuscule weight. Lemma 7.3.
Assume W k ∈ C for any k = 1 , . . . , ℓ . Then κ + W ( t ) ∈ C for any t ∈ [0 , ℓ ] . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 29
Proof.
Since κ is a dominant weight, we can consider π κ any path from 0 to κ which remainsin C . First observe that the hypothesis W k ∈ C for any k = 1 , . . . , ℓ is equivalent to κ + W ( k ) ∈ κ + C for any k = 1 , . . . , ℓ. We also know by Assertion 8 of Theorem 3.6 that B ( π κ ) ⊗ B ( π κ ) ⊗ ℓ iscontained in P min Z for any ℓ ≥
1. Set W ( ℓ ) = π ⊗· · ·⊗ π ℓ . By Assertion 3 of Proposition 3.4, wehave to prove that ˜ e i ( π κ ⊗ π ⊗· · ·⊗ π ℓ ) = 0 for any i = 1 , . . . , n providing wt( π κ ⊗ π ⊗· · ·⊗ π k ) = π κ ⊗ π ⊗ · · · ⊗ π k (1) ∈ κ + P + for any k = 1 , . . . , ℓ . Fix i = 1 , . . . , n . Set κ ( i ) = m ( i ) − . Weproceed by induction.Assume ℓ = 1. Since we have ˜ e i ( π κ ) = 0, it suffices to prove by using Assertion 2 ofProposition 3.5 that ε i ( π ) ≤ ϕ i ( π κ ). By definition of the dominant weight π κ , we have ϕ i ( π κ ) = κ ( i ). So we have to prove that ε i ( π ) ≤ κ ( i ). Assertion 7 of Theorem 3.6 and thehypothesis wt( π κ ⊗ π ) ∈ κ + P + permits to write(34) wt( π κ ) i + wt( π ) i = wt( π κ ⊗ π ) i ≥ κ ( i ) . Recall that π belongs to B ( π κ ). So ε i ( π ) ≤ κ ( i ) + 1 because ε i ( π ) gives the distance of π from the source vertex of its i -chain. When ε i ( π ) < κ ( i ) + 1 we are done. So assume ε i ( π ) = κ ( i ) + 1. This means that π satisfies ϕ i ( π ) = 0. Therefore, wt( π ) i = − κ ( i ) − π κ ⊗ π ) i = κ ( i ) − ( κ ( i ) + 1) = − ≥ κ ( i )hence a contradiction.Now assume ˜ e i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) = 0 for any k = 1 , . . . , ℓ −
1. Observe that wt( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) i = ϕ i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) ≥ κ ( i ) since π κ ⊗ π ⊗ · · · ⊗ π ℓ − ∈ κ + P + and˜ e i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) = 0. We also have(35) wt( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ⊗ π ℓ ) i = wt( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) i + wt( π ℓ ) i ≥ κ ( i ) . We proceed as in the case ℓ = 1 . Assume first ε i ( π ℓ ) ≤ ϕ i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ). Thenby Proposition 3.5 and the induction equality ˜ e i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) = 0, we will have˜ e i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ ) = 0.Now assume ε i ( π ℓ ) > ϕ i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ). Since ϕ i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) ≥ κ ( i ) and π ℓ ∈ B ( π κ ), we must have ε i ( π ℓ ) = κ ( i ) + 1 , ϕ i ( π ℓ ) = 0 and ϕ i ( π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) = κ ( i ).Therefore, we get wt( π ℓ ) i = − κ ( i ) − π κ ⊗ π ⊗ · · · ⊗ π ℓ − ) i = κ ( i ). Then (35) yieldsyet the contradiction − ≥ κ ( i ) . (cid:3) Remark:
In general the assertion W k ∈ C for any k = 1 , . . . , ℓ is not equivalent to the assertion κ + W ( t ) ∈ κ + C for any t ∈ [0 , ℓ ] . This is nevertheless true when κ is a minuscule weightsince κ = 0 in this case and the paths in B ( π κ ) are lines.We deduce from the previous lemma the inequality P µ ( W k ∈ C for any k = 0 , . . . , ℓ ) ≤ P µ + κ ( W ( t ) ∈ C for any t ∈ [0 , ℓ ]) . When ℓ tends to infinity, this yields P µ ( W ℓ ∈ C for any ℓ ≥ ≤ P µ + κ ( W ( t ) ∈ C for any t ≥ . By using Theorem 6.2, this implies the
Theorem 7.4.
Assume g is of finite type (then m α = 1 for any α ∈ R + ). Then, for any µ ∈ P + we have Y α ∈ R + (1 − τ α ) S µ ( τ ) ≤ P µ ( W ℓ ∈ C for any ℓ ≥ ≤ Y α ∈ R + (1 − τ α ) S µ + κ ( τ ) . In particular, we recover the result of Corollary 7.4.3 in [10] : P µ ( W ℓ ∈ C for any ℓ ≥
1) = Y α ∈ R + (1 − τ α ) S µ ( τ ) when κ is minuscule. Remark:
The inequality obtained in the previous theorem can also be rewritten1 ≤ P µ ( W ℓ ∈ C for any ℓ ≥ P µ ( W ( t ) ∈ C for any t ∈ [0 , + ∞ [) ≤ S µ + κ ( τ ) S µ ( τ ) . When µ tends to infinity, we thus have P µ ( W ℓ ∈ C ∀ ℓ ≥ ∼ P µ ( W ( t ) ∈ C for any t ≥
0) asexpected. 8.
