# Affine Lie algebras and conditioned space-time Brownian motions in affine Weyl chambers

## Abstract

We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra \(\mathfrak {g}\) considering tensor product of irreducible highest weight modules of \(\mathfrak {g}\) and specializations of the characters involving the Weyl vector \(\rho \). We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of \(\mathfrak {g}\). This extends in particular the results of Defosseux (arXiv:1401.3115, 2014) to any affine Lie algebras, in the case with a drift.

### Mathematics Subject Classification

17B67 35R37 60J65## 1 Introduction

*h*in a Cartan subalgebra \(\mathfrak {h}\) of \(\mathfrak {g}\) by

*h*in a subset of the Cartan subalgebra which doesn’t depend on \(\lambda \). A particular choice of an element \(h\in \mathfrak {h}\) in the region of convergence of the characters is called a specialization. Let us fix a dominant weight \(\omega \) once for all. For a dominant weight \(\lambda \), the following decomposition

*h*is chosen in the region of convergence of the characters. Such a Markov chain has been recently considered by C. Lecouvey, E. Lesigne, and M. Peigné in [4].

The paper is organized as follows. In Sect. 2 we describe the conditioned process occuring in our setting when representations of affine Lie algebra \(\hat{\mathfrak {sl}_2}\) are considered. This is a space-time Brownian motion with a positive drift conditioned (in Doob’s sense) to remain forever in the time-dependent domain *D*. We show by purely probabilistic arguments that the theta functions play a crucial role in the construction of the process, which is unlighted by the algebraic point of view developed in the following sections. The vocation of this section is to give an idea of the probabilistic aspects of mathematical objects occuring in the paper. In Sect. 3 we briefly recall the necessary background on representation theory of affine Lie algebras. We introduce in Sect. 4 random walks on the set of integral weights of an affine Lie algebra \(\mathfrak {g}\), and Markov chains on the set of its dominant integral weights, considering tensor products of irreducible highest weight representations of \(\mathfrak {g}\). We show that the Weyl character formula implies that they satisfy a reflection principle. In Sect. 5 we consider a sequence of random walks obtained for particular specializations involving the Weyl vector \(\rho \) of the affine Lie algebra, and prove that its scaling limit is a space-time standard Brownian motion with drift \(\rho \), living on the Cartan subalgebra of \(\mathfrak {g}\). We introduce in Sect. 6 a space-time Brownian motion with drift \(\rho \), conditioned to remain in an affine Weyl chamber and prove that it satisfies a reflection principle. We prove in Sect. 7 that this conditioned space-time Brownian motion is the scaling limit of a sequence of Markov processes constructed in Sect. 4 for particular specializations involving \(\rho \).

## 2 A moving boundary problem

*x*, and \(\tau _t=u+t\), for all \(t\ge 0\). Consider the subset

*D*of \(\mathbb {R}^2\) defined by

*h*defined on the closure \(\bar{D}\) of

*D*by

*h*is the unique bounded harmonic positive function for the space-time Brownian motion killed on the boundary \(\partial D\), i.e.

*h*can be determined using a reflection principle involving the group of tranformations

*W*generated by linear transformations \(s_k\), \(k\in \mathbb {Z}\), defined on \(\mathbb {R}^2\) by

*W*is actually a semi-direct product

*W*on \(\mathbb {R}^2\). The following proposition is immediate.

**Proposition 2.1**

The probability semi-group of the space-time Brownian motion killed on the boundary of *D* satisfies a reflection principle. Let \(\{e_1,e_2\}\) be the canonical basis of \(\mathbb {R}^2\) and (., .) be the usual inner product on \(\mathbb {R}^2\). Let *T* denote the first exit time of *D*. The reflection principle is the following.

**Proposition 2.2**

*Proof*

One obtains for the function *h* the following expression.

