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 Online first
Types et contragrÃ©dientes Soit $\mathrm{G}$ un groupe rÃ©ductif $p$-adique, et soit $\mathrm{R}$
un corps algÃ©briquement clos.
Soit $\pi$ une reprÃ©sentation lisse de $\mathrm{G}$ dans un espace
vectoriel $\mathrm{V}$ sur
$\mathrm{R}$.
Fixons un sous-groupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une reprÃ©sentation
lisse irrÃ©ductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel
$\mathrm{W}$ de dimension
finie sur $\mathrm{R}$.
Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algÃ¨bre
d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$.
Nous examinons la compatibilitÃ© de ces constructions avec le passage aux
reprÃ©sentations contragrÃ©dientes $\mathrm{V}^Äe$ et $\mathrm{W}^Äe$, et donnons en
particulier des conditions sur $\mathrm{W}$ ou sur la caractÃ©ristique
de $\mathrm{R}$ pour que
le comportement soit semblable au cas des reprÃ©sentations complexes.
Nous prenons un point de vue abstrait, n'utilisant que des propriÃ©tÃ©s
gÃ©nÃ©rales de $\mathrm{G}$.
Nous terminons par une application Ã la thÃ©orie des types pour le groupe
$\mathrm{GL}_n$ et ses formes intÃ©rieures sur un corps local non archimÃ©dien.
Keywords:modular representations of p-adic reductive groups, types, contragredient, intertwining Category:22E50 |
3. CJM Online first
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction Categories:11G18, 14G35, 14G05 |
4. CJM 2012 (vol 66 pp. 3)
On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the GÃ¶ttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.
Keywords:binary forms, covariants, $SL_2$-representations Categories:14L30, 13A50 |
5. CJM 2011 (vol 63 pp. 1107)
Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$,
where $F$ is a non-Archimedean local field
of characteristic
$0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$-case of a
conjecture of
Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$-function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.
Keywords:generic representations, local Langlands parameters Categories:22E50, 11S37 |
6. CJM 2009 (vol 62 pp. 439)
On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.
Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius group Categories:32A25, 32A35, 32A37, 47B35 |
7. CJM 2009 (vol 62 pp. 34)
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field |
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series
representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.
Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups Categories:20G25, 20G05 |
8. CJM 2006 (vol 58 pp. 23)
Constructing Representations of Finite Simple Groups and Covers Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient
and simple method to construct representations of finite groups is applicable
whenever $G$ has a subgroup $H$ such that $\chi_H$
has a linear constituent with multiplicity $1$.
In this paper we show (with a few exceptions) that if $G$
is a simple group or a covering group of a simple group and
$\chi$ is an irreducible character of $G$ of degree less than 32,
then there exists a subgroup $H$ (often a Sylow subgroup) of $G$
such that $\chi_H$ has a linear constituent with multiplicity $1$.
Keywords:group representations, simple groups, central covers, irreducible representations Categories:20C40, 20C15 |
9. CJM 2005 (vol 57 pp. 648)
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of
the restriction of the principal
series representations of $SL(2,k)$, for $k$ a $p$-adic field,
to each of its two maximal compact subgroups (up to conjugacy).
We identify these irreducible subrepresentations in the
Kirillov-type classification
of Shalika. We go on to explicitly describe the decomposition
of the reducible principal series of $SL(2,k)$ in terms of the
restrictions of its irreducible constituents to a maximal compact
subgroup.
Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types Categories:20G25, 22E35, 20H25 |
10. CJM 2000 (vol 52 pp. 1121)
Ramanujan Type Buildings We will construct a finite union of finite quotients of the affine
building of the group $\GL_3$ over the field of $p$-adic numbers
$\mathbb{Q}_p$. We will view this object as a hypergraph and estimate
the spectrum of its underlying graph.
Keywords:automorphic representations, buildings Category:11F70 |
11. CJM 1997 (vol 49 pp. 543)
Some summation theorems and transformations for $q$-series We introduce a double sum extension of a very well-poised series and
extend to this the transformations of Bailey and Sears as well as the
${}_6\f_5$ summation formula of F.~H.~Jackson and the $q$-Dixon sum.
We also give $q$-integral representations of the double sum.
Generalizations of the Nassrallah-Rahman integral are also found.
Keywords:Basic hypergeometric series, balanced series,, very well-poised series, integral representations,, Al-Salam-Chihara polynomials. Categories:33D20, 33D60 |