Probability Theory and Related Fields

, Volume 165, Issue 3–4, pp 649–665 | Cite as

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

Article

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

In [2] we have studied a conditioned space-time Brownian motion which appears naturally in the framework of representation theory of the affine Lie algebra \(\hat{\mathfrak {sl}_2}\): a space-time Brownian motion \((t,B_t)_{t\ge 0}\) conditioned (in Doob’s sense) to remain in a moving boundary domain
$$\begin{aligned} D=\{(r,z)\in \mathbb {R}_+\times \mathbb {R}_+: 0< z< r\}, \end{aligned}$$
which can be seen as the Weyl chamber associated to the root system of the affine Lie algebra \(\hat{\mathfrak {sl}_2}\). The present paper deals with the case of any affine Lie algebras. Let us briefly describe the framework of the paper. First we need an affine Lie algebra \(\mathfrak {g}\). As in the finite dimensional case, for a dominant integral weight \(\lambda \) of \(\mathfrak {g}\) one defines the character of an irreducible highest-weight representation \(V(\lambda )\) of \(\mathfrak {g}\) with highest weight \(\lambda \), as a formal series defined for h in a Cartan subalgebra \(\mathfrak {h}\) of \(\mathfrak {g}\) by
$$\begin{aligned} \text{ ch }_\lambda (h)=\sum _{\mu }\text{ dim } (V(\lambda )_\mu )e^{\langle \mu ,h\rangle }, \end{aligned}$$
where \(V(\lambda )_\mu \) is the weight space of \(V(\lambda )\) corresponding to the weight \(\mu \). This formal series converges for every 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
$$\begin{aligned} \text{ ch }_{\omega }\text{ ch }_{\lambda }=\sum _{\beta \in P_+} M_{\lambda }(\beta )\text{ ch }_{\beta }, \end{aligned}$$
where \(M_{\lambda }(\beta )\) is the multiplicity of the module with highest weight \(\beta \) in the decomposition of \(V(\omega )\otimes V(\lambda )\), allows to define a transition probability \(Q_\omega \) on the set of dominant weights, letting for \(\beta \) and \(\lambda \) two dominant weights of \(\mathfrak {g}\),
$$\begin{aligned} Q_\omega (\lambda ,\beta )=\frac{\text{ ch }_\beta (h)}{\text{ ch }_\lambda (h) \text{ ch }_\omega (h)}M_{\lambda }(\beta ), \end{aligned}$$
(1)
where 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].
It is a natural question to ask if there exists a sequence \((h_n)_n\) of elements of \(\mathfrak {h}\) such that the corresponding sequence of Markov chains converges towards a continuous process and what the limit is. One could show that there are basically three cases depending on the scaling factor. Roughly speaking the three cases are the following. When the scaling factor is \(n^{-\alpha }\), with \(\alpha \in (0,1)\) (resp. \(\alpha >1\)), the limiting process has to do with a Brownian motion conditioned—in Doob’s sense—to remain in a Weyl chamber (resp. an alcove) associated to the root system of an underlying finite dimensional Lie algebra. When \(\alpha =1\), the limiting process has to do with a space time Brownian motion conditioned to remain in a Weyl chamber associated to the root system of the affine Lie algebra. Figure 1 below illustrates three distinct asymptotic behaviors in the case when the affine Lie algebra is \(\hat{\mathfrak {sl}_2}\). The Weyl chamber \({\mathcal {C}}\) is the area delimited by gray and light gray half-planes. Essentially, when the scaling factor is \(n^{-\alpha }\) with \(\alpha \in (0,1)\), one could show that the \(\Lambda _0\)-component of the limiting process is \(+\infty \) and that its projection on \(\mathbb {R}\alpha _1\) is a Brownian motion conditioned to remain positive. When the scaling factor is \(n^{-\alpha }\) with \(\alpha >1\), one could show that the projection of the limiting process on \(\mathbb {R}_+\Lambda _0+\mathbb {R}\alpha _1\) lives in an interval (dashed interval within Fig. 1) and that its projection on \(\mathbb {R}\alpha _1\) is a Brownian motion conditioned to remain in an interval. When the scaling factor is \(n^{-1}\), the projection of the limiting process on \(\mathbb {R}_+\Lambda _0+\mathbb {R}\alpha _1\) is a space-time Brownian motion conditioned to remain in \({\mathcal {C}}\), the time axis being \(\mathbb {R}_+\Lambda _0\) and the space axis being \(\mathbb {R}\alpha _1\). This is this last case which is considered in the paper, for any affine Lie algebras. The convergence for the other values of \(\alpha \) could be obtained with similar arguments as the ones developed in this paper. Nevertheless the case when \(\alpha =1\) seems the most interesting case in our context as the limiting process in this case, is the only one that is really specific to the affine framework. Thus we prefer to focus on this case. In this way we lose in generality but hope to win in clarity.
Fig. 1

The affine Weyl Chamber corresponding to \(A_{1}^{(1)}\)