**Proposition 2.3**

*Proof*

Summing over *y* such that \((t+u,y)\in D\) in Proposition 2.2 and letting *t* go to infinity gives the proposition. \(\square \)

Actually *W* can be identified with the Weyl group associated to an affine Lie algebra \(\hat{\mathfrak {sl}_2}\). Writing \(X_t=\tau _t\Lambda _0+B_t^\gamma \frac{\alpha _1}{2}\), \(t\ge 0\), where \(\Lambda _0\) and \(\alpha _1\) are defined below, the Doob’s *h*-transform of \((X_t)_{t\ge 0}\) is a Markov process conditioned to remain in a Weyl chamber associated to the root system of the affine Lie algebra \(\hat{\mathfrak {sl}_2}\). The following sections extend this construction to any affine Lie algebras and relate identities from Propostions 2.2 and 2.3, which are particular cases of Propositions 6.4 and 6.1, to representations theory of affine Lie algebras.

## 3 Affine Lie algebras and their representations

In order to make the reading more pleasant, we have tried to emphasize only on definitions and properties that we need for our purpose. For more details, we refer the reader to [3], which is our main reference for the whole paper.

### 3.1 Affine Lie algebras

*A*are positive and \(\det A=0\). Suppose that rows and columns of

*A*are ordered such that \(\det \mathring{A}\ne 0\), where \(\mathring{A}= (a_{i,j})_{1\le i,j\le l}\). Let \((\mathfrak {h},\Pi ,\Pi ^\vee )\) be a realization of

*A*with \(\Pi =\{\alpha _0,\dots ,\alpha _l\}\subset \mathfrak {h}^*\) the set of simple roots, \(\Pi ^\vee =\{\alpha _0^\vee ,\ldots ,\alpha _l^\vee \}\subset \mathfrak {h}\), the set of simple coroots, which satisfy the following condition

*Q*and \(Q^\vee \) the root and the coroot lattices. We denote \(a_i,\)\(i,\ldots ,l\) the labels of the Dynkin diagram of

*A*and \(a^\vee _i,\)\(i=0,\ldots ,l\) the labels of the Dynkin diagram of \(^tA\). The numbers

*W*-invariant, for

*W*the Weyl group of the affine Lie algebra \(\mathfrak {g}\), i.e. the subgroup of \(GL(\mathfrak {h}^*)\) generated by fundamental reflections \(s_\alpha \), \(\alpha \in \Pi \), defined by

**Notation**

For \(\lambda \in \mathfrak {h}^*\) such that \(\lambda =a\Lambda _0+z+b\delta \), \(a,b\in \mathbb {C}\), \(z\in \mathring{\mathfrak {h}}^*_\mathbb {R}\), denote \(\bar{\lambda }\) the projection of \(\lambda \) on \(\mathbb {C}\Lambda _0+\mathring{\mathfrak {h}}^*\) defined by \(\bar{\lambda }=a\Lambda _0+z\), and by \(\bar{\bar{\lambda }}\) its projection on \(\mathring{\mathfrak {h}}^*\) defined by \(\bar{\bar{\lambda }}=z\).

*W*is the semi-direct product \(T\ltimes \mathring{W}\) (Proposition 6.5 chapter 6 of [3]) where

*T*is the group of transformations \(t_{\alpha }\), \(\alpha \in M\), defined by

### 3.2 Weights, highest-weight modules, characters

*P*(resp. \(P_+\)) the set of integral (resp. dominant) weights defined by

*k*defined by

*V*is called \(\mathfrak {h}\)-diagonalizable if it admits a weight space decomposition \(V=\oplus _{\lambda \in \mathfrak {h}^*}V_\lambda \) by weight spaces \(V_\lambda \) defined by

*V*which are \(\mathfrak {h}\)-diagonalizable with finite dimensional weight spaces and such that there exists a finite number of elements \(\lambda _1,\ldots ,\lambda _s\in \mathfrak {h}^*\) such that

*V*from \({\mathcal {O}}\) by

### 3.3 Theta functions

*k*by the series

## 4 Markov chains on the sets of integral or dominant weights

Let us choose for this section a dominant weight \(\omega \in P_+\) and \(h\in \mathfrak {h}_\mathbb {R}\) such that \( \delta (h)\in \mathbb {R}_+^*\).

