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2,800 |
The moment mapping for a unitary representation
|
math.RT
|
For any unitary representation of an arbitrary Lie group I construct a moment
mapping from the space of smooth vectors of the representation into the dual of
the Lie algebra. This moment mapping is equivariant and smooth. For the space
of analytic vectors the same construction is possible and leads to a real
analytic moment mapping.
|
math
|
2,801 |
All unitary representations admit moment mappings
|
math.RT
|
This is a review of [Michor, Peter W.: The moment mapping for a unitary
representation, Ann. Global Anal. Geometry, 8, No 3(1990), 299--313] including
a careful description of calculus in infinite dimensions. For any unitary
representation of an arbitrary Lie group I construct a moment mapping from the
space of smooth vectors of the representation into the dual of the Lie algebra.
This moment mapping is equivariant and smooth. For the space of analytic
vectors the same construction is possible and leads to a real analytic moment
mapping.
|
math
|
2,802 |
Nilpotent orbits, normality, and Hamiltonian group actions
|
math.RT
|
Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex
semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson
bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the
unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The
action of $\g'\simeq R[2]$ exponentiates to an action of the corresponding Lie
group $G'$ on a $G'$-cover $M'$ of a nilpotent orbit in $\g'$ such that $M$ is
open dense in $M'$. We determine all such pairs $(\g\subset\g')$.
|
math
|
2,803 |
Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations
|
math.RT
|
In this paper we study the reducibility, composition series and unitarity of
the components of some degenerate principal series representations of
$\RMO(p,q)$, $\RMU(p,q)$ and $\SP(p,q)$. This is done by realizing these
representations in paces of homogeneous functions on light cones and writing
down the explicit actions of the universal enveloping algebra of the group
concerned.
|
math
|
2,804 |
An external approach to unitary representations
|
math.RT
|
The main aim of this paper is to present the ideas which lead first to the
solution of the unitarizability problem for $\GL(n)$ over nonarchimedean local
fields and to the recognition that the same result holds over archimedean local
fields, a result which was proved by Vogan using an internal approach. Let us
say that the approach that we are going to present may be characterized as
external. At no point do we go into the internal structure of representations.
|
math
|
2,805 |
On the convergence of the zeta function for certain prehomogeneous vector spaces
|
math.RT
|
Let (G,V) be an irreducible prehomogeneous vector space defined over a number
field k, P in k[V] a relative invariant polynomial, and X a rational character
of G such that P(gx)=X(g)P(x). Let V_k^{ss}={x \in V_k such that P(x) is not
equal to 0}. For x in V_k^{ss}, let G_x be the stabilizer of x, and G_x^0 the
connected component of 1 of G_x. We define L_0 to be the set of x in V_k^{ss}
such that G_x^0 does not have a non-trivial rational character. We study the
zeta function for (G,V).
|
math
|
2,806 |
The moment map for a multiplicity free action
|
math.RT
|
Let $K$ be a compact connected Lie group acting unitarily on a
finite-dimensional complex vector space $V$. One calls this a {\em
multiplicity-free} action whenever the $K$-isotypic components of $\C[V]$ are
$K$-irreducible. We have shown that this is the case if and only if the moment
map $\tau:V\rightarrow\k^*$ for the action is finite-to-one on $K$-orbits. This
is equivalent to a result concerning \gp s associated with Heisenberg groups
that is motivated by the Orbit Method. Further details of this work will be
published elsewhere.
|
math
|
2,807 |
Exceptional Theta-correspondences I
|
math.RT
|
Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In
this paper we study the restriction of the minimal representation of $G$ to
various dual pairs in $G$. For example, the restriction of the minimal
representation of $E_7$ to the dual pair $G_2 \times{}$Sp(6) gives the
non-endoscopic Langlands lift of irreducible representations of $G_2$ to Sp(6).
|
math
|
2,808 |
Inversion of an integral transform and ladder representations of U(1,q)
|
math.RT
|
An integral transform for G=U(1,q) is studied. The transform maps the
positive spin ladder representations of G on a Bargmann-Segal-Fock space
F_n^1,q into a space of polynomial-valued functions on the bounded realization
B^q of G/K. An inversion is given for the transform and unitary structures are
given for the geometric realization of the positive spin ladder representations
over G/K.
|
math
|
2,809 |
On the automorphism groups of complex homogeneous spaces
|
math.RT
|
If G is a (connected) complex Lie Group and Z is a generalized flag manifold
for G, the the open orbits D of a (connected) real form G_0 of G form an
interesting class of complex homogeneous spaces, which play an important role
in the representation theory of G_0. We find that the group of automorphisms,
i.e., the holomorphic diffeomorphisms, is a finite-dimensional Lie group,
except for a small number of open orbits, where it is infinite dimensional. In
the finite-dimensional case, we determine its structure. Our results have some
consequences in representation theory.
|
math
|
2,810 |
On the Shintani zeta function for the space of binary tri-Hermitian forms
|
math.RT
|
In this paper, we consider the most non-split parabolic D_4 type
prehomogeneous vector space. The vector space is an analogue of the space of
Hermitian forms. We determine the principal part of the zeta function.
|
math
|
2,811 |
On Shintani zeta functions for GL(2)
|
math.RT
|
In this paper we consider an analogue of the zeta function for not
necessarily prehomogeneous representations of GL(2) and compute some of the
poles.
|
math
|
2,812 |
Harmonic Analysis on the Finite Twisted Poincaré Upper Half Plane
|
math.RT
|
We prove that the induced representation from a non trivial character of the
Coxeter torus of GL$(2,F)$, for a finite field $F$, is multiplicity-free; we
give an explicit description of the corresponding (twisted) spherical functions
and a version of the Heisenberg Uncertainty Principle.
|
math
|
2,813 |
Classification of $N$-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of $Spin(p,q)$
|
math.RT
|
We classify extended Poincar\'e Lie super algebras and Lie algebras of any
signature (p,q), that is Lie super algebras and Z_2-graded Lie algebras g = g_0
+ g_1, where g_0 = so(V) + V is the (generalized) Poincar\'e Lie algebra of the
pseudo Euclidean vector space V = R^{p,q} of signature (p,q) and g_1 = S is the
spinor so(V)-module extended to a g_0-module with kernel V. The remaining super
commutators {g_1,g_1} (respectively, commutators [g_1, g_1]) are defined by an
so(V)-equivariant linear mapping vee^2 g_1 -> V (respectively, wedge^2 g_1 ->
V). Denote by P^+(n,s) (respectively, P^-(n,s)) the vector space of all such
Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s
= p - q is the signature. The description of P^+-(n,s) reduces to the
construction of all so(V)-invariant bilinear forms on S and to the calculation
of three Z_2-valued invariants for some of them.