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

Let \(\gamma \in \mathbb {R}\), and \((X_t,t\ge 0)=((\tau _t,B^\gamma _t),t\ge 0)\) be a space-time Brownian motion. For \((u,x)\in \mathbb {R}\times \mathbb {R}\), \(\mathbb {P}_{(u,x)}\) denotes a probability under which \((B_t^{\gamma })_{t\ge 0}\) is a standard Brownian motion with drift \(\gamma \), starting from x, and \(\tau _t=u+t\), for all \(t\ge 0\). Consider the subset D of \(\mathbb {R}^2\) defined by
$$\begin{aligned} D=\{(r,z)\in \mathbb {R}_+\times \mathbb {R}_+: 0< z< r\}, \end{aligned}$$
and consider an application h defined on the closure \(\bar{D}\) of D by
$$\begin{aligned} h(u,x)&:=\mathbb {P}_{(u,x)}(\forall t\ge 0, 0<B_t^\gamma <\tau _t), \end{aligned}$$
\((u,x)\in \bar{D}\). When \(\gamma \in (0,1),\) a classical martingale argument shows that the function h is the unique bounded harmonic positive function for the space-time Brownian motion killed on the boundary \(\partial D\), i.e.
$$\begin{aligned} \forall (t,x)\in D,\quad \left( \frac{1}{2}\partial _{xx}+\gamma \partial _x+\partial _t\right) h(t,x)=0, \end{aligned}$$
which satisfies the following boundary conditions
$$\begin{aligned} \forall t\ge 0,\quad h(t,0)=h(t,t)=0, \end{aligned}$$
and the condition at infinity
$$\begin{aligned} \underset{\tiny {\begin{array}{ll} (t,x)\rightarrow +\infty :\\ \frac{x}{t}\rightarrow \gamma \end{array}}}{\lim } h(t,x)=1. \end{aligned}$$
Such a problem is usually referred to as a moving boundary problem (see for instance [1] for a review of various problems specifically related to time-dependent boundaries). Actually the function 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
$$\begin{aligned} s_k(t,x)=(t,2kt-x),\quad (t,x)\in \mathbb {R}^2. \end{aligned}$$
Let us explain how. For \(k\in \mathbb {Z}\), define \(t_k\) as the transformation on \(\mathbb {R}^2\) given by
$$\begin{aligned} t_k(t,x)=(t,2kt+x), \end{aligned}$$
\((t,x)\in \mathbb {R}^2\). The group W is actually a semi-direct product
$$\begin{aligned} \{\text{ Id },s_0\}\ltimes \{t_k,k\in \mathbb {Z}\}, \end{aligned}$$
and \(\bar{D}\) is a fundamental domain for the action of W on \(\mathbb {R}^2\). The following proposition is immediate.

Proposition 2.1

For \((u,x),(u+t,y)\in \mathbb {R}^2\),
$$\begin{aligned} \mathbb {P}_{s_0(u,x)}(X_t=s_0(u+t,y))&=e^{-2\gamma (y-x)}\mathbb {P}_{(u,x)}(X_t=(u+t,y))\\ \mathbb {P}_{t_k(u,x)}(X_t=t_k(u+t,y))&=e^{-2k(y-x)-2k^2t+2k\gamma t}\mathbb {P}_{(u,x)}(X_t=(u+t,y)), \end{aligned}$$
for \(k\in \mathbb {Z}\), where \(\mathbb {P}_{(u,x)}(X_t=(u+t,y))\) stands, by a usual abuse of notation, for the probability semi-group of \((X_t)_{t\ge 0}\).

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

$$\begin{aligned} \mathbb {P}_{(u,x)}(X_t=(u+t,y),\, T\ge t)&=e^{-\gamma x}\sum \det (r) e^{-2k^2u-2kx+(\gamma rt_k(u,x),e_1)}\\&\quad \times \mathbb {P}_{rt_k(u,x)}(X_t=(u+t,y)), \end{aligned}$$
where the sum runs over \(r\in \{\text{ Id },s_0\}, k\in \mathbb {Z}\).

Proof

As \(X_T\in \bar{D}\), one obtains using a strong Markov property and Proposition 2.1 that for \((u,x),(u+t,y)\in D\),
$$\begin{aligned} e^{-\gamma x} \sum _{r\in \{\text{ Id },s_0\}, k\in \mathbb {Z}}\det (r)&e^{-2k^2u-2kx+(\gamma rt_k(u,x),e_1)} \mathbb {P}_{rt_k(u,x)}(X_t=(u+t,y), T\le t) \end{aligned}$$
equals 0. Moreover
$$\begin{aligned} e^{-\gamma x}\sum _{r\in \{\text{ Id },s_0\}, k\in \mathbb {Z}}\det (r)&e^{-2k^2u-2kx+(\gamma rt_k(u,x),e_1)} \mathbb {P}_{rt_k(u,x)}(X_t=(u+t,y), T>t) \end{aligned}$$
equals
$$\begin{aligned} \mathbb {P}_{(u,x)}(X_t=(u+t,y),\, T\ge t). \end{aligned}$$
Proposition follows by summing the two identities. \(\square \)

One obtains for the function h the following expression.

Proposition 2.3

For \((u,x)\in D\),
$$\begin{aligned} h(u,x)=2\sum _{k\in \mathbb {Z}}\text{ sh }(\gamma (x+2ku))e^{-2(kx+k^2u)-\gamma x}. \end{aligned}$$