*Random walks on*

*P*We define a probability measure \(\mu _\omega \) on

*P*letting

*Remark 4.1*

*P*whose increments are distributed according to \(\mu _\omega \), keep in mind that the function

*X*(

*n*) on \(\mathring{\mathfrak {h}}^* _\mathbb {R}\).

*Markov chains on*\(P_+\) Given two irreducible representations \(V(\lambda )\) and \(V(\omega )\), the tensor product of \(\mathfrak {g}\)-modules \(V(\lambda )\otimes V(\beta )\) decomposes has a direct sum of irreducible modules. The following decomposition

**Lemma 4.2**

*Proof*

See Proposition 2.1 of [5] and remark below. The proof is exactly the same in the framework of Kac-Moody algebras.

Let us consider the random walk \((X(n))_{n\ge 0}\) defined above and its projection \((\bar{X}(n))_{n\ge 0}\) on \((\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\). Denote \(\bar{P}_\omega \) the transition kernel of this last random walk. The next property is immediate.

**Lemma 4.3**

Let us consider a Markov process \((\Lambda (n))_{n\ge 0}\) whose Markov kernel is given by (5). If \(\lambda _1\) and \(\lambda _2\) are two dominant weights such that \(\lambda _1= \lambda _2\,(mod\, \delta )\) then the irreducible modules \(V(\lambda _1)\) and \(V(\lambda _2)\) are isomorphic. Thus if we consider the random process \((\bar{\Lambda }(n),n\ge 0)\), where \(\bar{\Lambda }(n)\) is the projection of \(\Lambda (n)\) on \((\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\), then \((\bar{\Lambda }(n),n\ge 1)\) is a Markov process whose transition kernel is denoted \(\bar{Q}_\omega \).

**Proposition 4.4**

*n*be a positive integer. The transition kernel \(\bar{Q}_\omega \) satisfies

*Proof*

## 5 Scaling limit of Random walks on *P*

Let us fix \(\rho =h^\vee \Lambda _0+\bar{\bar{\rho }}\), where \(\bar{\bar{\rho }}\) is half the sum of positive roots in \(\mathring{\mathfrak {h}}^*\). For \(n\in {\mathbb {N}}^*\), we consider a random walk \((X^{n}(k),k\ge 0)\) starting from 0, whose increments are distributed according to a probability measure \(\mu _{\omega }\) defined by (4) with \(\omega \in P_+^{h^\vee }\) and \(h=\frac{1}{n}\nu ^{-1}( \rho )\). In particular \(X^n(k)\) is an integral weight of level \(h^\vee k\) for \(k\in {\mathbb {N}}\). Proposition 5.1 gives the scaling limit of the process \((\bar{\bar{X}}^{n}(k),k\ge 0)\),

**Proposition 5.1**

The sequence of processes \((\frac{1}{n}\bar{\bar{X}}^{n}([nt]),t\ge 0)_{n\ge 0}\) converges towards a standard Brownian motion on \(\mathring{\mathfrak {h}}_\mathbb {R}^*\) with drift \(\bar{\bar{\rho }}\).