This calculation is based on a simple explicit model of an irreducible
Clifford module S for the Clifford algebra Cl_{p,q} of arbitrary signature
(p,q). As a result of the classification, we obtain the numbers L^+-(n,s) =
\dim P^+-(n,s) of independent Lie super algebras and algebras, which take
values 0,1,2,3,4 or 6. Due to Bott periodicity, L^+-(n,s) may be considered as
periodic functions with period 8 in each argument. They are invariant under the
group Gamma generated by the four reflections with respect to the axes n=-2,
n=2, s-1 = -2 and s-1 = 2. Moreover, the reflection (n,s) -> (-n,s) with
respect to the axis s=0 interchanges L^+ and L^- : L^+(-n,s) = L^-(n,s).
|
math
|
2,814 |
Some results on the admissible representations of non-connected reductive p-adic groups
|
math.RT
|
We examine the theory of induced representations for non-connected reductive
$p$-adic groups for which $G/G^0$ is abelian. We first examine the structure of
those representations of the form $\Ind_{P^0}^G(\sigma),$ where $P^0$ is a
parabolic subgroup of $G^0$ and $\s$ is a discrete series representation of the
Levi component of $P^0.$ Here we develop a theory of $R$--groups, extending the
theory in the connected case. We then prove some general results in the theory
of representations of non-connected $p$-adic groups whose component group is
abelian. We define the notion of cuspidal parabolic for $G$ in order to give a
context for this discussion. Intertwining operators for the non-connected case
are examined and the notions of supercuspidal and discrete series are defined.
Finally, we examine parabolic induction from a cuspidal parabolic subgroup of
$G.$ Here we also develop a theory of $R$--groups, and show that these groups
parameterize the induced representations in a manner that is consistent with
the connected case and with the first set of results as well.
|
math
|
2,815 |
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups G(r,p,n)
|
math.RT
|
Iwahori-Hecke algebras for the infinite series of complex reflection groups
$G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Brou\'e
and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas
for computing the irreducible characters of these algebras. Our method is a
generalization of that in our earlier paper in which we derived
Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of
the classical Weyl groups. In both papers we have been motivated by C. Greene,
who gave a new derivation of the Murnaghan-Nakayama formula for irreducible
symmetric group characters by summing diagonal matrix entries in Young's
seminormal representations. We use the analogous representations of the
Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike.
|
math
|
2,816 |
Seminormal representations of Weyl groups and Iwahori-Hecke algebras
|
math.RT
|
The purpose of this paper is to describe a general procedure for computing
analogues of Young's seminormal representations of the symmetric groups. The
method is to generalize the Jucys-Murphy elements in the group algebras of the
symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The
combinatorics of these elements allows one to compute irreducible
representations explicitly and often very easily. In this paper we do these
computations for Weyl groups and Iwahori-Hecke algebras of types $A_n$, $B_n$,
$D_n$, $G_2$. Although these computations are in reach for types $F_4$, $E_6$,
and $E_7$, we shall, in view of the length of the current paper, postpone this
to another work.
|
math
|
2,817 |
On local coefficients for non-generic representations of some classical groups
|
math.RT
|
This paper is concerned with representations of split orthogonal and
quasi-split unitary groups over a nonarchimedean local field which are not
generic, but which support a unique model of a different kind, the generalized
Bessel model. The properties of the Bessel models under induction are studied,
and an analogue of Rodier's theorem concerning the induction of Whittaker
models is proved for Bessel models which are minimal in a suitable sense. The
holomorphicity in the induction parameter of the Bessel functional is
established. Last, local coefficients are defined for each irreducible
supercuspidal representation which carries a Bessel functional and also for a
certain component of each representation parabolically induced from such a
supercuspidal.
|
math
|
2,818 |
Groupoids: unifying internal and external symmetry
|
math.RT
|
The aim of this paper is to explain, mostly through examples, what groupoids
are and how they describe symmetry. We will begin with elementary examples,
with discrete symmetry, and end with examples in the differentiable setting
which involve Lie groupoids and their corresponding infinitesimal objects, Lie
algebroids.
|
math
|
2,819 |
Prehomogeneous vector spaces and ergodic theory I
|
math.RT
|
This is part one of a series of papers. In this series of papers, we consider
problems analogous to the Oppenheim conjecture from the viewpoint of
prehomogeneous vector spaces.
|
math
|
2,820 |
Prehomogeneous vector spaces, Eisenstein series, and invariant theory
|
math.RT
|
In this paper, we give an introduction to the rationality of the equivariant
Morse stratification, and state the author's results on zeta functions of
prehomogeneous vector spaces.
|
math
|
2,821 |
Prehomogeneous vector spaces and ergodic theory II
|
math.RT
|
We apply M. Ratner's theorem on closures of unipotent orbits to the study of
three families of prehomogeneous vector spaces. As a result, we prove analogues
of the Oppenheim Conjecture for simultaneous approximation by values of certain
alternating bilinear forms in an even number of variables and certain
alternating trilinear forms in six and seven variables.
|
math
|
2,822 |
Correction to "Measurable quotients of unipotent translations on homogeneous spaces
|
math.RT
|
The statements of Main~Theorem~1.1 and Theorem~2.1 of the author's paper
[\emph{Trans.\ Amer.\ Math.\ Soc.}\ {\bf 345} (1994) 577--594] should assume
that $\Gamma $~is discrete and $G$~is connected. (Cors.~1.3, 5.6, and~5.8 are
affected similarly.) These restrictions can be removed if the conclusions of
the results are weakened to allow for the possible existence of transitive,
proper subgroups of~$G$. In this form, the results can be extended to the
setting where $G$ is a product of real and $p$-adic Lie groups.
|
math
|
2,823 |
Superrigid subgroups of solvable Lie groups
|
math.RT
|
Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie
group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If
$\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional
representation of~$\Gamma $,we show that $\alpha$ virtually extends to a
continuous representation~$\sigma $ of~$G$. Furthermore, the image of~$\sigma$
is contained in the Zariski closure of the image of~$\alpha $.
