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1. CJM Online first

Graham, Robert; Pichot, Mikael
 A Free Product Formula for the Sofic Dimension It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality $s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3)$ where $\mathfrak{h}$ is the normalized Haar measure on $G$. Keywords:sofic groups, dynamical systems, orbit equivalence, free entropyCategory:20E06

2. CJM 2012 (vol 65 pp. 575)

 The Geometry and Fundamental Group of Permutation Products and Fat Diagonals Permutation products and their various fat diagonal'' subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included. Keywords:symmetric products, fundamental group, orbit stratificationCategories:14F35, 57F80

3. CJM 2009 (vol 62 pp. 94)

Kuo, Wentang
 The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects. Keywords:Langlands correspondence, nilpotent orbits, principal seriesCategories:22E50, 22E35

4. CJM 2006 (vol 58 pp. 1229)

Henniart, Guy; Lemaire, Bertrand
 IntÃ©grales orbitales tordues sur $\GL(n,F)$ et corps locaux proches\,: applications Soient $F$ un corps commutatif localement compact non archim\'edien, $G=\GL (n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\ere de $F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour les $\kappa$-int\'egrales orbitales r\'eguli\eres sur $G$ permettant, si $F$ est de caract\'eristique $>0$, de les relever \a la caract\'eristique nulle. On en d\'eduit deux r\'esultats nouveaux en caract\'eristique $>0$\,: le lemme fondamental'' pour l'induction automorphe, et une version simple de la formule des traces tordue locale d'Arthur reliant $\kappa$-int\'egrales orbitales elliptiques et caract\eres $\kappa$-tordus. Cette formule donne en particulier, pour une s\'erie $\kappa$-discr\ete de $G$, les $\kappa$-int\'egrales orbitales elliptiques d'un pseudo-coefficient comme valeurs du caract\ere $\kappa$-tordu. Keywords:corps local, reprÃ©sentation lisse, intÃ©grale orbitale tordue, induction automorphe, lemme fondamental, formule des traces locale, pseudo-coefficientCategory:22E50

5. CJM 2006 (vol 58 pp. 897)

Courtès, François
 Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \a l'espace de leurs restrictions \a l'alg\ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis. Keywords:reductive $p$-adic groups, orbital integrals, invariant distributionsCategories:22E35, 20G40

6. CJM 2005 (vol 57 pp. 750)

Sabourin, Hervé
 Sur la structure transverse Ã  une orbite nilpotente adjointe We are interested in Poisson structures to transverse nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $sl_n$, we construct some families of nilpotent orbits with quadratic transverse structures. Keywords:nilpotent adjoint orbits, conormal orbits, Poisson transverse structureCategories:22E, 53D

7. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

8. CJM 2003 (vol 55 pp. 91)

Choi, Man-Duen; Li, Chi-Kwong; Poon, Yiu-Tung
 Some Convexity Features Associated with Unitary Orbits Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U} (C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$ such that every matrix in the convex hull of $\mathcal{U}(C)$ can be written as the average of two matrices in $\mathcal{U}(C)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U} (C)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed. Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical rangeCategories:15A60, 15A42

9. CJM 2002 (vol 54 pp. 571)

Li, Chi-Kwong; Poon, Yiu-Tung
 Diagonals and Partial Diagonals of Sum of Matrices Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$\mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\}$$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed. Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular valuesCategories:15A42, 15A18

10. CJM 2001 (vol 53 pp. 944)

Ludwig, J.; Molitor-Braun, C.
 ReprÃ©sentations irrÃ©ductibles bornÃ©es des groupes de Lie exponentiels Let $G$ be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations $(T, \calU)$ of $G$ on a Banach space $\calU$ by giving a $G$-orbit in $\frn^*$ ($\frn$ being the nilradical of $\frg$), a topologically irreducible representation of $L^1(\RR^n, \o)$, for a certain weight $\o$ and a certain $n \in \NN$, and a topologically simple extension norm. If $G$ is not symmetric, \ie, if the weight $\o$ is exponential, we get a new type of representations which are fundamentally different from the induced representations. Soit $G$ un groupe de Lie r\'esoluble exponentiel. Nous caract\'erisons toutes les repr\'esentations $(T, \calU)$ continues born\'ees topologiquement irr\'eductibles de $G$ dans un espace de Banach $\calU$ \a l'aide d'une $G$-orbite dans $\frn^*$ ($\frn$ \'etant le radical nilpotent de $\frg$), d'une repr\'esentation topologiquement irr\'eductible de $L^1(\RR^n, \o)$, pour un certain poids $\o$ et un certain $n \in \NN$, d'une norme d'extension topologiquement simple. Si $G$ n'est pas sym\'etrique, c. \a d. si le poids $\o$ est exponentiel, nous obtenons un nouveau type de repr\'esentations qui sont fondamentalement diff\'erentes des repr\'esentations induites. Keywords:groupe de Lie rÃ©soluble exponentiel, reprÃ©sentation bornÃ©e topologiquement irrÃ©ductible, orbite, norme d'extension, sous-espace invariant, idÃ©al premier, idÃ©al primitifCategory:43A20

11. CJM 1998 (vol 50 pp. 525)

Brockman, William; Haiman, Mark
 Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here $\mu'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka polynomial~$\Klmt(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$. \LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition of~$\Klmt(q)$ as a sum of atomic'' polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum decomposition of the ideals yielding the $k[\Cmubar\cap \hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials. Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varietiesCategories:05E10, 14M99, 20G05, 05E15