*Proof*

*z*, \(m_\Lambda =\frac{\vert \vert \Lambda +\rho \vert \vert ^2}{4h^\vee }-\frac{\vert \vert \rho \vert \vert ^2}{2h^\vee }\) and \(S_{\omega ,\Lambda }\) is a coefficient independent of

*z*and

*n*, for \(\Lambda \in P^{h^\vee }_+\). Notice that the sum is well-defined as for \(\lambda _1=\lambda _2\,\text{ mod }\, \mathbb {C}\delta \) one has

*n*. Besides, one easily verifies that for such a \(\Lambda \) one has \(m_\Lambda \ge m_{h^\vee \Lambda _0}\) and that \(m_\Lambda =m_{h^\vee \Lambda _0}\) implies \(\Lambda =h^\vee \Lambda _0\). Thus

*n*goes to infinity. The last convergence and Theorem 13.8 of [3], recalled at the beginning of the proof, imply

## 6 A conditioned space-time Brownian motion

*T*defined by

**Proposition 6.1**

*Proof*

*h*defined on \((\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*)\) by

*h*is the unique bounded harmonic function for the killed process \((X_{t\wedge T})_{t\ge 0}\) under \(\mathbb {P}_x^\rho \) such that

*x*is in the interior of \({\mathcal {C}}\), formula (3) implies that

*g*is bounded by 1 on \(\{(t,x)\in \mathbb {R}_+^*\times \mathbb {R}^l: t\Lambda _0+x_1v_1+\cdots +x_lv_l\in {\mathcal {C}}\}\), \((g((\tau _{t\wedge T},B_{t\wedge T}))_{t\ge 0}\) is a martingale, i.e.

*g*is harmonic for the killed process under \(\mathbb {P}_x^\rho \). It remains to prove that the condition (8) is satisfied. For this, we notice that for any \(w\in W\) distinct from the identity, \(\rho -w(\rho )= \sum _{i=0}^lk_i\alpha _{i},\) where the \(k_i\) are non negative integers not simultaneously equal to zero. As almost surely

*g*is analytic on \(\mathbb {R}_+^*\times \mathbb {R}^l\), the expected convergence follows. \(\square \)

The following lemma is needed to prove a reflection principle for a Brownian motion killed on the boundary of the affine Weyl chamber.

**Lemma 6.2**

*Proof*

**Lemma 6.3**

*Proof*

**Proposition 6.4**

*Proof*

The result follows in a standard way from Lemma 6.3 from a Girsanov’s theorem. \(\square \)

## 7 Scaling limit of the Markov chain on \(P_+\)

For \(x\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\), Proposition 6.1 and identity (3) imply in particular that the probability \(P^\rho _{x}(T=+\infty )\) is positive when *x* is in the interior of \({\mathcal {C}}\). Let \(({\mathcal {F}}_t)_{t\ge 0}\) be the natural filtration of \((X_t)_{t\ge 0}\). Let us fix \( x\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\) in the interior of \({\mathcal {C}}\). One considers the following conditioned process.

**Definition 7.1**

Under the probability \(\mathbb {Q}^\rho _{x}\), the process \((X_t)_{t\ge 0}\) is a space-time Brownian motion with drift \( \rho \), conditioned to remain forever in the affine Weyl chamber. Let \((x_n)_{n\ge 0}\) be a sequence of elements of \(P_+\) such that the sequence \((\frac{x_n}{n})_{n\ge 0}\) converges towards *x* when *n* goes to infinity. For any \(n\in {\mathbb {N}}^*\), we consider a Markov process \((\Lambda ^{n}(k),k\ge 0)\) starting from \(x_n\), with a transition probability \(Q_\omega \) defined by (5), with \(\omega \in P_+^{h^\vee }\) and \(h=\frac{1}{n}\nu ^{-1}( \rho )\). Notice that for \(n,k\in {\mathbb {N}}\), \(\Lambda ^{n}(k)\) is a dominant weight of level \(kh^\vee +(x_n\vert \delta )\). Then the following convergence holds.

**Theorem 7.2**

The sequence of processes \((\frac{1}{n}\bar{\Lambda }^{n}([nt]), t\ge 0)\) converges when *n* goes to infinity towards the process \((X_t,t\ge 0)\) under \(\mathbb {Q}^\rho _{x}\).

*Proof*

*n*goes to infinity towards a Markov process with transition probability semi-group \((q_t)_{t\ge 0}\) defined by

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