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Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula
No full textInternational audienceLet G be reductive group over a non-Archimedean local field (e.g., GL(n)(Q(p))) and G(boolean OR)(C) its Langlands dual. Jacquet's Whittaker function on G is essentially proportional to the character of an irreducible representation of G(boolean OR)(C) (a Schur function if G = GL(n)(Q(p))). We propose a probabilistic approach to this claim, known as the Shintani-Casselman-Shalika formula, when the group G has at least one minuscule cocharacter. Thanks to random walks on the group, we start by establishing a Poisson kernel formula for the non-Archimedean Whittaker function. The expression and its ingredients are similar to the one previously obtained by the author in the Archimedean case, hence a unified point of view
Whittaker processes and Landau-Ginzburg potentials for flag manifolds
No full text61 pages, 2 figures; preliminary version; extends and replaces chapter 6 from the author's phD thesis, available as arXiv:1302.0902. Added description of Poisson boundary and consequencesThe Robinson-Schensted (RS) correspondence and its variants naturally give rise to integrable dynamics of non-intersecting particle systems. In previous work, the author exhibited a RS correspondence for geometric crystals by constructing a Littelmann path model, in general Lie type. Since this representation-theoretic map takes as input a continuous path on a Euclidian space, the natural starting measure is the Wiener measure. In this paper, we characterize the measures induced by Brownian motion through the RS map. On the one hand, the highest weight in the output is a remarkable Markov process (the Whittaker process), and can be interpreted as a weakly non-intersecting particle system which deforms Brownian motion in a Weyl chamber. One the other hand, the measure induced on geometric crystals is given by the Landau-Ginzburg potential for complete flag manifolds which appear in mirror symmetry. The measure deforms the uniform measure on string polytopes. Whittaker functions, which appear as volumes of geometric crystals, play the role of characters in the theory
A note on a Poissonian functional and a -deformed Dufresne identity
No full textInternational audienceIn this note, we compute the Mellin transform of a Poissonian exponential functional, the underlying process being a simple continuous time random walk. It shows that the Poissonian functional can be expressed in term of the inverse of a q-gamma random variable.The result interpolates between two known results. When the random walk has only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor. In the Brownian limit (q -> 1(-)), one recovers Dufresne's identity involving an inverse gamma random variable. Hence, one can see it as a q-deformed Dufresne identity
Littelmann path model for geometric crystals
No full text90 pages, 5 figures, 1 appendix; fixed typos and submitted; extends and replaces chapter 4 from the author's phD thesis, available as arXiv:1302.0902We construct a path model for geometric crystals in the sense of Berenstein and Kazhdan. Our model is in every way similar to Littelmann's and tropicalizes to his path model. This paper lays the foundational material for a subsequent work where we examine the measure induced on geometric crystals by Brownian motion. If we call Berenstein and Kazhdan's realization of geometric crystals the group picture, we prove that the path model projects onto the group picture thanks to a morphism of crystals that restricts to an isomorphism on connected components. This projection is in fact the geometric analogue of the Robinson-Schensted correspondence and involves solving a left-invariant differential equation on the Borel subgroup. Moreover, we identify the geometric Pitman transform introduced by Biane, Bougerol and O'Connell as the transform giving the path with highest weight, in the geometric crystal path model. This allows to prove a geometric version of Littelmann's independence theorem. The geometric Robinson-Schensted correspondence is detailed in a special section, because of its importance. Finally, we exhibit the Kashiwara and Sch\"utzenberger involutions in both the group picture and the path model. In an appendix, we explain how the left-invariant flow is related to the image of the Casimir element in Kostant's Whittaker model
Beta-gamma algebra identities and Lie-theoretic exponential functionals of Brownian motion
No full textInternational audienc
Archimedean Whittaker functions and geometric crystals
No full textNon UBCUnreviewedAuthor affiliation: Universität ZürichPostdoctora
Scattering of the Toda system and the Gaussian -ensemble
Full text linkThe classical Toda flow is a well-known integrable Hamiltonian system that
diagonalizes matrices. By keeping track of the distribution of entries and
precise scattering asymptotics, one can exhibit matrix models for log-gases on
the real line. These types of scattering asymptotics date back to fundamental
work of Moser.
More precisely, using the classical Toda flow acting on symmetric real
tridiagonal matrices, we give a "symplectic" proof of the fact that the
Dumitriu-Edelman tridiagonal model has a spectrum following the Gaussian
-ensemble.Comment: 13 pages, v1: Submitte
Some topics at the interface between probability theory, algebro-geometric structures,and mathematical physics
No full textA probabilistic approach to the Shintani-Casselman-Shalika formula
No full textRecall that Jacquet's Whittaker function for a group , in the non-Archimedean case, is essentially proportional to a character of an irreducible representation of the Langlands dual group - a Schur function in the case of . This statement is known as the Shintani-Casselman-Shalika formula.
In my opinion, Shintani's proof for is remarkably different from the more general proof by Casselman-Shalika. In this talk, I will present a probabilistic proof that is the natural generalisation of Shintani's. It explains the appearance of the Weyl character formula from a reflection principle for random walks.Non UBCUnreviewedAuthor affiliation: Institut de Mathématiques de ToulousePostdoctora
Some topics at the interface between probability theory, algebro-geometric structures,and mathematical physics
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