Expand all Collapse all | Results 1 - 10 of 10 |
1. CJM 2011 (vol 63 pp. 1307)
A Bott-Borel-Weil Theorem for Diagonal Ind-groups A diagonal ind-group is a direct limit of classical affine algebraic
groups of growing rank under a class of
inclusions that contains the inclusion
$$
SL(n)\to SL(2n), \quad
M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix}
$$
as a typical special case. If $G$ is a diagonal ind-group and
$B\subset G$ is a Borel ind-subgroup,
we consider the ind-variety $G/B$ and compute the cohomology
$H^\ell(G/B,\mathcal{O}_{-\lambda})$
of any $G$-equivariant line bundle $\mathcal{O}_{-\lambda}$ on
$G/B$. It has been known that, for a generic $\lambda$,
all cohomology groups of $\mathcal{O}_{-\lambda}$ vanish, and that a
non-generic equivariant
line bundle $\mathcal{O}_{-\lambda}$ has at most one
nonzero cohomology group. The new result of this paper is a
precise description of when
$H^j(G/B,\mathcal{O}_{-\lambda})$ is nonzero and the proof of the fact
that, whenever nonzero,
$H^j(G/B, \mathcal{O}_{-\lambda})$ is a $G$-module dual to a highest
weight module.
The main difficulty is in defining an appropriate analog $W_B$ of the
Weyl group, so that the action of $W_B$
on weights of $G$ is compatible with the analog of the Demazure
``action" of the Weyl group on the cohomology
of line bundles. The highest weight corresponding to $H^j(G/B,
\mathcal{O}_{-\lambda})$ is then computed
by a procedure similar to that in the classical Bott-Borel-Weil theorem.
Categories:22E65, 20G05 |
2. 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 |
3. CJM 2009 (vol 61 pp. 691)
Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a non-Archimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is non-abelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.
Categories:22E50, 20G05 |
4. CJM 2007 (vol 59 pp. 449)
$\SL_n$, Orthogonality Relations and Transfer Let $\pi$ be a square integrable representation of
$G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension
over a local field $F$ \emph{of non-zero characteristic}. We prove
that, on the elliptic set, the character of $\pi$ equals the complex
conjugate of the orbital integral of one of the pseudocoefficients
of~$\pi$. We prove also the orthogonality relations for characters of
square integrable representations of $G'$. We prove the stable
transfer of orbital integrals between $\SL_n(F)$ and its inner forms.
Category:20G05 |
5. CJM 2002 (vol 54 pp. 1229)
The Weil Character of the Unitary Group Associated to a Finite Local Ring Let $\mathbf{R}/R$ be a quadratic extension of finite, commutative,
local and principal rings of odd characteristic. Denote by
$\mathbf{U}_n (\mathbf{R})$ the unitary group of rank $n$ associated
to $\mathbf{R}/R$. The Weil representation of $\mathbf{U}_n
(\mathbf{R})$ is defined and its character is explicitly computed.
Category:20G05 |
6. CJM 2000 (vol 52 pp. 449)
An Intertwining Result for $p$-adic Groups For a reductive $p$-adic group $G$, we compute the supports of the Hecke
algebras for the $K$-types for $G$ lying in a certain frequently-occurring
class. When $G$ is classical, we compute the intertwining between any
two such $K$-types.
Categories:22E50, 20G05 |
7. CJM 2000 (vol 52 pp. 438)
On Some $q$-Analogs of a Theorem of Kostant-Rallis In the first part of this paper generalizations of Hesselink's
$q$-analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the Kostant-Rallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$-analog of the
Kostant-Rallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.
Categories:22E47, 20G05 |
8. CJM 2000 (vol 52 pp. 265)
On Orbit Closures of Symmetric Subgroups in Flag Varieties We study $K$-orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$-stable
(resp. $\theta$-split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$-double coset in $G$ into
$(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$-module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$-module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$-types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 |
9. CJM 1998 (vol 50 pp. 525)
Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
scheme-theoretic
intersections of nilpotent orbit closures with the diagonal matrices.
Here $\mu'$ gives the Jordan block structure of the nilpotent matrix.
de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of
Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology
rings of the varieties constructed by
Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka
polynomial~$\Klmt(q)$ is the Hilbert series for the
multiplicity of the irreducible symmetric group representation indexed
by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$.
\LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition
of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with
non-negative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobenius-splitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of
the ideals of nilpotent orbit closures, arising from the inclusions of
the corresponding sets. We carry out the restriction to the diagonal
using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum
decomposition of the ideals yielding the $k[\Cmubar\cap
\hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of
the $q$-Kostka polynomials.
Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties Categories:05E10, 14M99, 20G05, 05E15 |
10. CJM 1997 (vol 49 pp. 133)
Exterior powers of the adjoint representation Exterior powers of the adjoint representation of a complex semisimple Lie
algebra are decomposed into irreducible representations, to varying
degrees of satisfaction.
Keywords:Lie algebras, adjoint representation, exterior algebra Categories:20G05, 20C30, 22E10, 22E60 |