Expand all Collapse all | Results 26 - 45 of 45 |
26. CMB 2003 (vol 46 pp. 332)
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 |
27. CMB 2002 (vol 45 pp. 466)
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 |
28. CMB 2002 (vol 45 pp. 436)
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 |
29. CMB 2002 (vol 45 pp. 364)
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 |
30. CMB 2002 (vol 45 pp. 220)
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 |
31. CMB 2001 (vol 44 pp. 408)
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 |
32. CMB 2001 (vol 44 pp. 491)
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 |
33. CMB 2001 (vol 44 pp. 482)
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 |
34. CMB 2001 (vol 44 pp. 429)
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 |
35. CMB 2001 (vol 44 pp. 298)
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 |
36. CMB 2000 (vol 43 pp. 459)
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 |
37. CMB 2000 (vol 43 pp. 380)
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 |
38. CMB 2000 (vol 43 pp. 90)
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 |
39. 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 |
40. CMB 1999 (vol 42 pp. 393)
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 |
41. CMB 1998 (vol 41 pp. 463)
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 |
42. CMB 1998 (vol 41 pp. 368)
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 |
43. CMB 1997 (vol 40 pp. 376)
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 |
44. CMB 1997 (vol 40 pp. 183)
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 |
45. CMB 1997 (vol 40 pp. 72)
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 |