In this paper we give a new flavour to what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population of constant size N, assume that each individual at time 0 carries a distinct type, or allele. Consider the joint dynamics of these N alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time t. Call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent. Stimulated by this result, we investigate whether it extends (i) to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times, and (ii) to the general case of Λ-Fleming‒Viot processes.

Soit $\text{G}$ un groupe réductif $p$-adique, et soit $\text{R}$ un corps algébriquement clos. Soit $\text{ }\!\!\pi\!\!\text{ }$ une représentation lisse de $\text{G}$ dans un espace vectoriel $\text{V}$ sur $\text{R}$. Fixons un sous-groupe ouvert et compact $\text{K}$ de $\text{G}$ et une représentation lisse irréductible $\varrho$ de $\text{K}$ dans un espace vectoriel $\text{W}$ de dimension finie sur $\text{R}$. Sur l'espace $\text{Ho}{{\text{m}}_{\text{K}}}(\text{W,}\,\text{V)}$ agit l'algèbre d'entrelacement $\mathcal{H}(\text{G,}\,\text{K,}\,\text{W)}$. Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes ${{\text{V}}^{\vee }}$ et ${{\text{W}}^{\vee }}$, et donnons en particulier des conditions sur $\text{W}$ ou sur la caractéristique de $\text{R}$ pour que le comportement soit semblable au cas des représentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriétés générales de $\text{G}$. Nous terminons par une application á la théorie des types pour le groupe $\text{G}{{\text{L}}_{n}}$ et ses formes intérieures sur un corps local non archimédien.

We supply some relations that establish intertwining from duality and give a probabilistic interpretation. This is carried out in the context of discrete Markov chains, fixing up the background of previous relations established for monotone chains and their Siegmund duals. We revisit the duality for birth-and-death chains and the nonneutral Moran model, and we also explore the duality relations in an ultrametric-type dual that extends the Siegmund kernel. Finally, we discuss the sharp dual, following closely the Diaconis-Fill study.

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chainwhich is irreducible and reversible outside the absorbing point.This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues.In the discrete time setting, if the underlying Dirichlet eigenvalues(namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point)are nonnegative, we show that T is distributed as a mixture ofsums of independent geometric laws whose parameters are successiveDirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law.This result leads to a probabilistic interpretation of the spectrum,in terms of strong random times and local equilibria through a simple intertwining relation.Next this study is extended to the continuous time framework,wheregeometric laws have to be replaced by exponential distributions having the (opposite)Dirichlet eigenvalues of the generator as parameters.Returning to the discrete time setting we consider the influence ofnegative eigenvalues which are given another probabilistic meaning.These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

Let $F$ be a non-Archimedean local field. In an earlier paper, we constructed certain types (in the sense of Bushnell and Kutzko) in $SL_n (F)$, in fact enough to describe the non-supercuspidal components of the Bernstein decomposition of this group. We now give an almost fully explicit presentation of the Hecke algebras of these types.