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26. CMB 2009 (vol 40 pp. 376)

Gross, Benedict H.; Savin, Gordan
The dual pair $PGL_3 \times G_2$
Let $H$ be the split, adjoint group of type $E_6$ over a $p$-adic field. In this paper we study the restriction of the minimal representation of $H$ to the closed subgroup $PGL_3 \times G_2$.

Categories:22E35, and, 50, 11F70

27. CMB 2009 (vol 40 pp. 72)

Lee, Min Ho
Generalized Siegel modular forms and cohomology of locally symmetric varieties
We generalize Siegel modular forms and construct an exact sequence for the cohomology of locally symmetric varieties which plays the role of the Eichler-Shimura isomorphism for such generalized Siegel modular forms.

Categories:11F46, 11F75, 22E40

28. CMB 2008 (vol 51 pp. 60)

Janzen, David
F{\o}lner Nets for Semidirect Products of Amenable Groups
For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided.

Categories:22D05, 43A07, 22D15, 43A20

29. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.

Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion,
Categories:05D10, 20D60, 22A10

30. CMB 2007 (vol 50 pp. 460)

Spielberg, Jack
Weak Semiprojectivity for Purely Infinite $C^*$-Algebras
We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra
Categories:46L05, 46L80, 22A22

31. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
A Künneth Theorem for $p$-Adic Groups
Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$.

Categories:22E50, 18G15, 55U25

32. CMB 2007 (vol 50 pp. 291)

Sarkar, Rudra P.; Sengupta, Jyoti
Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type
We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principle
Categories:22E30, 43A85

33. CMB 2007 (vol 50 pp. 48)

Dvorsky, Alexander
Tensor Square of the Minimal Representation of $O(p,q)$
In this paper, we study the tensor product $\pi=\sigma^{\min}\otimes \sigma^{\min}$ of the minimal representation $\sigma^{\min}$ of $O(p,q)$ with itself, and decompose $\pi$ into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

Category:22e46

34. CMB 2006 (vol 49 pp. 578)

Muić, Goran
On the Structure of the Full Lift for the Howe Correspondence of $(Sp(n), O(V))$ for Rank-One Reducibilities
In this paper we determine the structure of the full lift for the Howe correspondence of $(Sp(n),O(V))$ for rank-one reducibilities.

Categories:22E35, 22E50, 11F70

35. CMB 2005 (vol 48 pp. 505)

Bouikhalene, Belaid
On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group
Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.
Categories:39B32, 39B42, 22D10, 22D12, 22D15

36. CMB 2004 (vol 47 pp. 439)

Parker, John R.
On the Stable Basin Theorem
The stable basin theorem was introduced by Basmajian and Miner as a key step in their necessary condition for the discreteness of a non-elementary group of complex hyperbolic isometries. In this paper we improve several of Basmajian and Miner's key estimates and so give a substantial improvement on the main inequality in the stable basin theorem.

Categories:22E40, 20H10, 57S30

37. CMB 2004 (vol 47 pp. 215)

Jaworski, Wojciech
Countable Amenable Identity Excluding Groups
A discrete group $G$ is called \emph{identity excluding\/} if the only irreducible unitary representation of $G$ which weakly contains the $1$-dimensional identity representation is the $1$-dimensional identity representation itself. Given a unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let $P_\mu$ denote the $\mu$-average $\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu$ on an identity excluding group and every unitary representation $\pi$ there exists and orthogonal projection $E_\mu$ onto a $\pi$-invariant subspace such that $s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50

38. CMB 2003 (vol 46 pp. 332)

Đoković, Dragomir Z.; Tam, Tin-Yau
Some Questions about Semisimple Lie Groups Originating in Matrix Theory
We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\cdot x)$ is star-shaped with respect to the origin.

Categories:15A45, 20G20, 22E60

39. CMB 2002 (vol 45 pp. 466)

Arthur, James
A Note on the Automorphic Langlands Group
Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

Categories:11R39, 22E55

40. CMB 2002 (vol 45 pp. 436)

Sawyer, P.
The Spherical Functions Related to the Root System $B_2$
In this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system $B_2$.

Categories:43A90, 22E30, 33C80

41. CMB 2002 (vol 45 pp. 364)

Deitmar, Anton
Mellin Transforms of Whittaker Functions
In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

Categories:11F30, 22E30, 11F70, 22E45

42. CMB 2002 (vol 45 pp. 220)

Hakim, Jeffrey; Murnaghan, Fiona
Globalization of Distinguished Supercuspidal Representations of $\GL(n)$
An irreducible supercuspidal representation $\pi$ of $G= \GL(n,F)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be ``distinguished'' by a subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if $\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue of this notion for and irreducible, automorphic, cuspidal representation associated to $\GL(n)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

Categories:22E50, 22E35, 11F70

43. CMB 2001 (vol 44 pp. 491)

Wang, Weiqiang
Resolution of Singularities of Null Cones
We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

Categories:14L35, 22G

44. CMB 2001 (vol 44 pp. 482)

Mezo, Paul
Matching of Weighted Orbital Integrals for Metaplectic Correspondences
We prove an identity between weighted orbital integrals of the unit elements in the Hecke algebras of $\GL(r)$ and its $n$-fold metaplectic covering, under the assumption that $n$ is relatively prime to any proper divisor of every $1 \leq j \leq r$.

Category:22E35

45. CMB 2001 (vol 44 pp. 429)

Henniger, J. P.
Ergodic Rotations of Nilmanifolds Conjugate to Their Inverses
In answer to a question posed in \cite{G}, we give sufficient conditions on a Lie nilmanifold so that any ergodic rotation of the nilmanifold is metrically conjugate to its inverse. The condition is that the Lie algebra be what we call quasi-graded, and is weaker than the property of being graded. Furthermore, the conjugating map can be chosen to be an involution. It is shown that for a special class of groups, the condition of quasi-graded is also necessary. In certain examples there is a continuum of conjugacies.

Categories:28Dxx, 22E25

46. CMB 2001 (vol 44 pp. 408)

Falbel, E.
Finite Groups Generated by Involutions on Lagrangian Planes of $\mathbf{C}^2$
We classify finite subgroups of $\SO(4)$ generated by anti-unitary involutions. They correspond to involutions fixing pointwise a Lagrangian plane. Explicit descriptions of the finite groups and the configurations of Lagrangian planes are obtained.

Categories:22E40, 53D99

47. CMB 2001 (vol 44 pp. 298)

Muić, Goran
A Proof of Casselman-Shahidi's Conjecture for Quasi-split Classical Groups
In this paper the author prove that standard modules of classical groups whose Langlands quotients are generic are irreducible. This establishes a conjecture of Casselman and Shahidi for this important class of groups.

Category:22E35

48. CMB 2000 (vol 43 pp. 459)

Ndogmo, J. C.
Properties of the Invariants of Solvable Lie Algebras
We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\bbC$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

Categories:17B30, 22E70

49. CMB 2000 (vol 43 pp. 380)

Shahidi, Freydoon
Twists of a General Class of $L$-Functions by Highly Ramified Characters
It is shown that given a local $L$-function defined by Langlands-Shahidi method, there exists a highly ramified character of the group which when is twisted with the original representation leads to a trivial $L$-function.

Categories:11F70, 22E35, 22E50

50. CMB 2000 (vol 43 pp. 90)

Muić, Goran; Savin, Gordan
Complementary Series for Hermitian Quaternionic Groups
Let $G$ be a hermitian quaternionic group. We determine complementary series for representations of $G$ induced from super-cuspidal representations of a Levi factor of the Siegel maximal parabolic subgroup of $G$.

Category:22E35
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