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

The following definitions mainly come from chapters 1 and 6 of [3]. Let \(A=(a_{i,j})_{0\le i,j\le l}\) be a generalized Cartan matrix of affine type. That is all the proper principal minors of 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
$$\begin{aligned} \alpha _j(\alpha _i^\vee )=a_{i,j}, \, i,j\in \{0,\ldots ,l\}. \end{aligned}$$
Let us consider the affine Lie algebra \(\mathfrak {g}\) with generators \(e_i\), \(f_i\), \(i=0,\ldots ,l\), \(\mathfrak {h}\) and the following defining relations:
$$\begin{aligned}&[e_i,f_i]=\delta _{ij}\alpha _i^\vee , \quad [h,e_i]=\alpha _i(h)e_i,\quad [h,f_i]=-\alpha _i(h)f_i,\\&[h,h']=0, \text { for } h,h'\in \mathfrak {h}, \\&(\text{ ad } e_i)^{1-a_{ij}}e_{j}=0, \quad (\text{ ad } f_i)^{1-a_{ij}}f_{j}=0, \end{aligned}$$
for all \(i,j=0,\ldots ,l\). Let \(\Delta \) (resp. \(\Delta _+\)) denote the set of roots (resp. positive roots) of \(\mathfrak {g}\), 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
$$\begin{aligned} h=\sum _{i=0}^la_i\quad and \quad h^\vee =\sum _{i=0}^la_i^\vee , \end{aligned}$$
are called, respectively, the Coxeter number and the dual Coxeter number. The element
$$\begin{aligned} K=\sum _{i=0}^na_i^\vee \alpha _i^\vee , \end{aligned}$$
is called the canonical central element. The element \(\delta \) defined by
$$\begin{aligned} \delta =\sum _{i=0}^na_i\alpha _i, \end{aligned}$$
is the smallest positive imaginary root. Fix an element \(d\in \mathfrak {h}\) which satisfies the following condition
$$\begin{aligned} \alpha _i(d)=0, \text { for } i=1,\ldots ,l,\quad \alpha _0(d)=1. \end{aligned}$$
The elements \(\alpha _0^\vee ,\ldots ,\alpha _l^\vee ,d\), form a basis of \(\mathfrak {h}\). We denote \(\mathfrak {h}_\mathbb {R}\) the linear span over \(\mathbb {R}\) of \(\alpha _0^\vee ,\ldots ,\alpha _l^\vee ,d\). We define a nondegenerate symmetric bilinear \(\mathbb {C}\)-valued form \((.\vert .)\) on \(\mathfrak {h}\) as follows
$$\begin{aligned} \left\{ \begin{array}{ll} (\alpha _i^\vee \vert \alpha _j^\vee )=\frac{a_j}{a_j^{\vee }}a_{ij}&{} i,j=0,\ldots ,l \\ (\alpha _i^\vee \vert d)=0&{} i=1,\ldots ,l \\ (\alpha _0^\vee \vert d)=a_0 &{} (d\vert d)=0. \end{array} \right. \end{aligned}$$
We define an element \(\Lambda _0\in \mathfrak {h}^*\) by
$$\begin{aligned} \Lambda _0(\alpha _i^\vee )=\delta _{0i}, \quad i=0,\ldots ,l;\quad \Lambda _0(d)=0. \end{aligned}$$
The linear isomorphism
$$\begin{aligned} \nu :\, \,&\mathfrak {h}\rightarrow \mathfrak {h}^*, \\&h\mapsto (h\vert .) \end{aligned}$$
identifies \(\mathfrak {h}\) and \(\mathfrak {h}^*\). We still denote \((.\vert .)\) the induced inner product on \(\mathfrak {h}^*\). We record that
$$\begin{aligned}&(\delta \vert \alpha _i)=0,\quad i=0,\ldots ,l,\quad (\delta \vert \delta )=0,\quad (\delta \vert \Lambda _0)=1\\&(K\vert \alpha _i)=0, \quad i=0,\ldots ,l,\quad (K\vert K)=0,\quad (K\vert d)=a_0. \end{aligned}$$
The form \((.\vert .)\) is 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
$$\begin{aligned} s_\alpha (\beta )=\beta - \beta (\alpha ^\vee ) \alpha ,\ \quad \beta \in \mathfrak {h}^*. \end{aligned}$$
We denote \(\mathring{\mathfrak {h}}\) (resp. \(\mathring{\mathfrak {h}}_\mathbb {R}\)) the linear span over \(\mathbb {C}\) (resp. \(\mathbb {R}\)) of \(\alpha _1^\vee ,\ldots ,\alpha _l^\vee \). The dual notions \(\mathring{\mathfrak {h}}^*\) and \(\mathring{\mathfrak {h}}^*_\mathbb {R}\) are defined similarly. Then we have an orthogonal direct sum of subspaces:
$$\begin{aligned} \mathfrak {h}=\mathring{\mathfrak {h}}_\mathbb {R}\oplus (\mathbb {C}K+\mathbb {C}d);\quad \mathfrak {h}^*=\mathring{\mathfrak {h}}^*_\mathbb {R}\oplus (\mathbb {C}\delta +\mathbb {C}\Lambda _0). \end{aligned}$$
We set \(\mathfrak {h}_\mathbb {R}=\mathring{\mathfrak {h}}_\mathbb {R}+ \mathbb {R}K+\mathbb {R}d,\) and \(\mathfrak {h}_\mathbb {R}^*=\mathring{\mathfrak {h}}^*_\mathbb {R}+\mathbb {R}\delta +\mathbb {R}\Lambda _0\).

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 \(\bar{\bar{\lambda }}=z\).

We denote \(\mathring{W}\) the subgroup of \(GL(\mathfrak {h}^*)\) generated by fundamental reflections \(s_{\alpha _i}\), \(i=1,\ldots ,l\). Let \(\mathbb {Z}(\mathring{W}.\theta ^\vee )\) denote the lattice in \(\mathring{\mathfrak {h}}_\mathbb {R}\) spanned over \(\mathbb {Z}\) by the set \(\mathring{W}.\theta ^\vee \), where
$$\begin{aligned} \theta ^\vee =\sum _{i=1}^la_i^\vee \alpha _i^\vee , \end{aligned}$$
and set \(M=\nu (\mathbb {Z}(\mathring{W}.\theta ^\vee )).\) Then 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
$$\begin{aligned} t_\alpha (\lambda )=\lambda +\lambda (K)\alpha -\left( (\lambda \vert \alpha )+\frac{1}{2}(\alpha \vert \alpha )\lambda (K)\right) \delta , \quad \lambda \in \mathfrak {h}^*. \end{aligned}$$