Appendix (proof of Proposition 2.1)
By definition of the probability Q , for any ℓ ≥ µ , · · · , µ ℓ , λ ∈ C , one gets Q ( Y ℓ +1 = λ | Y ℓ = µ ℓ , · · · , Y = µ ) = Q ( Y ℓ +1 = λ, Y ℓ = µ ℓ , . . . , Y = µ ) Q ( Y ℓ = µ ℓ , . . . , Y = µ )= P ( E, Y ℓ +1 = λ, Y ℓ = µ ℓ , . . . , Y = µ ) P ( E, Y ℓ = µ ℓ , . . . , Y = µ ) =: N ℓ D ℓ . We first have, using the Markov property N ℓ = P ( Y ( t ) ∈ C for t ≥ , Y ℓ +1 = λ, Y ℓ = µ ℓ , . . . , Y = µ )= P ( Y ( t ) ∈ C for t ≥ ℓ + 1 | Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [0 , ℓ + 1] , Y ℓ = µ ℓ , . . . , Y = µ ) × P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [0 , ℓ + 1] , Y ℓ = µ ℓ , . . . , Y = µ )= P ( Y ( t ) ∈ C for t ≥ ℓ + 1 | Y ℓ +1 = λ ) × P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [0 , ℓ + 1[ , Y ℓ = µ ℓ , . . . , Y = µ )= P ( Y ( t ) ∈ C for t ≥ | Y = λ ) × P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [0 , ℓ + 1[ , Y ℓ = µ ℓ , . . . , Y = µ )with P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [0 , ℓ + 1[ , Y ℓ = µ ℓ , . . . , Y = µ )= P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1[ | Y ( t ) ∈ C for t ∈ [0 , ℓ [ , Y ℓ = µ ℓ , . . . , Y = µ ) × P ( Y ( t ) ∈ C for t ∈ [0 , ℓ [ , Y ℓ = µ ℓ , . . . , Y = µ )= P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1[ | Y ℓ = µ ℓ ) × P ( Y ( t ) ∈ C for t ∈ [0 , ℓ [ , Y ℓ = µ ℓ , . . . , Y = µ ) . We therefore obtain N ℓ = P ( E | Y = λ ) × P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1] | Y ℓ = µ ℓ ) × P ( Y ( t ) ∈ C for t ∈ [0 , ℓ [ , Y ℓ = µ ℓ , . . . , Y = µ ) . A similar computation yields D ℓ = P ( E | Y ℓ = µ ℓ ] × P [ Y ( t ) ∈ C for t ∈ [0 , ℓ [ , Y ℓ = µ ℓ , . . . , Y = µ ) . ONDITIONED RANDOM WALKS FROM KAC-MOODY ROOT SYSTEMS 31
Finally, we get Q ( Y ℓ +1 = λ | Y ℓ = µ ℓ , · · · , Y = µ ) = P ( Y ℓ +1 = λ, Y ( t ) ∈ C for t ∈ [ ℓ, ℓ + 1] | Y ℓ = µ ℓ ) × P ( E | Y = λ ) P ( E | Y = µ ℓ ) . Laboratoire de Math´ematiques et Physique Th´eorique (UMR CNRS 7350)Universit´e Fran¸cois-Rabelais, ToursF´ed´eration de Recherche Denis Poisson - CNRSParc de Grandmont, 37200 Tours, [email protected]@[email protected]
References [1]
P. Biane, P. Bougerol and N. O’Connell , Littelmann paths and Brownian paths, Duke Math. J., (2005), no. 1, 127-167.[2]
Bourbaki,
Groupes et alg`ebres de Lie, Chapitres 4,5 et 6, Hermann (1968).[3]
B. Hall,
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , Springer (2004).[4]
J. Hong , S. J. Kang , Introduction to quantum groups and crystals bases , Graduate studies in Mathematics, Amer. Math. Soc. (2002).[5]
A Joseph,
Lie Algebras, their representation and crystals, Lecture Notes Weizman Institute.[6]
V. G. Kac,
Infinite dimensional Lie algebras, Cambridge University Press, third edition (1989).[7]
M. Kashiwara , On crystal bases, Canadian Mathematical Society, Conference Proceedings, (1995),155-197.[8] S. V. Kerov , Asymptotic representation theory of the symmetric group and its applications in analysis , AMSTranslation of Mathematical Monographs, vol 219 (2003).[9]
C. Lecouvey,
Combinatorics of crystal graphs for the root systems of types A n , B n , C n , D n and G . Combinatorial aspect of integrable systems , 11–41, MSJ Mem., 17, Math. Soc. Japan, Tokyo, 2007.[10]
C. Lecouvey, E. Lesigne and M. Peign´e , Random walks in Weyl chambers and crystals, Proc. LondonMath. Soc. (2012) 104(2): 323-358.[11]
C. Lecouvey, E. Lesigne and M. Peign´e , Conditioned one-way simple random walks and representationtheory, preprint arXiv 1202.3604 (2012), to appear in Seminaire Lotharingien de Combinatoire.[12]
P. Littelmann , A Littlewood-Richardson type rule for symmetrizable Kac-Moody algebras, Inventionesmathematicae 116, 329-346 (1994).[13]
P. Littelmann , Paths and root operators in representation theory, Annals of Mathematics 142, 499–525(1995).[14]
P. Littelmann , The path model, the quantum Frobenius map and standard monomial theory, AlgebraicGroups and Their Representations NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 517, Kluwer, Dordrecht,Germany,175–212 (1998).[15]
N. O’ Connell , A path-transformation for random walks and the Robinson-Schensted correspondence,Trans. Amer. Math. Soc., , 3669-3697 (2003).[16]
N. O’ Connell , Conditioned random walks and the RSK correspondence, J. Phys. A, , 3049-3066 (2003).[17] W. Woess,