When $\Gamma$ is not discrete, the same conclusions are true if we make the
additional assumption that the closure of $[\Gamma, \Gamma]$ is a finite-index
subgroup of $[G,G] \cap \Gamma$ (and $\Gamma$ is closed and $\alpha$ is
continuous).
|
math
|
2,824 |
Correction to "Zero-entropy affine maps on homogeneous spaces
|
math.RT
|
Proposition~6.4 of the author's paper {\origpaper} is incorrect. This invalid
proposition was used in the proof of Corollary~6.5, so we provide a new proof
of the latter result.
|
math
|
2,825 |
Cohomology at infinity and the well-rounded retract for general Linear Groups
|
math.RT
|
Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let
$\Gamma$ be an arithmetic subgroup of $\bold G(\Q)$. Let $X$ be the symmetric
space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology
(mod torsion) of the space $X/\Gamma$ is the same as the cohomology of
$\Gamma$. In turn, $X/\Gamma$ will have the same cohomology as $W/\Gamma$, if
$W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation
retract of $X$ by a $\Gamma$-equivariant deformation retraction, that
$W/\Gamma$ is compact, and that $\dim W$ equals the virtual cohomological
dimension (vcd) of $\Gamma$. Then $W$ can be given the structure of a cell
complex on which $\Gamma$ acts cellularly, and the cohomology of $W/\Gamma$ can
be found combinatorially.
|
math
|
2,826 |
Prehomogeneous vector spaces and field extensions III
|
math.RT
|
In this paper, we determine the rational orbit decomposition for two
prehomogeneous vector spaces associated with the simple group of type G_2.
|
math
|
2,827 |
Prehomogeneous vector spaces and ergodic theory III
|
math.RT
|
Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a
prehomogeneous vector space (see [22], [26], [25], for the definition of
prehomogeneous vector spaces). A non-constant polynomial \delta(x) on V is
called a relative invariant polynomial if there exists a character \chi such
that \delta(gx)=\chi(g)\delta(x). Such \delta(x) exists for our case and is
essentially unique. So we define V^{ss}={x in V such that \delta(x) is not
equal to 0}. For x in V_R^{ss}, let H_{x R+}^0 be the connected component of 1
in classical topology of the stabilizer H_{x R}. We will prove that if x in
V_R^ss is "sufficiently irrational", H_{x R+}^0 H_Z is dense in H_R.
|
math
|
2,828 |
A remark on the regularity of prehomogeneous vector spaces
|
math.RT
|
In this note, we prove that if $(G,V)$ is a prehomogeneous vector space over
any field $k$ such that the stabilizer of a generic point is reductive, the set
of semi-stable points is a single orbit over the separable closure of $k$.
|
math
|
2,829 |
Ratner's theorem and invariant theory
|
math.RT
|
In this note, we consider applications of Ratner's theorem to constructions
of families of polynomials with dense values on the set of primitive integer
points from the viewpoint of invariant theory.
|
math
|
2,830 |
Simple Lie algebras which generalize Witt algebras
|
math.RT
|
We introduce a new class of simple Lie algebras $W(n,m)$ that generalize the
Witt algebra by using "exponential" functions, and also a subalgebra $W^*(n,m)$
thereof; and we show each derivation of $W^*(1,0)$ can be written as a sum of
an inner derivation and a scalar derivation. The Lie algebra $W(n,m)$ is
$Z$-graded and is infinite growth.
|
math
|
2,831 |
Generalized W-type and H-type algebras
|
math.RT
|
It is well known that the Poisson Lie algebra is isomorphic to the
Hamiltonian Lie algebra. We show that the Poisson Lie algebra can be embedded
properly in the special type Lie algebra. We also generalize the Hamiltonian
Lie algebra using exponential functions, and we show that these Lie algebras
are simple.
|
math
|
2,832 |
Down-up Algebras
|
math.RT
|
The algebra generated by the down and up operators on a differential
partially ordered set (poset) encodes essential enumerative and structural
properties of the poset. Motivated by the algebras generated by the down and up
operators on posets, we introduce here a family of infinite-dimensional
associative algebras called down-up algebras. We show that down-up algebras
exhibit many of the important features of the universal enveloping algebra
$U(\fsl)$ of the Lie algebra $\fsl$ including a Poincar\'e-Birkhoff-Witt type
basis and a well-behaved representation theory. We investigate the structure
and representations of down-up algebras and focus especially on Verma modules,
highest weight representations, and category $\mathcal O$ modules for them. We
calculate the exact expressions for all the weights, since that information has
proven to be particularly useful in determining structural results about
posets.
|
math
|
2,833 |
Approximate representations and Virasoro algebra
|
math.RT
|
In the paper there are investigated various approximate representations of
the infinite dimensional $\Bbb Z$--graded Lie algebras: the Witt algebra of all
Laurent polynomial vector fields on a circle and its one-dimensional nontrivial
central extension, the Virasoro algebra, by the infinite dimensional hidden
symmetries in the Verma modules over the Lie algebra $\sl(2,C)$. There are
considered as asymptotic representations "$mod O(\hbar^n)$" and representations
up to a class $\frak S$ of operators (compact operators, Hilbert-Schmidt
operators and finite-rank operators) as cases, which combine both types of
approximations and in these cases an effect of noncommutativity of the order of
their perform is explicated, that perhaps is underlied by a more general
fundamental fact of deviations between the asymptotical theory of
pseudodifferential operators and the pseudodifferential calculus on the
asymptotic manifolds.
|
math
|
2,834 |
Tensor products of singular representations and an extension of the theta-correspondence
|
math.RT
|
In this paper we consider the problem of decomposing tensor products of
certain singular unitary representations of a semisimple Lie group G. Using
explicit models for these representations (constructed earlier by one of us) we
show that the decomposition is controlled by a reductive homogeneous space
G'/H'. Our procedure establishes a correspondence between certain unitary
representations of G and those of G'. This extends the usual
theta--correspondence for dual reductive pairs. As a special case we obtain a
correspondence between certain representations of real forms of E_7 and F_4.
|
math
|
2,835 |
Generalized Spencer Cohomology and filtered Deformations of Z-graded Lie Superalgebras
|
math.RT
|
In this paper we introduce generalized Spencer cohomology for finite depth
Z-graded Lie (super)algebras. We develop a method of finding filtered
deformations of such Z-graded Lie (super)algebras based on this cohomology. As
an application we determine all simple filtered deformations of certain
Z-graded Lie superalgebras classified in [K3], thus completing the last step in
the classification of simple infinite-dimensional linearly compact Lie
superalgebras.
|
math
|
2,836 |
The generalized Witt algebras using additive maps
|
math.RT
|
Wawamoto generalized the Witt algebra using Laurent extension of polynomial
ring. We construct the generalized Witt algebra $W(g_p,n)$ by using an additive
map $g_p$ from a set of integers into a field of characteristic zero where
$1\leq p \leq n.$
|
math
|
2,837 |
Cross-projective representations of pairs of anticommutative algebras, alloys and finite-dimensional irreducible representations of some infinite-dimensional Lie algebras
|
math.RT
|
The article is devoted to some ``strange'' phenomena of representation theory
and their interrelations. Cross-projective representations of pairs of
anticommutative algebras, alloys, their universal envelopping Lie algebras and
their representations, quaternary algebras and their alloyability are
discussed. Considered examples allow to conclude that new representations have
some intriguing features (continuous moduli of finite-dimensional irreducible
representations, sophisticated Clebsch-Gordan coefficient calculus, etc.).
