Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Abstract

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations.By theorem 5.5, J λ (z) = E(e z,η(s) |σ(P w0 η(a), a ≤ s) when P w0 η(s) = λ and, since τ s η and η have the same law, J µ (z) = E(e z,τsη(t) |σ(P w0 τ s η(b), b ≤ t)) when P w0 τ s η(t) = µ. Therefore E(e z,η(s+t) |G s,t ) = J λ (z)J µ (z).On the other hand, by lemma 4.12, G s,t is contained in σ(P w0 η(r), r ≤ s + t), thus E(e z,η(s+t) |G s,t ) = E(E(e z,η(s+t) |σ(P w0 η(r), r ≤ s + t))|G s,t ) = E(J z (P w0 η(s + t))|G s,t ). It thus follows from theorem 5.15 thatTherefore, for all z ∈ V * ,By injectivity of the Fourier-Laplace transform this implies thatThe positive product formula gives a positive answer to a question of Rösler [35] for the radial Dunkl kernel. It shows that one can generalize the structure

A pentagonal crystal, the golden section, alcove packing and aperiodic tilings

“…For this reason we had been rather reluctant to push this particular point further. On the other hand we learned recently from F. Biane that he and his colleagues had constructed a Littelmann path model for all finite Coxeter groups [2], [3] but that they did not realize that the resulting crystals could be of Kashiwara type for a larger rank Lie algebra. For the moment we can only say that we expect that this to be true of their crystals for reflection groups in the plane via our construction and then perhaps to hold in general.…”

“…Incidentally we do not know how to use the Littelmann path model to give our desired explicit description of B(∞), though we believe this should be possible. Thus for the moment we would not expect the results of [3] to give Proposition 5.4 which is just one such example in a simple case which is surprisingly difficult to describe.…”

“…These integers determine the places in b at which the crystal operators enter via the rules (i), (ii) of 3. 3. As in 4.2, it is only certain differences that matter.…”

“…Consider the pair T 1 , T 2 of triangles given by 8. 3. Their sides have lengths {1, g, 1} and {g, g, 1} respectively, where g is the Golden Section.…”

System of Complex Brownian Motions Associated with the O’Connell Process

“…In the present paper, the limit a → 0 is called the combinatorial limit and an inverse of this procedure is said to be a geometric lifting in the sense of [4]. (See also [3].)…”