3.2 Weights, highest-weight modules, characters

The following definitions and properties mainly come from chapter 9 and 10 of [3]. We denote P (resp. \(P_+\)) the set of integral (resp. dominant) weights defined by
$$\begin{aligned}&P=\{\lambda \in \mathfrak {h}^*: \langle \lambda ,\alpha ^\vee _i \rangle \in \mathbb {Z}, \, i=0,\ldots , l\},\\&(\text {resp. } P_+=\{\lambda \in P: \langle \lambda ,\alpha ^\vee _i \rangle \ge 0, \, i=0,\ldots ,l\}), \end{aligned}$$
where \(\langle .,.\rangle \) is the pairing between \(\mathfrak h\) and its dual \(\mathfrak h^*\). The level of an integral weight \(\lambda \in P\), is defined as the integer \((\delta \vert \lambda )\). For \(k\in {\mathbb {N}}\), we denote \(P^k\) (resp. \(P^k_+)\) the set of integral (resp. dominant) weights of level k defined by
$$\begin{aligned}&P^k=\{\lambda \in P: (\delta \vert \lambda )=k\}.\\&(\text {resp. }P^k_+=\{\lambda \in P_+: (\delta \vert \lambda )=k\}.) \end{aligned}$$
Recall that a \(\mathfrak {g}\)-module 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
$$\begin{aligned} V_\lambda =\{v\in V: \forall h\in \mathfrak {h},\, h.v=\lambda (h)v\}. \end{aligned}$$
The category \({\mathcal {O}}\) is defined as the set of \(\mathfrak {g}\)-modules 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
$$\begin{aligned} P(V)\subset \cup _{i=1}^s\{\mu \in \mathfrak {h}^*: \lambda _i-\mu \in {\mathbb {N}}\Delta _+\}, \end{aligned}$$
where \(P(V)=\{\lambda \in \mathfrak {h}^*: V_\lambda \ne \{0\}\}\). One defines the formal character \(\text{ ch }(V)\) of a module V from \({\mathcal {O}}\) by
$$\begin{aligned} \text{ ch }(V)=\sum _{\mu \in P(V)}\dim (V_\mu )e^{\mu }. \end{aligned}$$
For \(\lambda \in P_+\) we denote \(V(\lambda )\) the irreducible module with highest weight \(\lambda \). It belongs to the category \({\mathcal {O}}\). The Weyl character’s formula (Theorem 10.4, chapter 10 of [3]) states that
$$\begin{aligned} \text{ ch }(V(\lambda ))=\frac{\sum _{w\in W}\det (w)e^{w(\lambda +\rho )-\rho }}{\prod _{\alpha \in \Delta _+}(1-e^{-\alpha })^{\text{ mult }(\alpha )}}, \end{aligned}$$
(2)
where \(\text{ mult }(\alpha )\) is the dimension of the root space \(\mathfrak {g}_\alpha \) defined by
$$\begin{aligned} \mathfrak {g}_\alpha =\{x\in \mathfrak {g}: \, \forall h\in \mathfrak {h},\, [h,x]=\alpha (h)x\}, \end{aligned}$$
for \(\alpha \in \Delta \) and \(\rho \in \mathfrak {h}^*\) is chosen such that \(\rho (\alpha _i^\vee )=1\), for all \(i\in \{0,\ldots ,l\}\). In particular
$$\begin{aligned} \prod _{\alpha \in \Delta _+}(1-e^{-\alpha })^{\text{ mult }(\alpha )}=\sum _{w\in W}\det (w)e^{w(\rho )-\rho }. \end{aligned}$$
(3)
Letting \(e^{\mu }(h)=e^{\mu (h)}\), \(h\in \mathfrak {h}\), the formal character \(\text{ ch }(V(\lambda ))\) can be seen as a function defined on its region of convergence. Actually the series
$$\begin{aligned} \sum _{\mu \in P}\text{ dim } (V(\lambda )_\mu )e^{\langle \mu ,h\rangle } \end{aligned}$$
converges absolutely for every \(h\in \mathfrak {h}\) such that \(\text{ Re }(\delta (h))>0\) (see chapter 11 of [3]). We denote \(\text{ ch }_\lambda (h)\) its limit. For \(\beta \in \mathfrak {h}\) such that \({\text {Re}}(\beta \vert \delta )>0\), let \(\text{ ch }_\lambda (\beta )=\text{ ch }_\lambda (\nu ^{-1}(\beta ))\).

3.3 Theta functions

Connections between affine Lie algebras and theta functions are developed in chapter 13 of [3]. We recall properties that we need for our purpose. For \(\lambda \in P\) such that \((\delta \vert \lambda )=k\) one defines the classical theta function \(\Theta _\lambda \) of degree k by the series
$$\begin{aligned} \Theta _\lambda =e^{-\frac{(\lambda \vert \lambda )}{2k}\delta }\sum _{\alpha \in M}e^{t_\alpha (\lambda )}. \end{aligned}$$
This series converges absolutely on \(\{h\in \mathfrak {h}: \text{ Re }(\delta (h))>0\}\) to an analytic function. As
$$\begin{aligned} e^{\frac{(\lambda \vert \lambda )}{2k}\delta }\sum _{w\in \mathring{W}}\det (w)\Theta _{w(\lambda )}=\sum _{w\in W}\det (w)e^{w(\lambda )}, \end{aligned}$$
this last series converges absolutely on \(\{h\in \mathfrak {h}: \text{ Re }(\delta (h))>0\}\) to an analytic function too.