|
math
|
2,838 |
Weight modules of direct limit Lie algebras
|
math.RT
|
In this article we initiate a systematic study of irreducible weight modules
over direct limits of reductive Lie algebras, and in particular over the simple
Lie algebras $A(\infty)$, $B(\infty)$, $C(\infty)$ and $D(\infty)$. Our main
tool is the shadow method introduced recently in \cite{DMP}. The integrable
irreducible modules are an important particular class and we give an explicit
parametrization of the finite integrable modules which are analogues of
finite-dimensional irreducible modules over reductive Lie algebras. We then
introduce the more general class of pseudo highest weight modules. Our most
general result is the description of the support of any irreducible weight
module.
|
math
|
2,839 |
The Fine Structure of Translation Functors
|
math.RT
|
Let E be a simple finite dimensional representation of a semisimple Lie
algebra with extremal weight nu and choose nonzero e in E_{nu}. Let M(tau) be
the Verma module with highest weight tau and v_{tau} in M(tau)_{tau} its
canonical generator. We investigate the projection of e \otimes v_{tau} in E
\otimes M(tau) on the central character chi(tau + nu). This is a rational
function in tau and we calculate its poles and zeros. We then apply this result
in order to compare translation functors.
|
math
|
2,840 |
Gröbner-Shirshov Bases for Lie Superalgebras and Their Universal Enveloping Algebras
|
math.RT
|
We show that a set of monic polynomials in the free Lie superalgebra is a
Gr\"obner-Shirshov basis for a Lie superalgebra if and only if it is a
Gr\"obner-Shirshov basis for its universal enveloping algebra. We investigate
the structure of Gr\"obner-Shirshov bases for Kac-Moody superalgebras and give
explicit constructions of Gr\"obner-Shirshov bases for classical Lie
superalgebras.
|
math
|
2,841 |
Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras
|
math.RT
|
Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian
group satisfying a certain finiteness condition. Suppose that a group $G$ acts
on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak
L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie
superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this
paper, we show that the Euler-Poincar\'e principle yields the generalized
denominator identity for ${\frak L}$ and derive a closed form formula for the
supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a)
\in \Gamma\times A$. We discuss the applications of our supertrace formula to
various classes of infinite dimensional Lie superalgebras such as free Lie
superalgebras and generalized Kac-Moody superalgebras. In particular, we
determine the decomposition of free Lie superalgebras into a direct sum of
irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous
Lie superalgebras with group actions. Finally, we prove that the generalized
characters of Verma modules and the irreducible highest weight modules over a
generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin
diagram automorphism $\sigma$ are the same as the usual characters of Verma
modules and irreducible highest weight modules over the orbit Lie superalgebra
$\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.
|
math
|
2,842 |
The Polynomial Behavior of Weight Multiplicities for the Affine Kac-Moody Algebras $A^{(1)}_r$
|
math.RT
|
We prove that the multiplicity of an arbitrary dominant weight for an
integrable highest weight representation of the affine Kac-Moody algebra
$A_{r}^{(1)}$ is a polynomial in the rank $r$. In the process we show that the
degree of this polynomial is less than or equal to the depth of the weight with
respect to the highest weight. These results allow weight multiplicity
information for small ranks to be transferred to arbitrary ranks.
|
math
|
2,843 |
Discrete Series for Loop Groups.I. An algebraic Realization of Standard Modules
|
math.RT
|
In this paper we consider the category $C (\tilde k, \tilde H)$ of the
$(\tilde k, \tilde H)$-modules, including all the Verma modules, where $k$ is
some compact Lie algebra and H some Cartan subgroup, $\tilde k$ and $\tilde H$
are the corresponding affine Lie algebra and the affine Cartan group,
respectively. To this category we apply the Zuckerman functor and its
derivatives. By using the determinant bundle structure, we prove the natural
duality of the Zuckerman derived functors, and deduce a Borel-Weil-Bott type
theorem on decomposition of the nilpotent part cohomology.
|
math
|
2,844 |
Irreducible representations of solvable Lie superalgebras
|
math.RT
|
The description of irreducible finite dimensional representations of finite
dimensional solvable Lie superalgebras over complex numbers given by V.~Kac is
refined. In reality these representations are not just induced from a
polarization but twisted, as infinite dimensional representations of solvable
Lie algebras. Various cases of irreducibility (general and of type Q) are
classified.
|
math
|
2,845 |
Orthogonal polynomials and Lie superalgebras
|
math.RT
|
For the orthogonal Lie algebra O(2n+1), in addition to the conventional set
of orthogonal polynomials, another set is produced with the help of the Lie
superalgebra OSP(1|2n). Difficulties related with expression of Dyson's
constant for the Lie superalgebras are discussed.
|
math
|
2,846 |
The Howe duality and the Projective Representations of Symmetric Groups
|
math.RT
|
The symmetric group S_n possesses a nontrivial central extension, whose
irreducible representations, different from the irreducible representations of
S_n itself, coincide with the irreducible representations of a certain algebra
A_n. Recently M.~Nazarov realized irreducible representations of A_n and Young
symmetrizers by means of the Howe duality between the Lie superalgebra q(n) and
the Hecke algebra H_n, the semidirect product of S_n with the Clifford algebra
C_n on n indeterminates.
Here I construct one more analog of Young symmetrizers in H_n as well as the
analogs of Specht modules for A_n and H_n.
|
math
|
2,847 |
On a new class of the commutative subalgebras of the maximal Gel'fand-Kirillov dimension in the universal enveloping algebra of a simple Lie algebra
|
math.RT
|
New commutative subalgebras of the maximal Gel'fand-Kirillov dimension in the
universal enveloping algebras of classical Lie algebras gl(n) and so(n) are
constructed. In the case of sp(n) Gel'fand-Tsetlin algebra is extended to a
maximally commutative one.
|
math
|
2,848 |
A duality of the twisted group algebra of the symmetric group and a Lie superalgebra
|
math.RT
|
Let A_k denote the twisted group algebra of the symmetric group S_k, whose
representations correspond to the nonlinear projective representations of S_k.
We establish a duality relation between A_k and a Lie superalgebra q(n),
sometimes called the ``queer'' Lie superalgebra. This duality clarifies Schur's
formula, in the sense that the Schur-Weyl duality clarifies Frobenius' formula.
|
math
|
2,849 |
On the classification of unitary representations of reductive Lie groups
|
math.RT
|
Suppose G is a real reductive Lie group in Harish-Chandra's class. We propose
here a structure for the set \Pi_u(G) of equivalence classes of irreducible
unitary representations of G. (The subscript u will be used throughout to
indicate structures related to unitary representations.) We decompose
\Pi_{u}(G) into disjoint subsets with a (very explicit) discrete parameter set
\Lambda_u: \Pi_u(G) = \bigcup_{\lambda_u \in \Lambda_u} \Pi_u^{\lambda_u}(G).
