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1. CMB Online first
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Free Groups Generated by Two Heisenberg Translations In this paper, we will discuss the groups generated by two
Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free.
Keywords:free group, Heisenberg group, complex triangle group Categories:30F40, 22E40, 20H10 |
2. CMB Online first
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On Induced Representations Distinguished by Orthogonal Groups Let $F$ be a local non-archimedean field of characteristic zero. We
prove that a representation of $GL(n,F)$ obtained from irreducible
parabolic induction of supercuspidal representations is distinguished
by an orthogonal group only if the inducing data is distinguished by
appropriate orthogonal groups. As a corollary, we get that an
irreducible representation induced from supercuspidals that is
distinguished by an orthogonal group is metic.
Keywords:distinguished representation, parabolic induction Category:22E50 |
3. CMB 2011 (vol 55 pp. 870)
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Left Invariant Einstein-Randers Metrics on Compact Lie Groups In this paper we study left invariant Einstein-Randers metrics on compact Lie
groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple
Lie groups with the underlying Riemannian metric naturally reductive.
Further, we completely determine the identity component of the group of
isometries for this type of metrics on simple groups. Finally, we study some
geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature
of such metrics.
Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature Categories:17B20, 22E46, 53C12 |
4. CMB 2011 (vol 56 pp. 116)
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Central Extensions of Loop Groups and Obstruction to Pre-Quantization An explicit construction of a pre-quantum line bundle for the moduli
space of flat $G$-bundles over a Riemann surface is given, where $G$
is any non-simply connected compact simple Lie group. This work helps
to explain a curious coincidence previously observed between
Toledano Laredo's work classifying central extensions of loop groups
$LG$ and the author's previous work on the obstruction to
pre-quantization of the moduli space of flat $G$-bundles.
Keywords:loop group, central extension, prequantization Categories:53D, 22E |
5. CMB 2011 (vol 54 pp. 663)
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Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$
related to a basis $\Delta$ for the root system $\Phi$ associated with
$W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We
show that the set of $\theta$-twisted involutions in $W$,
$\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one
to one correspondence with the set of regular involutions
$\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are
characterized by sequences in $\Sigma$ which induce an ordering called
the Richardson-Springer Poset. In particular, for $\Phi$ irreducible,
the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$,
for nontrivial $\theta$ is identical to the descending
Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.
Categories:20G15, 20G20, 22E15, 22E46, 43A85 |
6. CMB 2010 (vol 54 pp. 44)
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Star-Shapedness and $K$-Orbits in Complex Semisimple Lie Algebras
Given a complex semisimple Lie algebra
$\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact
real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to
\mathfrak{h}$ be the orthogonal projection (with respect to the
Killing form) onto the Cartan subalgebra
$\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a
maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in
\mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is
the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show
that it is star-shaped. The result extends a result of Tsing. We also
consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$,
where $f$ is a linear functional on $\mathfrak{g}$. We establish the
star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras
of type $B$.
Categories:22E10, 17B20 |
7. CMB 2010 (vol 54 pp. 126)
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Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups
We prove that the fundamental solutions
of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$
on the anisotropic Heisenberg groups are tempered distributions and have
meromorphic continuation in $\alpha$ with simple poles. We compute the
residues and find the partial fundamental solutions
at the poles. We also find formulas for the
fundamental solutions for some matrix-valued
Kohn type sub-Laplacians
on H-type groups.
Categories:22E30, 35R03, 43A80 |
8. CMB 2007 (vol 50 pp. 440)
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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 |
9. CMB 2007 (vol 50 pp. 291)
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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 |
10. CMB 2006 (vol 49 pp. 578)
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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 |
11. CMB 2004 (vol 47 pp. 439)
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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 |
12. CMB 2003 (vol 46 pp. 332)
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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 |
13. CMB 2002 (vol 45 pp. 466)
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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 |
14. CMB 2002 (vol 45 pp. 436)
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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 |
15. CMB 2002 (vol 45 pp. 364)
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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 |
16. CMB 2002 (vol 45 pp. 220)
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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 |
17. CMB 2001 (vol 44 pp. 482)
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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 |
18. CMB 2001 (vol 44 pp. 429)
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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 |
19. CMB 2001 (vol 44 pp. 408)
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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 |
20. CMB 2001 (vol 44 pp. 298)
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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 |
21. CMB 2000 (vol 43 pp. 459)
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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 |
22. CMB 2000 (vol 43 pp. 380)
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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 |
23. CMB 2000 (vol 43 pp. 90)
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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 |
24. CMB 2000 (vol 43 pp. 47)
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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 |
25. CMB 1999 (vol 42 pp. 393)
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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 |
