Expand all Collapse all | Results 1 - 11 of 11 |
1. CJM Online first
Symplectic Degenerate Flag Varieties A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semi-direct product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
In general this is a
singular variety, but we conjecture that it has many nice properties similar to
that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group.
The symplectic case is important for the conjecture
because it is the first known case where even for fundamental weights $\omega$ the varieties
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
are normal locally complete intersections with terminal and rational singularities.
We also show that these varieties are Frobenius split. Using the above mentioned results, we
prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula
for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed
points formula.
Keywords:Lie algebras, flag varieties, symplectic groups, representations Categories:14M15, 22E46 |
2. CJM 2012 (vol 64 pp. 721)
Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski
and B. Kostant of minimal representations of simple Lie groups. We
start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$
a homogeneous polynomial of degree 4 on $V$.
The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$,
the conformal group of the pair $(V,Q)$, in a finite dimensional
representation space.
By a generalized Kantor-Koecher-Tits construction we obtain a complex
simple Lie algebra $\mathfrak g$, and furthermore a real
form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie
group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak
g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic
functions defined on the manifold $\Xi $.
Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-function Categories:17C36, 22E46, 32M15, 33C80 |
3. CJM 2012 (vol 65 pp. 66)
On Flag Curvature of Homogeneous Randers Spaces In this paper we give an explicit formula for the flag curvature of
homogeneous Randers spaces of Douglas type and apply this formula to
obtain some interesting results. We first deduce an explicit formula
for the flag curvature of an arbitrary left invariant Randers metric
on a two-step nilpotent Lie group. Then we obtain a classification of
negatively curved homogeneous Randers spaces of Douglas type. This
results, in particular, in many examples of homogeneous non-Riemannian
Finsler spaces with negative flag curvature. Finally, we prove a
rigidity result that a homogeneous Randers space of Berwald type whose
flag curvature is everywhere nonzero must be Riemannian.
Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groups Categories:22E46, 53C30 |
4. CJM 2011 (vol 64 pp. 669)
The Genuine Omega-regular Unitary Dual of the Metaplectic Group We classify all genuine unitary representations of the metaplectic group whose
infinitesimal character is real and at least as regular as that of the
oscillator representation. In a previous paper we exhibited a certain family
of representations satisfying these conditions, obtained by cohomological
induction from the tensor product of a one-dimensional representation and an
oscillator representation. Our main theorem asserts that this family exhausts
the genuine omega-regular unitary dual of the metaplectic group.
Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal series Category:22E46 |
5. CJM 2009 (vol 62 pp. 52)
An Algebraic Approach to Weakly Symmetric Finsler Spaces In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the non-Riemannian reversible weakly
symmetric Finsler spaces we find are non-Berwaldian and with vanishing
S-curvature. This means that reversible non-Berwaldian Finsler spaces
with vanishing S-curvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature Categories:53C60, 58B20, 22E46, 22E60 |
6. CJM 2009 (vol 61 pp. 961)
Transfert des intÃ©grales orbitales pour les algÃ¨bres de Lie classiques Dans cet article, on consid\`ere un groupe semi-simple $\rmG$ classique
r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un
sous-groupe de Cartan compact. On d\'efinit une famille de
sous-alg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang
que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$.
Soit $\g'$ une telle sous-alg\`ebre de Lie. On construit alors une
application de transfert des int\'egrales orbitales de $\g'$ dans
l'espace des int\'egrales orbitales de $\g$. On montre que cette
application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur
simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$.
Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de
rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette
application de transfert.
On utilise cette application de transfert pour obtenir une formule de
r\'eduction de l'int\'egrale de Cauchy Harish-Chandra pour les paires
duales d'alg\`ebres de Lie r\'eductives $\bigl( \Ug(p,q),\Ug(r,s)
\bigr)$ et $\bigl( \Sp(p,q),\Og^*(2n) \bigr)$ avec $p+q = r+s = n$.
Categories:22E30, 22E46 |
7. CJM 2009 (vol 61 pp. 351)
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicity-free. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual Littlewood--Richardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Categories:14L30, 22E46 |
8. CJM 2004 (vol 56 pp. 945)
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.
Categories:20G15, 20G20, 22E15, 22E46 |
9. CJM 2002 (vol 54 pp. 769)
Nilpotent Orbits and Whittaker Functions for Derived Functor Modules of $\Sp(2,\mathbb{R})$ We study the moderate growth generalized Whittaker functions,
associated to a unitary character $\psi$ of a unipotent subgroup,
for the non-tempered cohomological representation of $G = \Sp
(2,\mathbb{R})$. Through an explicit calculation of a holonomic
system which characterizes these functions we observe that their
existence is determined by the including relation between the real
nilpotent coadjoint $G$-orbit of $\psi$ in
$\mathfrak{g}_{\mathbb{R}}^\ast$ and the asymptotic support of the
cohomological representation.
Categories:22E46, 22E30 |
10. CJM 1999 (vol 51 pp. 1135)
Endoscopic $L$-Functions and a Combinatorial Identity The trace formula contains terms on the spectral side that are
constructed from unramified automorphic $L$-functions. We shall
establish an identify that relates these terms with corresponding
terms attached to endoscopic groups of $G$. In the process, we
shall show that the $L$-functions of $G$ that come from automorphic
representations of endoscopic groups have meromorphic continuation.
Categories:22E45, 22E46 |
11. CJM 1999 (vol 51 pp. 636)
First Occurrence for the Dual Pairs $\bigl(U(p,q),U(r,s)\bigr)$ We prove a conjecture of Kudla and Rallis about the first occurrence
in the theta correspondence, for dual pairs of the form
$\bigl(U(p,q),U(r,s)\bigr)$ and most representations.
Category:22E46 |