Each subset is identified conjecturally with a collection of unitary
representations of a certain subgroup G(\lambda_u) of G. (We will give strong
evidence and partial results for this conjecture.) In this way the problem of
classifying \Pi_u(G) would be reduced (by induction on the dimension of G) to
the case G(\lambda_u) = G.
|
math
|
2,850 |
A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra
|
math.RT
|
We establish a duality relation between one of the twisted group algebras of
the hyperoctahedral groupf H_k and a Lie superalgebra q(n_0) \oplus q(n_1) for
any integers k and n_0, n_1, where q(n_0) and q(n_1) denote the ``queer''
Liesuperalgebras. Note that this twisted group algebra \B'_k belongs to a
different cocycle from the one \B_k used by A. N. Sergeev in [8] and by the
present author in [11]. We will use the supertensor product \C_k \otimes \B'_k
of the 2^k-dimensional Clifford algebra \C_k and \B'_k, as an intermediary for
establishing our duality. We show that the algebra \C_k \otimes B'_k and q(n_0)
\oplus q(n_1) act on the k-fold tensor product W=V^{\otimes k} of the natural
representation V of q(n_0+n_1) ``as mutual centralizers of each other''
(Theorem 4.1). Moreover, we show that \B'_k and q(n_0) \oplus q(n_1) act on a
subspace W' of W ``as mutual centralizers of each other'' (Theorem 4.2). This
duality relation gives a formula for character values of simple B'_k-modules.
This formula is di fferent from a formula (Theorem D) obtained by J. R.
Stembridge (cf. [10, Lem 7.5]).
|
math
|
2,851 |
Explicit Hilbert spaces for certain unipotent representations II
|
math.RT
|
We construct an explicit realization of a minimal representation of G, where
G is the conformal group of a real Jordan algebra N. We characterize spherical
vectors for these representation and prove that they are closely related to the
Bessel K-function $K_\tau (z)$. The resulting construction can be used to study
tensor powers of the minimal representation and establish an extension of the
Howe duality correspondence to some exceptional groups.
|
math
|
2,852 |
Raising operators for the Whittaker wave functions of the Toda chain and intertwining operators
|
math.RT
|
Intertwiners between representations of Lie groups can be used to obtain
relations for matrix elements. We apply this technique to obtain different
identities for the wave functions of the open Toda chain, in particular raising
operators and bilinear relations for the wave functions at different energy
levels. We also recall the group theory approach to the Toda chain: treating
the wave functions as matrix elements in irreducible representations between
the so-called Whittaker vectors, integral representations of the wave
functions, etc.
|
math
|
2,853 |
Classification of finite dimensional modules of singly atypical type over the Lie superalgebras sl(m/n)
|
math.RT
|
We classify the finite dimensional indecomposable sl(m/n)-modules with at
least a typical or singly atypical primitive weight. We do this classification
not only for weight modules, but also for generalized weight modules. We obtain
that such a generalized weight module is simply a module obtained by
``linking'' sub-quotient modules of generalized Kac-modules. By applying our
results to sl(m/1), we have in fact completely classified all finite
dimensional sl(m/1)-modules.
|
math
|
2,854 |
NilCoxeter algebras categorify the Weyl algebra
|
math.RT
|
We show that induction and restriction functors for inclusions of nilCoxeter
algebras provide a categorical realization of the algebra of polynomial
differential operators in one variable.
|
math
|
2,855 |
Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras
|
math.RT
|
In this paper we prove theorems that describe how the representation theory
of the affine Hecke algebra of type A and of related algebras such as the group
algebra of the symmetric group are controlled by integrable highest weight
representations of the characteristic zero affine Lie algebra \hat{sl}_l. In
particular we parameterise the representations of these algebras by the nodes
of the crystal graph, and give various Hecke theoretic descriptions of the
edges.
As a consequence we find for each prime p a basis of the integrable
representations of \hat{sl}_l which shares many of the remarkable properties,
such as positivity, of the global crystal basis/canonical basis of Lusztig and
Kashiwara. This {\it $p$-canonical basis} is the usual one when p = 0, and the
crystal of the p-canonical basis is always the usual one.
The paper is self-contained, and our techniques are elementary (no perverse
sheaves or algebraic geometry is invoked).
|
math
|
2,856 |
Dimension of a minimal nilpotent orbit
|
math.RT
|
We show that the dimension of the minimal nilpotent coadjoint orbit for a
complex simple Lie algebra is equal to twice the dual Coxeter number minus two.
|
math
|
2,857 |
Analytic continuation of representations and estimates of automorphic forms
|
math.RT
|
Properties of analytic vectors in representations of SL(2,R) are used to give
new bounds for the triple products recently considered by P. Sarnak. A
conjecture of Sarnak about such products is proved. The results of this paper
generalize results of A. Good and M. Jutila about special cases, but the
techniques are entirely different. One consequence of these results is a new
estimate of the magnitude of the Fourier coefficients of cusp forms for
non-arithmetic sub-groups of SL(2,R).
|
math
|
2,858 |
Steinberg modules and Donkin pairs
|
math.RT
|
We prove that in positive characteristic a module with good filtration for a
group of type E6 restricts to a module with good filtration for a subgroup of
type F4. (Recall that a filtration of a module for a semisimple algebraic group
is called good if its layers are dual Weyl modules.) Our result confirms a
conjecture of Brundan for one more case. The method relies on the canonical
Frobenius splittings of Mathieu. Next we settle the remaining cases, in
characteristic not 2, with a computer-aided variation on the old method of
Donkin.
|
math
|
2,859 |
Reciprocity Theorems for Holomorphic Representations of Some Infinite-Dimensional Groups
|
math.RT
|
In this article we prove several reciprocity theorems for some
infinite-dimensional dual pairs of representations on Bargmann-Segal-Fock
spaces.
|
math
|
2,860 |
Weyl's character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras
|
math.RT
|
We generalize I. Frenkel's orbital theory for non twisted affine Lie algebras
to the case of twisted affine Lie algebras using a character formula for
certain non-connected compact Lie groups.
|
math
|
2,861 |
Localization of elementary systems in the theory of Wigner
|
math.RT
|
Starting from Wigner's theory of elementary systems and following a recent
approach of Schroer we define certain subspaces of localized wave functions in
the underlying Hilbert space with the help of the theory of modular von-Neumann
algebras of Tomita and Takesaki. We characterize the elements of these
subspaces as boundary values of holomorphic functions in the sense of
distribution theory and show that the corresponding holomorphic functions
satisfy the sufficient conditions of the theorems of Paley-Wiener-Schwartz and
H\"{o}rmander.
|
math
|
2,862 |
Generalized Loop Groups of Complex Manifolds, Gaussian Quasi-Invariant Measures on them and their Representations
|
math.RT
|
Loop groups G as families of mappings of the complex manifold M into another
complex manifold N preserving marked points $s_0\in M$ and $y_0\in N$ are
investigated. Quasi-invariant measures $\mu $ on G relative to dense subgroups
$G'$ are constructed. These measures are used for the studying of irreducible
representations of such groups.
|
math
|
2,863 |
Poisson measures for topological groups and their representations
|
math.RT
|
Gaussian quasi-invariant measures on groups of diffeomorphisms and loop
groups G relative to dense subgroups G' were constructed. In the
non-Archimedean case the wider class of measures was investigated, than in the
real case. The cases of Riemann and non-Archimedean manifolds were considered.
