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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. CMB 2000 (vol 43 pp. 47)

 A Property of Lie Group Orbits Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action. Categories:32B20, 22E15

39. CMB 1999 (vol 42 pp. 393)

Savin, Gordan
 A Class of Supercuspidal Representations of $G_2(k)$ Let $H$ be an exceptional, adjoint group of type $E_6$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of $G_2(k)$. Categories:22E35, 22E50, 11F70

40. CMB 1998 (vol 41 pp. 463)

Moran, Alan
 The right regular representation of a compact right topological group We show that for certain compact right topological groups, $\overline{r(G)}$, the strong operator topology closure of the image of the right regular representation of $G$ in ${\cal L}({\cal H})$, where ${\cal H} = \L2$, is a compact topological group and introduce a class of representations, ${\cal R}$, which effectively transfers the representation theory of $\overline{r(G)}$ over to $G$. Amongst the groups for which this holds is the class of equicontinuous groups which have been studied by Ruppert in [10]. We use familiar examples to illustrate these features of the theory and to provide a counter-example. Finally we remark that every equicontinuous group which is at the same time a Borel group is in fact a topological group. Category:22D99

41. CMB 1998 (vol 41 pp. 368)

Moskowitz, Martin; Wüstner, Michael
 Exponentiality of certain real solvable Lie groups In this article, making use of the second author's criterion for exponentiality of a connected solvable Lie group, we give a rather simple necessary and sufficient condition for the semidirect product of a torus acting on certain connected solvable Lie groups to be exponential. Categories:22E25, 22E15

42. CMB 1997 (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

43. CMB 1997 (vol 40 pp. 183)

Kepert, Andrew G.
 The range of group algebra homomorphisms A characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P.~J.~Cohen. Categories:43A22, 22B10, 46J99

44. CMB 1997 (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
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