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 onP We define a probability measure \(\mu _\omega \) on P letting
$$\begin{aligned} \mu _\omega (\beta )=\frac{\dim (V(\omega )_\beta )}{\text{ ch }_\omega (h) }e^{\langle \beta ,h\rangle }, \quad \beta \in P. \end{aligned}$$
(4)

Remark 4.1

If \((X(n),n \ge 0)\) is a random walk on P whose increments are distributed according to \(\mu _\omega \), keep in mind that the function
$$\begin{aligned} z\in \mathring{\mathfrak {h}} _\mathbb {R}\mapsto \Big (\frac{\text{ ch }_\omega (iz+h)}{\text{ ch }_\omega (h)}\Big )^n, \end{aligned}$$
is the Fourier transform of the projection of 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
$$\begin{aligned} V(\lambda )\otimes V(\omega )=\sum _{\beta \in P_+} M_{\lambda }(\beta )V(\beta ), \end{aligned}$$
where \(M_{\lambda }(\beta )\) is the multiplicity of the module with highest weight \(\beta \) in the decomposition of \(V(\omega )\otimes V(\lambda )\), leads to the definition a transition probability \(Q_\omega \) on \(P_+\) given by
$$\begin{aligned} Q_\omega (\lambda ,\beta )=\frac{\text{ ch }_\beta (h)}{\text{ ch }_\lambda (h) \text{ ch }_\omega (h)}M_{\lambda }(\beta ),\quad \lambda ,\beta \in P_+. \end{aligned}$$
(5)
For \(n\in {\mathbb {N}}\), \(\omega \in P_+\), \(\beta \in P,\) denote \(m_{\omega ^{\otimes n}}(\beta )\) the multiplicity of the weight \(\beta \) in \(V(\omega )^{\otimes n}\). For \(n\in {\mathbb {N}}\), \(\lambda ,\beta \in P_+,\) denote \(M_{\lambda ,\omega ^{\otimes n}}(\beta )\) the multiplicity defined by
$$\begin{aligned} V(\lambda )\otimes V(\omega )^{\otimes n}=\sum _{\beta \in P_+}M_{\lambda \otimes \omega ^{\otimes n}}(\beta )V(\beta ). \end{aligned}$$
The Weyl character formula implies the following lemma, which is known as a consequence of the Brauer-Klimyk rule when \(\mathfrak {g}\) is a complex semi-simple Lie algebra.

Lemma 4.2

For \(n\in {\mathbb {N}}\), \(\lambda ,\beta \in P_+\) one has
$$\begin{aligned} M_{\lambda \otimes \omega ^{\otimes n}}(\beta )=\sum _{w\in W}\det (w)m_{\omega ^{\otimes n}}(w(\beta +\rho )-(\lambda +\rho )), \end{aligned}$$

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 \(\beta _0,\lambda _0\) be two weights in \( (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\). The transition kernel \(\bar{P}_\omega \) satisfies for every \(n\in {\mathbb {N}}\),
$$\begin{aligned} \bar{P}_\omega ^n(\lambda _0,\beta _0) =\sum _{\beta \in P:\bar{\beta }=\beta _0}e^{\langle \beta -\lambda _0,h\rangle }\frac{m_{\omega ^{\otimes n}}(\beta -\lambda _0)}{\text{ ch }^n_{\omega }(h)}\\ \end{aligned}$$

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

Let \(\beta _0,\lambda _0\) be two dominant weights in \(( \mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\), and n be a positive integer. The transition kernel \(\bar{Q}_\omega \) satisfies
$$\begin{aligned} \bar{Q}_\omega ^n(\lambda _0,\beta _0)&=\frac{{\text {ch}}_{\beta _0}(h)e^{-\langle \beta _0,h\rangle }}{{\text {ch}}_{\lambda _0}(h)e^{-\langle \lambda _0,h\rangle } } \sum _{w\in W}\det (w)e^{\langle w(\lambda _0+\rho )-(\lambda _0+\rho ),h\rangle }\bar{P}_\omega ^n(\overline{w(\lambda _0+\rho )-\rho },\beta _0) \end{aligned}$$

Proof

Using Lemma (4.2), one obtains for any dominant weight \(\lambda _0,\beta _0\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {g}}_\mathbb {R}^*)\),
$$\begin{aligned} \bar{Q}_\omega ^n(\lambda _0,\beta _0)&=\frac{\text{ ch }_{\beta _0}(h)}{\text{ ch }_{\lambda _0}(h) \text{ ch }^n_{\omega }(h)}\sum _{\beta \in P_+:\bar{\beta }=\beta _0}e^{\langle \beta -\bar{\beta },h\rangle }M_{\lambda ,\omega ^{\otimes n}}(\beta )\\&=\frac{\text{ ch }_{\beta _0}(h)}{\text{ ch }_{\lambda _0}(h) \text{ ch }^n_{\omega }(h)}\sum _{\beta \in P:\bar{\beta }=\beta _0}e^{\langle \beta -\bar{\beta },h\rangle }\sum _{w\in W}\det (w)m_{\omega ^{\otimes n}}(w(\beta +\rho )-(\lambda +\rho )).\\&=\frac{\text{ ch }_{\beta _0}(h)e^{-\langle \beta _0,h\rangle }}{\text{ ch }_{\lambda _0}(h)e^{-(\lambda _0,h)} } \sum _{w\in W}\det (w)e^{\langle w(\lambda _0+\rho )-(\lambda _0+\rho ),h\rangle }\bar{P}_\omega ^n(\overline{w(\lambda _0+\rho )-\rho },\beta _0). \end{aligned}$$