This article is related with unitary representations of G' associated with
Poisson measures on $G^{\bf N}$ and uses quasi-invariant measures on G from the
previous works. Several groups are considered: (1) (a) diffeomorphisms and (b)
loop groups of real manifolds, (2) (a) diffeomorphisms and (b) loop groups of
non-Archimedean manifolds over local fields. Besides these four cases further
the fifth and the sixth cases are considered: for (3) (a) real and (b)
non-Archimedean groups of diffeomorphisms Diff(M) representations associated
with Poisson measures on configuration spaces $\Gamma_M$ contained in products
of manifolds $M^{\bf N}$ are investigated. Here the cases of
infinite-dimensional Banach manifold M (3) (a), non-Archimdean locally compact
and non-locally compact Banach manifolds (3) (b) are investigated.
|
math
|
2,864 |
On the Ghost Centre of Lie Superalgebras
|
math.RT
|
We define a notion of ghost centre of a Lie superalgebra g=g_0+g_1 which is a
sum of invariants with respect to the usual adjoint action (centre) and
invariants with respect to a twisted adjoint action (``anticentre''). We
calculate the anticentre in the case when the top external degree of g_1 is a
trivial g_0-module. We describe the Harish-Chandra projection of the ghost
centre for basic classical Lie superalgebras and show that for these cases the
ghost centre coincides with the centralizer of the even part of the enveloping
algebra.
The ghost centre of a Lie superalgebra plays a role of the usual centre of a
Lie algebra in some problems of representation theory. For instance, for
osp(1,2l) the annihilator of a Verma module is generated by the intersection
with the ghost centre.
|
math
|
2,865 |
On extensions of representations for compact Lie groups
|
math.RT
|
Let $H$ be a closed normal subgroup of a compact Lie group $G$ such that
$G/H$ is connected. This paper provides a necessary and sufficient condition
for every complex representation of $H$ to be extendible to $G$, and also for
every complex $G$-vector bundle over the homogeneous space $G/H$ to be trivial.
In particular, we show that the condition holds when the fundamental group of
$G/H$ is torsion free.
|
math
|
2,866 |
Introduction to the Alexandru Conjecture
|
math.RT
|
Is a Verma module transformed into another Verma module by a selfequivalence?
The answer is affirmative and the proof suggests a notion of standard object in
the category of Harish-Chandra modules that coincides often, but not always,
with the usual one.
|
math
|
2,867 |
Statement of the Alexandru Conjecture
|
math.RT
|
The Vogan Conjectures (sometimes called Kazhdan-Lusztig Conjectures) say that
a certain algorithm works both on the category of BGG modules and on the
category of Harish-Chandra modules. The Alexandru Conjecture tries to uncover
the general property common to these two categories which makes Vogan's
algorithm work.
|
math
|
2,868 |
A two-box-shift morphism between Specht modules
|
math.RT
|
Let n geq 1, let lambda be a partition of n, let mu be a partition arising
from lambda by a downwards shift of two boxes situated at the bottom of a
column. We give a formula for a ZS_n-linear morphism of order m between the
corresponding Specht modules over Z/(m), where m is the box shift length
(divided by two in certain combinatorially specified cases). Reformulated, this
yields an extension of the corresponding Specht modules over Z of order m in
Ext^1.
|
math
|
2,869 |
A simple question about a complicated object
|
math.RT
|
Let n and k be positive integers with and k < n.
Then of course SU(k,1) is contained into SU(n,1).
Moreover, which is less clear - but proved by Khoroshkin -, the
representation theory of SU(k,1) at the generalized infinitesimal character of
the trivial module can be fully (and even Ext-fully) embedded into that of
SU(n,1). Here is the obvious bet: This embedding is implemented by the
cohomological induction functor. I conjecture that a similar phenomenon occurs
whenever SU(k,1) is a Levi factor of a theta stable parabolic subalgebra of a
reductive group.
|
math
|
2,870 |
Branching rules for modular fundamental representations of symplectic groups
|
math.RT
|
In this paper branching rules for the fundamental representations of the
symplectic groups in positive characteristic are found. The submodule structure
of the restrictions of the fundamental modules for the group $Sp_{2n}(K)$ to
the naturally embedded subgroup $Sp_{2n-2}(K)$ is determined. As a corollary,
inductive systems of fundamental representations for $Sp_{\infty}(K)$ are
classified. The submodule structure of the fundamental Weyl modules is refined.
|
math
|
2,871 |
On an asymptotic behavior of elements of order p in irreducible representations of the classical algebraic groups with large enough highest weights
|
math.RT
|
The behavior of the images of a fixed element of order p in irreducible
representations of a classical algebraic group in odd characteristic p with
highest weights large enough with respect to p and this element is
investigated. Lower estimates for the number of Jordan blocks of size p in
images of such elements that lie in naturally embedded subgroups of the same
type as the initial group and smaller ranks are obtained.
|
math
|
2,872 |
Normalisation des opérateurs d'entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques p-adiques
|
math.RT
|
On montre comment les conjectures d'Arthur permettent de calculer les points
de r\'eductibilit\'e pour les induites de cuspidales des groupes classiques.
Les conjectures d'Arthur utilis\'ees portent sur l'existence d'un rel\`evement
faible des repr\'esentations automorphes cuspidales d'un groupe classique vers
un groupe lin\'eaire convenable. Et les points de r\'eductibilit\'es sont
decrits en terme de la g\'eom\'etrie des orbites unipotentes des groupe dual.
On en d\'eduit des r\'esultats sur la normalisation des operateurs
d'entrelacement.
We show how the global conjectures of Arthur allow us to calculate the points
of irreducibility for representations induced from cuspidal representations of
classical groups. The conjectures of Arthur used are concerned with the
existence of weak liftings of automorphic representations from classical groups
to suitable linear ones. And the points of irreducibility are described in
terms of the geometry of unipotent orbits on the dual group. We deduce certain
results about the normalization of intertwining operators.
|
math
|
2,873 |
Discrete group actions on Stein domains in complex Lie groups
|
math.RT
|
This paper deals with the analytic continuation of holomorphic automorphic
forms on a Lie group $G$. We prove that for any discrete subgroup $\Gamma$ of
$G$ there always exists a non-trivial holomorphic automorphic form, i.e., there
exists a $\Gamma$-spherical unitary highest weight representation of $G$.
