Random walks in Weyl chambers and crystals

Abstract

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined in [1] for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. For the defining representation V of gl n , we notably recover the main result of [19]. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice. This theorem also have applications in representation theory since it permits to precise the behavior of some outer multiplicities for large dominant weights.

Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula

“…This is exactly corollary 7.7 in [LLP12], which is proved using the Littelmann path model, also in the minuscule case. We give an elementary proof using the reflection principle.…”

“…, T with increments in W Λ ∨ is a crystal element which is obtained by successive tensor products of elements from the crystal with highest weight Λ ∨ . For further explanations, we refer to [LLP12]. In fact, Proposition 3.10 holds in the context of general Kac-Moody groups as proved in the subsequent paper [LLP13] of Lesigne et al, but one has to observe Littelmann paths continuously in time.…”

t-Martin Boundary of Killed Random Walks in the Quadrant

“…To compute the exact values of the constants α and β , a series expansion of (12) around 0 is enough. In the case t > t 0 but p = X(y 2 ), the pole of 1 w(x)−w(p) at p is of order 2 (indeed, around p ∈ {X(y 1 ), X(y 2 )}, the equality w(x) = w(x) yields w ′ (p) = 0), and the same expression as (15) can be obtained.…”

“…has no pole in D M (since w is injective in D M , see Definition 1, the function 1 w(x)−w(p) has a pole of order 1 at p, as soon as p ∈ D M ) and is continuous on D M . The function (15) also satisfies the condition f (x) = f (x) on the boundary. Hence we can use the above remark to conclude that (15) is a constant function.…”

“…The new conformal mapping is an algebraic function of w. To find it we can proceed as in the proof of Theorem 2, by compensating the poles of w in the new curve M (see (15)). Changing accordingly the values of the constants α and β and replacing L(x, 0) by x 2 L(1/x, 0) yields the correct statement of Theorem 2 for the step set {p −k,ℓ }.…”