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

The key ingredients for the proof are Theorems 13.8 and 13.9 of [3], which provide a transformation law for normalized characters. The two theorems deal with two different classes of affine Lie algebras. Let us make the proof in the framework of Theorem 13.8. The proof is similar in the framework of Theorem 13.9. For the affine Lie algebras considered in Theorem 13.8 one has that for \(n\ge 1\) and \(z\in \mathring{\mathfrak {h}}^*\),
$$\begin{aligned} \text{ ch }_{\omega }&\left( \frac{1}{n}(\rho +z)\right) \\&\quad \quad =C_ne^{\frac{1}{2 n}\vert \vert \bar{\bar{\rho }}+z\vert \vert ^2}\sum _{\Lambda \in P^{h^\vee }_+\text{ mod }\, \mathbb {C}\delta }S_{\omega ,\Lambda }e^{-m_\Lambda \frac{4\pi ^2 n}{h^\vee }}\text{ ch }_\Lambda \left( \frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) , \end{aligned}$$
where \(C_n\) is a constant independent of 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
$$\begin{aligned} e^{-m_{\lambda _1} \frac{4\pi ^2 n}{h^\vee }}\text{ ch }_{\lambda _1}\left( \frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) =e^{-m_{\lambda _2} \frac{4\pi ^2 n}{h^\vee }}\text{ ch }_{\lambda _2}\left( \frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) . \end{aligned}$$
Let us prove the convergence. Let \(i\in \{1,\ldots ,l\}\). One has \(\langle h^\vee \Lambda _0,\alpha _i^\vee \rangle =0\), which implies that \( V(h^\vee \Lambda _0)_{h^\vee \Lambda _0-\alpha _i}=\{0\}\). Consequentely,
$$\begin{aligned} \text {if }\beta \in P \hbox { and } \dim (V(h^\vee \Lambda )_\beta )\ne 0, \hbox { then } \beta =h^\vee \Lambda _0-\sum _{k=0}^li_k\alpha _k, \end{aligned}$$
where \(i_k\) is a nonnegative integer, for \(k\in \{1,\ldots ,l\}\), and \(i_0\) is a positive integer, which implies that \((\beta \vert \Lambda _0)\le -1\). Moreover, the action of \(f_k\), for \( k\in \{0,\ldots ,l\}\), on an integrable highest weight module being locally nilpotent, the number of weights \(\beta \) such that \(\dim (V(h^\vee \Lambda _0)_\beta )\ne 0\) and \((\beta \vert \Lambda _0)=-1\) is finite. As the characters are defined on the set
$$\begin{aligned} \{\lambda \in \mathfrak {h}^*: {\text {Re}}(\lambda \vert \delta )>0\}, \end{aligned}$$
by absolutely convergent series, it implies that
$$\begin{aligned} \text{ ch }_{h^\vee \Lambda _0} \left( \frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) \end{aligned}$$
is equal to
$$\begin{aligned} 1+(1+\epsilon (n))e^{-\frac{4n\pi ^2}{h^\vee }}\sum _{\beta : (\beta \vert \Lambda _0)=-1}\dim V(h^\vee \Lambda _0)_\beta e^{ \left( \beta \vert 2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) }, \end{aligned}$$
where \(\lim _{n\rightarrow \infty }\epsilon (n)=0\). Thus
$$\begin{aligned} \lim _{n\rightarrow \infty } \Big (\text{ ch }_{h^\vee \Lambda _0} (\frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee })\Big )^{[nt]}=1. \end{aligned}$$
(6)
Let \(\Lambda \in P_+^{h^\vee }\) such that \((\Lambda \vert \Lambda _0)=0\). As previously, if \(\beta \in P\) and \(\dim (V(\Lambda )_\beta )\ne 0\) then \(( \beta \vert \Lambda _0)\le 0\), and the number of weights \(\beta \) such that \(\dim (V(\Lambda )_\beta )\ne 0\) and \((\beta \vert \Lambda _0)= 0\) is finite. Thus,
$$\begin{aligned} \text{ ch }_\Lambda \left( \frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee }\right) , \end{aligned}$$
is bounded independently of 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
$$\begin{aligned} \left( 1+\sum _{\Lambda \in P^{h^\vee }_+\setminus \{h^\vee \Lambda _0\}\text{ mod }\, \mathbb {C}\delta }\frac{S_{\omega ,\Lambda }}{S_{\omega ,h^\vee \Lambda _0}}e^{-(m_\Lambda -m_{h^\vee \Lambda _0}) \frac{4\pi ^2 n}{h^\vee }}\frac{\text{ ch }_\Lambda (\frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee })}{\text{ ch }_{h^\vee \Lambda _0} (\frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee })}\right) ^{[nt]} \end{aligned}$$
converges towards 1 when n goes to infinity. The last convergence and Theorem 13.8 of [3], recalled at the beginning of the proof, imply
$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \frac{\text{ ch }_{\omega }(\frac{1}{n}(\rho +z))}{C_nS_{\omega ,h^\vee \Lambda _0}e^{-m_{h^\vee \Lambda _0}\frac{4\pi ^2n}{h^\vee }}\text{ ch }_{h^\vee \Lambda _0} (\frac{4\pi ^2n}{h^\vee }\Lambda _0+2i\pi \frac{\bar{\bar{\rho }}+z}{h^\vee })}\right) ^{[nt]}=e^{\frac{t}{2}\vert \vert \bar{\bar{\rho }}+z\vert \vert ^2}. \end{aligned}$$
Finally, using convergence (6) one obtains
$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \frac{\text{ ch }_{\omega }(\frac{1}{n}(\rho +z))}{\text{ ch }_{\omega }(\frac{1}{n}\rho )}\right) ^{[nt]}=e^{\frac{t}{2}(\vert \vert \bar{\bar{\rho }}+z\vert \vert ^2-\vert \vert \bar{\bar{\rho }}\vert \vert ^2)}, \end{aligned}$$
which achieves the proof by remark (4.1). \(\square \)