Holomorphic automorphic forms have the property that they analytically extend
to holomorphic functions on a complex Ol'shanski\u\i{} semigroup $S\subeq
G_\C$. As an application we prove that the bounded holomorphic functions on
$\Gamma\bs S\subseteq \Gamma\bs G_\C$ separate the points.
|
math
|
2,874 |
Formal dimension for semisimple symmetric spaces
|
math.RT
|
In this paper we generalize Harish Chandra's formula for the formal dimension
of a representation of the holomorphic discrete series of a hermitian Lie group
$G$ to semisimple symmetric spaces $G/H$.
|
math
|
2,875 |
The c-function for non-compactly causal symmetric spaces
|
math.RT
|
In this paper we prove the product formula for the c-function of
non-compactly causal symmetric spaces.
|
math
|
2,876 |
Grothendieck groups and tilting objects
|
math.RT
|
Let C be a connected noetherian hereditary abelian Ext-finite category with
Serre functor over an algebraically closed field k, with finite dimensional
homomorphism and extension spaces. Using the classification of such categories
from math.RT/9911242, we prove that if C has some object of infinite length,
then the Grothendieck group of C is finitely generated if and only if C has a
tilting object.
|
math
|
2,877 |
Affine Lie Algebras and Tame Quivers
|
math.RT
|
C.M. Ringel defined Hall algebra associated with the category of
representations of a quiver of Dynkin type and gave an explicit description of
the structure constants of the corresponding Lie algebra. We utilize functorial
properties of the Hall algebra to give a simple proof of Ringel's result, and
to generalize it to the case of a quiver of affine type.
|
math
|
2,878 |
Characteristic cycles and wave front cycles of representations of reductive Lie groups
|
math.RT
|
Vogan and Barbasch-Vogan attach two similar invariants to representations of
a reductive Lie group, one by an algebraic process, the other analytic. They
conjectured that the two invariants determine each other in a definite manner.
Here we prove the conjecture. Our arguments involve two finer invariants -- the
characteristic cycles of representations -- which are interesting in their own
right.
|
math
|
2,879 |
The second cohomology of small irreducible modules for simple algebraic groups
|
math.RT
|
Let G be a simple, simply connected and connected algebraic group over an
algebraically closed field of characteristic p>0, and let V be a rational
G-module such that dim V <= p. According to a result of Jantzen, V is
completely reducible, and H^1(G,V)=0. In this paper we show that H^2(G,V) = 0
unless some composition factor of V is a non-trivial Frobenius twist of the
adjoint representation of G.
|
math
|
2,880 |
Abelian Unipotent Subgroups of Reductive Groups
|
math.RT
|
Let G be a connected reductive group defined over an algebraically closed
field k of characteristic p > 0. The purpose of this paper is two-fold. First,
when p is a good prime, we give a new proof of the ``order formula'' of D.
Testerman for unipotent elements in G; moreover, we show that the same formula
determines the p-nilpotence degree of the corresponding nilpotent elements in
the Lie algebra of G.
Second, if G is semisimple and p is sufficiently large, we show that G always
has a faithful representation (r,V) with the property that the exponential of
dr(X) lies in r(G) for each p-nilpotent X in Lie(G). This property permits a
simplification of the description given by Suslin, Friedlander, and Bendel of
the (even) cohomology ring for the Frobenius kernels G_d, d > 1. The previous
authors already observed that the natural representation of a classical group
has the above property (with no restriction on p). Our methods apply to any
Chevalley group and hence give the result also for quasisimple groups with
``exceptional type'' root systems. The methods give explicit sufficient
conditions on p; for an adjoint semisimple G with Coxeter number h, the
condition p > 2h -2 is always good enough.
|
math
|
2,881 |
Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group
|
math.RT
|
The article establishes a long list of rigidity properties of lattices in G =
SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity
of lattices in higher-rank Lie groups. The arguments are set in the context of
strongly L^p unitary representations of G.
|
math
|
2,882 |
Howe Duality for Lie Superalgebras
|
math.RT
|
We study a dual pair of general linear Lie superalgebras in the sense of R.
Howe. We give an explicit multiplicity-free decomposition of a symmetric and
skew-symmetric algebra (in the super sense) under the action of the dual pair
and present explicit formulas for the highest weight vectors in each isotypic
subspace of the symmetric algebra. We give an explicit multiplicity-free
decomposition into irreducible $gl(m|n)$-modules of the symmetric and
skew-symmetric algebras of the symmetric square of the natural representation
of $gl(m|n)$. In the former case we find as well explicit formulas for the
highest weight vectors. Our work unifies and generalizes the classical results
in symmetric and skew-symmetric models and admits several applications.
|
math
|
2,883 |
Invariant differential operators on nonreductive homogeneous spaces
|
math.RT
|
A systematic exposition is given of the theory of invariant differential
operators on a not necessarily reductive homogeneous space. This exposition is
modelled on Helgason's treatment of the general reductive case and the special
non-reductive case of the space of horocycles. As a final application the
differential operators on (not a priori reductive) isotropic pseudo-Riemannian
spaces are characterized.
|
math
|
2,884 |
Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras
|
math.RT
|
We give a detailed calculation of the Hochschild and cyclic homology of the
algebra $\CIc(G)$ of locally constant, compactly supported functions on a
reductive p-adic group G. We use these calculations to extend to arbitrary
elements the definition the higher orbital integrals introduced in
\cite{Blanc-Brylinski} for regular semisimple elements. Then we extend to
higher orbital integrals some results of Shalika. We also investigate the
effect of the ``induction morphism'' on Hochschild homology.
|
math
|
2,885 |
Projectivity of modules for infinitesimal unipotent group schemes
|
math.RT
|
In this paper, it is shown that the projectivity of a rational module for an
infinitesimal unipotent group scheme over an algebraically closed field of
positive characteristic can be detected on a family of closed subgroups.
|
math
|
2,886 |
Cohomology and projectivity of modules for finite group schemes
|
math.RT
|
Let G be a finite group scheme over an algebraically closed field of positive
characteristic. Assume further that the connected component of G is unipotent.
It is shown that the projectivity of a rational G-module can be detected on a
family of closed subgroups. It is further shown that nilpotent cohomology or
extension classes can be detected on this family of subgroups.
|
math
|
2,887 |
Blocks of Lie Superalgebras of Type W(n)
|
math.RT
|
We calculate the blocks of the category of finite-dimensional representations
of W(0,n), with n > 2, and show that all are of wild type. As an application,
we show that the centre of the universal enveloping algebra is trivial.
|
math
|
2,888 |
Strongly typical representations of the basic classical Lie superalgebras
|
math.RT
|
The category of representations with a strongly typical central character of
a basic classical Lie superalgebra is proven to be equivalent to the category
of representations of its even part corresponding to an appropriate central
character.
