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Authors
Ahmed Moussaoui
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In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify this in those known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the cuspidal support of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters a la Bernstein. We check that these constructions match under the Langlands correspondence and as consequence, we obtain the compatibility of the Langlands correspondence with parabolic induction.
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We extend a result of Matucci on the number of conjugacy classes of finite order elements in the Thompson group $T$. According to Liousse, if $ gcd(m-1,q)$ is not a divisor of $r$ then there does not exist element of order $q$ in the Brown-Thompson group $T_{r,m}$. We show that if $ gcd(m-1,q)$ is a divisor of $r$ then there are exactly $varphi(q). gcd(m-1,q)$ conjugacy classes of elements of order $q$ in $T_{r,m}$, where $varphi$ is the Euler function phi. As a corollary, we obtain that the Thompson group $T$ is isomorphic to none of the groups $T_{r,m}$, for $m ot=2$ and any morphism from $T$ into $T_{r,m}$, with $m ot=2$ and $r ot= 0$ $mod (m-1)$, is trivial.
We study the dynamics of surface homeomorphisms around isolated fixed points whose Poincar{e}-Lefschetz index is not equal to 1. We construct a new conjugacy invariant, which is a cyclic word on the alphabet ${ua, ra, da, la}$. This invariant is a refinement of the P.-L. index. It can be seen as a canonical decomposition of the dynamics into a finite number of sectors of hyperbolic, elliptic or indifferent type. The contribution of each type of sector to the P.-L. index is respectively -1/2, $+1/2$ and 0. The construction of the invariant implies the existence of some canonical dynamical structures.
This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 case. Of course this is a more involved situation because the existence of the smooth stratum for the hypersurface {f = 0} forces to consider three strata for the nearby cycles. And we already know that the smooth stratum is always tangled if it is not alone (see [B.84b] and the introduction of [B.03]). The new phenomenon is the role played here by a new cohomology group, denote by $H^n_{ccap S}(F)_{=1}$, of the Milnors fiber of f at the origin. It has the same dimension as $H^n(F)_{=1}$ and $H^n_c(F)_{=1}$, and it leads to a non trivial factorization of the canonical map $$ can : H^n_{ccap S}(F)_{=1} to H^n_c(F)_{=1},$$ and to a monodromic isomorphism of variation $$ var :H^n_{ccap S}(F)_{=1}to H^n_c(F)_{=1}.$$ It gives a canonical hermitian form $$ mathcal{H} : H^n_{ccap S}(F)_{=1} times H^n(F )_{=1} to mathbb{C}$$ which is non degenerate. This generalizes the case of an isolated singularity for the eigenvalue 1 (see [B.90] and [B.97]). The overtangling phenomenon for strata associated to the eigenvalue 1 implies the existence of triple poles at negative integers (with big enough absolute value) for the meromorphic continuation of the distribution $int_X |f |^{2lambda}square $ for functions f having semi-simple local monodromies at each singular point of {f =0}.
We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
Let $F$ be a quadratic extension of $mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $mathrm{GL}_2(F)$ are not of finite presentation.
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