6 A conditioned space-time Brownian motion

Denote \({\mathcal {C}}\) the fundamental Weyl chamber defined by
$$\begin{aligned} {\mathcal {C}}=\{x\in \mathfrak {h}^* : \langle x,\alpha _i^\vee \rangle \ge 0,\, i=0,\ldots ,l\}. \end{aligned}$$
Let us consider a standard Brownian motion \((B_t)_{t\ge 0}\) on \( \mathring{\mathfrak {h}}^*_\mathbb {R}\). We consider a random process \((\tau _t\Lambda _0+B_t)_{t\ge 0}\) on \( (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\). For \(x\in ( \mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\), denote \(\mathbb {P}^0_{x}\) (resp. \(\mathbb {P}_{x}^\rho \)), a probability under which \(\tau _t=(x\vert \delta )+th^\vee ,\)\(\forall t\ge 0\), and \((B_t)_{t\ge 0}\) is a standard Brownian motion (resp. a standard Brownian motion with drift \( \bar{\bar{\rho }}\)) starting from \(\bar{\bar{x}}\). Under \(\mathbb {P}^0_{x}\) (resp. \(\mathbb {P}^\rho _{x}\)), the stochastic process \((\tau _t\Lambda _0+B_t)_{t\ge 0}\) has a transition probability semi-group \((p_t)_{t\ge 0}\) (resp. \((p^\rho _t)_{t\ge 0}\)) defined by
$$\begin{aligned}&p_t(x,y)=\frac{1}{(2\pi t)^{\frac{l}{2}}}e^{-\frac{1}{2t}\vert \vert y-x\vert \vert ^2}1_{(y\vert \delta )=th^\vee +(x\vert \delta )},\quad x,y\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^* _\mathbb {R}).\\&(\text {resp. } p^\rho _t(x,y)=\frac{1}{(2\pi t)^{\frac{l}{2}}}e^{-\frac{1}{2t}\vert \vert y-\bar{\bar{\rho }} t-x\vert \vert ^2}1_{(y\vert \delta )=th^\vee +(x\vert \delta )},\quad x,y\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^* _\mathbb {R}).) \end{aligned}$$
Let \(X_t=\tau _t\Lambda _0+B_t\), for \(t\ge 0\), and consider the stopping time T defined by
$$\begin{aligned} T=\inf \{t\ge 0: X_t\notin {\mathcal {C}}\}. \end{aligned}$$
The following proposition gives the probability for \((X_t)_{t\ge 0}\) to remain forever in \({\mathcal {C}}\), under \(\mathbb {P}_x^\rho \), for \(x\in {\mathcal {C}}\).

Proposition 6.1

Let \( x\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R})\cap {\mathcal {C}}\). One has
$$\begin{aligned} \mathbb {P}^\rho _{x}(T=+\infty ) =\sum _{w\in W}\det (w)e^{( x,w(\rho )-\rho )}. \end{aligned}$$

Proof

If we consider the function h defined on \((\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*)\) by
$$\begin{aligned} h(\lambda )=\mathbb {P}^\rho _{\lambda }(T=\infty ), \quad \lambda \in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*)\cap {\mathcal {C}}, \end{aligned}$$
usual martingal arguments state that 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
$$\begin{aligned} h(\lambda )=0,\, \text { for }\lambda \in \partial {\mathcal {C}}, \end{aligned}$$
(7)
and
$$\begin{aligned} \lim _{t\rightarrow \infty }h(X_{t\wedge T})=1_{T=\infty }. \end{aligned}$$
(8)
Let us proves that the function defined by the sum satisfies these properties. First notice that the boundary condition (7) is satisfied. Moreover, as x is in the interior of \({\mathcal {C}}\), formula (3) implies that
$$\begin{aligned} \sum _{w\in W}\det (w)e^{(x,w(\rho )-\rho )} \end{aligned}$$
is positive and bounded by 1. Choose an orthonormal basis \(v_1,\ldots ,v_l\) of \(\mathring{\mathfrak {h}}^*_\mathbb {R}\) and consider for \(w\in W\) a function \(g_w\) defined on \(\mathbb {R}_+^*\times \mathbb {R}^{l}\) by
$$\begin{aligned} g_w(t,x_1,\ldots ,x_l)=e^{( t\Lambda _0+x,w(\rho )-\rho )}, \end{aligned}$$
where \(x=x_1v_1+\cdots +x_lv_l\). Letting \(\Delta =\sum _{i=1}^l\partial _{x_ix_i}\), the function \(g_w\) satisfies
$$\begin{aligned} \left( \frac{1}{2}\Delta +h^\vee \partial _t+\sum _{i=1}^l(\rho ,v_i)\partial _{x_i}\right) g_w=\frac{1}{2 }\vert \vert w(\rho )-\rho \vert \vert ^2+(\rho \vert w(\rho )-\rho )=0. \end{aligned}$$
(9)
As the function \(g=\sum _w\det (w)g_w\) is analytic on \(\mathbb {R}_+^*\times \mathbb {R}^l\), it satisfies (9) too. Ito’s Lemma implies that \((g((\tau _{t\wedge T},B_{t\wedge T}))_{t\ge 0}\) is a local martingale. As the function 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
$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{X_t}{t}=\rho , \end{aligned}$$
one obtains
$$\begin{aligned} \lim _{t\rightarrow \infty } g_w(X_t)=0 \end{aligned}$$
for every \(w\in W\) distinct from the identity. As the function 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

For \(x,y\in {\mathfrak {h}}_\mathbb {R}^*\), \(t\in \mathbb {R}_+\), \(w\in W\), one has
$$\begin{aligned} p^0_t(\overline{wx},\overline{wy})=e^{(w(y-x)-(y-x),h^\vee \Lambda _0)}p^0_t(\bar{x},\bar{y}). \end{aligned}$$