For a Lie superalgebra $osp(1,2l)$ the category of representations with a
"generic" weakly atypical central character is described.
|
math
|
2,889 |
Méthode des orbites et formules du caractère pour les représentations tempérées d'un groupe algébrique réel réductif non connexe
|
math.RT
|
Let G be a non-connected reductive real Lie group. In this paper, I
parametrize the set of irreductible tempered characters of G. Afterwards, I
describe these characters by means of some ``Kirillov's formulas'', using the
descent method near each elliptic element in G.
If G is linear and connected, the parameters that I use are ``final basic''
parameters in the sense of Knapp and Zuckerman.
|
math
|
2,890 |
Degenerations of Schubert varieties of SL(n)/B to toric varities
|
math.RT
|
Using polytopes defined in an earlier paper, we show in this paper the
existence of degenerations of a large class of Schubert varieties of SL_n to
toric varieties by extending the method used by Gonciulea and Lakshmibai for a
minuscule G/P to Schubert varieties in SL_n
|
math
|
2,891 |
Toric degenerations of Schubert varieties
|
math.RT
|
Let $G$ be a simply connected semi-simple complex algebraic group. Fix a
maximal torus $T$ and a Borel subgroup $B$ such that $T\subset B\subset G$. Let
$W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\bar
{BwB/B}$ denote the Schubert variety corresponding to $w$. This talk is
concerned with the following problem : Is there a flat family over Spec${\bf
C}[t]$, such that the general fiber is $X_w$ and the special fiber is a toric
variety? Our approach of the problem is based on the canonical/global base of
Lusztig/Kashiwara and the so-called string parametrization of this base studied
by P. Littelmann and made precise by A. Berenstein and A. Zelevinsky. Fix $w$
in $W$ and let $P^+$ be the semigroup of dominant weights. For all $\lambda$ in
$P^+$, let ${\cal L}_\lambda$ be the line bundle on $G/B$ corresponding to
$\lambda$. Then, the direct sum of global sections $R_w:=\bigoplus_{\lambda\in
P^+}H^0(X_w,{\cal L}_\lambda)$ carries a natural structure of $P^+$-graded
${\bf C}$-algebra. Moreover, there exists a natural action of $T$ on $R_w$. Our
principal result can be stated as follows : There exists a filtration $({\cal
F}_m^w)_{m\in{\bf N}}$ of $R_w$ such that (i) for all $m$ in ${\bf N}$, ${\cal
F}_m^w$ is compatible with the $P^+$-grading of $R_w$, (ii) for all $m$ in
${\bf N}$, ${\cal F}_m^w$ is compatible with the action of $T$, (iii) the
associated graded algebra is the ${\bf C}$-algebra of the semigroup of integral
points in a rational convex polyhedral cone. Equations for this cone were
obtained by A. Berenstein and A. Zelevinski from $\tilde w_0$-trails in
fundamental Weyl modules of the Langlands dual of $G$. By standard arguments,
the previous theorem gives a positive answer to the Degeneration Problem.
|
math
|
2,892 |
A generating function for the trace of the Iwahori-Hecke algebra
|
math.RT
|
The Iwahori-Hecke algebra has a ``natural'' trace $\tau$. This trace is the
evaluation at the identity element in the usual interpretation of the
Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic
semi-simple group. The Iwahori-Hecke algebra contains an important commutative
sub-algebra ${\bf C}[\theta_x]$, that was described and studied by Bernstein,
Zelevinski and Lusztig. In this note we compute the generating function for the
value of $\tau$ on the basis $\theta_x$.
|
math
|
2,893 |
On the spectral decomposition of affine Hecke algebras
|
math.RT
|
An affine Hecke algebra H contains a large abelian subalgebra A. The center Z
of H is the subalgebra of Weyl group invariant elements in A. The natural trace
of the affine Hecke algebra can be written as an integral of a rational $n$
form (with values in the linear dual of H) over a certain cycle in the
algebraic torus T=spec(A). We derive the Plancherel formula of the affine Hecke
algebra by localization of this integral on a certain subset of spec(Z).
|
math
|
2,894 |
On certain representations of automorphism groups of an algebraically closed field
|
math.RT
|
Let k be an algebraically closed field of characteristic zero,
F its algebraically closed extension, and G be the group of k-automorphisms
of F endowed with a natural topology. One of the purposes of this paper is to
show that any non-faithful continuous representation of G factors through a
discrete quotient of G. Properties of representation of G arising from geometry
are studied. In some cases the groups of morphisms between geometric objects
are identified with the groups of morphisms between corresponding G-modules,
and the ${\rm Ext}^1$'s are related. In particular, the category of abelian
varieties over k with morphisms tensored with the rationals can be described as
a category of G-modules.
|
math
|
2,895 |
Centralizers of distinguished nilpotent pairs and related problems
|
math.RT
|
In this paper, by establishing an explicit and combinatorial description of
the centralizer of a distinguished nilpotent pair in a classical simple Lie
algebra, we solve in the classical case Panyushev's Conjecture which says that
distinguished nilpotent pairs are wonderful, and the classification problem on
almost principal nilpotent pairs. More precisely, we show that disinguished
nilpotent pairs are wonderful in types A, B and C, but they are not always
wonderful in type D. Also, as the corollary of the classification of almost
principal nilpotent pairs, we have that almost principal nilpotent pairs do not
exist in the simply-laced case and that the centralizer of an almost principal
nilpotent pair in a classical simple Lie algebra is always abelian.
|
math
|
2,896 |
Some New Applications of Weyl's Multipolarization Operators
|
math.RT
|
In Weyl's "The Classical Groups", he introduces some some remarkable
differential operators, which he calls "quasi-compositions" of the polarization
operators Dij. In the present paper, an equivalent combinatorial formulation is
obtained for these operators, and is then used to obtain explicit formulas for
the differentials in certain complexes (constucted by Zelevinsky, and further
studied by Verma, Akin et al.) which furnish higher syzygies for the Pluecker
equations, and also for the defining relations for Weyl modules.
|
math
|
2,897 |
On tensor categories attached to cells in affine Weyl groups II
|
math.RT
|
George Lusztig conjectured that asymptotic affine Hecke algebra of a simply
connected group can be explicitly described in terms of convolution algebras.
Main Theorem of this note (which is a continuation of RT/0010089) is a weak
version of this Conjecture. This version is strong enough to reprove all
previously known results (due to Nanhua Xi) in this direction, for example the
case of type $\tilde A_n$, see QA/0010159.
|
math
|
2,898 |
A generalization of Hall polynomials to ADE case
|
math.RT
|
Certain computable polynomials are described whose leading coefficients are
equal to multiplicities in the tensor product decomposition for representations
of a Lie algebra of ADE type.
|
math
|
2,899 |
Rationality properties of unipotent representations
|
math.RT
|
We describe those unipotent representations of a finite group of Lie type
which are defined over the rational numbers.
|
math
|
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