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We study a random walk on the subgroup of lower triangular matrices of SL$_2$, with i.i.d. increments. We prove that the process of the lower corner of the random walk satisfies a Rogers-Pitman criterion to be a Markov chain if and only if the increments are distributed according to a Generalized Inverse Gaussian (GIG) law on their diagonals. For this, we prove a new characterization of these laws. We prove a discrete-time version of the Dufresne identity. We show how to recover the Matsumoto-Yor theorem by taking the continuous limit of the random walk.

We study the space $S(X)^I$ of smooth functions on a symmetric space $X=G/H$ invariant to the action of an Iwahori subgroup $I$, as a module over $\mathcal{H}(G,I)$, the Iwahori Hecke algebra of a p-adic group $G$. We present a description of this module that generalizes the description given to $\mathcal{H}(G,I)$ by Iwahori and Matsumoto., Comment: 35 pages, 2 figures

The graded Iwahori--Matsumoto involution $\mathbb{IM}$ is an algebra involution on a graded Hecke algebra closely related to the more well-known Iwahori--Matsumoto involution on an affine Hecke algebra. It induces an involution on the Grothendieck group of complex finite-dimensional representations of $\mathbb{H}$. When $\mathbb{H}$ is a geometric graded Hecke algebra (in the sense of Lusztig) associated to a connected complex reductive group $G$, the irreducible representations of $\mathbb{H}$ are parametrised by a set $\mathcal{M}$ consisting of certain $G$-conjugacy classes of quadruples $(e,s,r_0,\psi)$ where $r_0 \in \mathbb{C}$, $e \in \mathrm{Lie}(G)$ is nilpotent, $s \in \mathrm{Lie}(G)$ is semisimple, and $\psi$ is some irreducible representation of the group of components of the simultaneous centraliser of $(e,s)$ in $G$. Let $\bar Y$ be an irreducible tempered representation of $\mathbb{H}$ with real infinitesimal character. Then $\mathbb{IM}(\bar Y) = \bar Y(e',s,r_0,\psi')$ for some $(e',s,r_0,\psi') \in \mathcal{M}$. The main result of this paper is to give an explicit algorithm that computes the $G$-orbit of $e'$ for $G = \mathrm{Sp}(2n,\mathbb{C})$ and $G = \mathrm{SO}(N,\mathbb{C})$. As a key ingredient of the main result, we also prove a generalisation of the main theorems of Waldspurger 2019 (for $\mathrm{Sp}(2n,\mathbb{C})$) and La 2024 (for $\mathrm{SO}(N,\mathbb{C})$) regarding certain maximality properties of generalised Springer representations., Comment: 37 pages, 4 tables, comments are welcome, updated version

We present a proof of a generalization of the theorem of H.~Matsumoto on Coxeter groups. Our generalized version is applicable to "graphs admitting geometric realization". The original version of the theorem for Coxeter groups is a special case when applied to the Cayley graph and the geometric representation of a Coxeter group. Our version of Matsumoto theorem is also applicable to skeleta, graphs that were defined in the recent paper by the authors on root Lie superalgebras., Comment: 10 pages, minor change in ntroduction

In 2017, Garunk\v{s}tis, Laurin\v{c}ikas and Macaitien\.{e} proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in various zeta-functions and $L$-functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions., Comment: 9 pages

This paper presents a multidimensional extension of the Matsumoto-Yor properties related to exponential functionals of drifted Brownian motion. The extension involves the interaction of geometric Brownian motions which are indexed by the vertices of a finite weighted graph, and the random potential associated with the Vertex Reinforced Jump process on this graph. We prove in this context a counterpart of Lamperti's transformation, of the Markov property of the Matsumoto-Yor process and of the intertwining relation., Comment: Extended version of arXiv:2004.10692 with a new author Keywords: exponential functional of Brownian motion, Inverse Gaussian law, vertex reinforced jump process, Pitman 2M-B theorem, Pitman transform

Recently, we have studied the Finsler space with h-Matsumoto change and found Cartan connection for the transformed space [2]. In this paper, we have discussed certain geometrical properties of the hypersurface of a Finsler space for the h-Matsumoto change.

In the present paper, we have studied the Matsumoto change $\overline{L}(x,y)= \frac{L^{2}(x,y)}{L(x,y) - \beta(x,y)} $ with an \textsl{h-}vector $b_{i}(x,y)$. We have derived some fundamental tensors for this transformation. We have also obtained the necessary and sufficient condition for which the Cartan connection coefficients for both the spaces $F^{n}=(M^n,L)$ and $\overline{F}^{\,n}=(M^{n},\overline{L})$ are same.

If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B} $$ are independent, we prove that this happens if and only if the $A$ and $B$ have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack (2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor (2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg - de Vries equation, where the above result was conjectured. We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open., Comment: 17 pages

We establish analogues of the geometric Pitman $2M-X$ theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. The Dufresne type identity provides another example of a stochastic matrix recursion, as considered by Chamayou and Letac (J. Theoret. Probab. 12, 1999), that admits an explicit solution., Comment: 31 pages

We give a Matsumoto-type presentation of $K_2$-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie's result for loop groups. Our main idea is due to U. Rehmann's approach in case of division rings., Comment: 30 pages

We study the Matsumoto-Yor property in free probability. We prove three characterizations of free-GIG and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are subordination and Boolean cumulants. In particular, we establish a new connection between additive subordination function and Boolean cumulants.

A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, the distribution of these hitting times is related to a random potential that appears in the study of the vertex-reinforced jump process. The aim of this paper is to prove a multivariate version of the Matsumoto-Yor opposite drift theorem, by applying a Lamperti-type time change to the previous family of interacting Brownian motions. Difficulties arise since the time change progresses at different speeds on different coordinates.

In this paper we consider the Matsumoto metric $F=\frac{\alpha^2}{\alpha-\beta}$, on the three dimensional real vector space and obtain the partial differential equations that characterize the minimal surfaces which are graphs of smooth functions and then we prove that plane is the only such surface. We also obtain the partial differential equation that characterizes the minimal translation surfaces and show that again plane is the only such surface.