Proof

Notice that \(\overline{wx}=\overline{w\bar{x}}\). For \(w\in \mathring{W}\), \(\overline{wx}=w\bar{x}\), \(p_t^0(w(\bar{x}),w(\bar{y}))=p_t^0(x,y)\) and \((wx-x\vert \Lambda _0)=(wy-t\vert \Lambda _0)=0\), which implies the identity. For \(w=t_\alpha ,\)\(\alpha \in M\), one has
$$\begin{aligned} p_t^0(\overline{wx},\overline{wy})&=p_t^0(h^\vee u\alpha + {\bar{x}},h^\vee (u+t)\alpha + {\bar{y}})\\&=\frac{1}{(2\pi t)^{\frac{l}{2}}}e^{-\frac{1}{2t}\vert \vert \bar{y}+th^\vee \alpha -\bar{x}\vert \vert ^2}1_{(y\vert \delta )=th^\vee +(x\vert \delta )}\\&=p^0_t(\bar{x},\bar{y})e^{-\frac{1}{2t}((h^\vee )^2t^2(\alpha \vert \alpha )+2h^\vee t(\alpha \vert y-x))}\\&=e^{(w(y-x)-(y-x),h^\vee \Lambda _0)}p^0_t(\bar{x},\bar{y}). \end{aligned}$$
\(\square \)
In the following, by a classical abuse of notation,
$$\begin{aligned} \mathbb {P}^\rho _x(X_t=y,T\ge t),\text { or } \, \mathbb {P}^0_x(X_t=y,T\ge t), \end{aligned}$$
\(x,y\in ( \mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*), t\ge 0\), stands for the semi-group of the process \((X_t)_{t\ge 0}\), with drift or not, killed on the boundary of \({\mathcal {C}}\). We first prove a reflection principle for a Brownian motion with no drift.

Lemma 6.3

For \(x,y\in ( \mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*)\) in the interior of \({\mathcal {C}}\), such that \((y\vert \delta )=(x\vert \delta )+th^\vee \), we have
$$\begin{aligned} \mathbb {P}^0_{x}(X_t=y, T> t)&=\sum _{w\in W}\det (w)e^{(wx-x,h^\vee \Lambda _0)}p_t^0(\overline{wx},y),\\&=\sum _{w\in W}\det (w)e^{(y-w(y),h^\vee \Lambda _0)}p_t^0(x,\overline{w(y)}). \end{aligned}$$

Proof

Lemma 6.2 implies in particular that we need to prove only one of the two identities. Let us prove the second one. Actually Lemma 6.2 implies that for \(\alpha \in \Pi \) such that \(s_\alpha (X_T)=0\)
$$\begin{aligned} {\mathbb {E}}_{X_T}(1_{X_r=\overline{wy}})=e^{(wy-s_\alpha wy\vert h^\vee \Lambda _0)}{\mathbb {E}}_{X_T}(1_{X_r}=\overline{s_\alpha wy}), \end{aligned}$$
which implies that
$$\begin{aligned} {\mathbb {E}}_x\left( \sum _{w\in W}\det (w)e^{(y-wy,h^\vee \Lambda _0)}1_{T\le t,\, X_t=wy}\right) =0. \end{aligned}$$
Then lemma follows from the fact that
$$\begin{aligned} {\mathbb {E}}_x\left( \sum _{w\in W}\det (w)e^{(y-w(y),h^\vee \Lambda _0)}1_{T> t,\, X_t=wy}\right) ={\mathbb {E}}_x(1_{X_t=y,\, T>t}). \end{aligned}$$
\(\square \)

Proposition 6.4

For \(x,y\in ( \mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}_\mathbb {R}^*)\) in the interior of \({\mathcal {C}}\), such that \((y\vert \delta )=(x\vert \delta )+th^\vee \), we have
$$\begin{aligned} \mathbb {P}^\rho _{x}(X_t=y, T> t)&=\sum _{w\in W}\det (w)e^{(w(x)-x,\rho )}p_t^\rho (\overline{w(x)},y)\\&=\sum _{w\in W}\det (w)e^{(y-w(y),\rho )}p_t^\rho (x,\overline{w(y)}), \end{aligned}$$

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

One defines a probability \(\mathbb {Q}^\rho _{x}\) letting
$$\begin{aligned} \mathbb {Q}^\rho _{x}(A)={\mathbb {E}}_{x}\left( \frac{\mathbb {P}^\rho _{X_t}(T=+\infty )}{\mathbb {P}^\rho _x(T=+\infty )}1_{T\ge t,\, A}\right) ,\quad \text { for } A\in {\mathcal {F}}_t,\, t\ge 0. \end{aligned}$$

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

Propositions 4.4 and 5.1 imply that the sequence of processes \((\frac{1}{n}\bar{\Lambda }^{n}([nt]), t\ge 0)\) converges when n goes to infinity towards a Markov process with transition probability semi-group \((q_t)_{t\ge 0}\) defined by
$$\begin{aligned} q_t(x,y)=\frac{\psi (y)}{\psi (x)}\sum _{w\in W}\det (w)e^{(w(x)-x\vert \rho )}p_t^\rho (\overline{w(x)},y), \quad x,y\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R}), \end{aligned}$$
where \(\psi (x)=\sum _{w\in W}\det (w)e^{ (x\vert w(\rho )-\rho )}\). Propositions 6.1 and 6.4 imply that
$$\begin{aligned} q_t(x,y)=\frac{\mathbb {P}^\rho _y(T=+\infty )}{\mathbb {P}^\rho _x(T=+\infty )}\mathbb {P}_x(X_t=y, \,T> t), \quad x,y\in (\mathbb {R}\Lambda _0+\mathring{\mathfrak {h}}^*_\mathbb {R}), \end{aligned}$$
which achieves the proof. \(\square \)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Appliquées à Paris 5, Université Paris 5ParisFrance

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