Expand all Collapse all | Results 1 - 6 of 6 |
1. CJM 2011 (vol 63 pp. 1083)
Decomposition of Splitting Invariants in Split Real Groups For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor Categories:11F70, 22E47, 11S37, 11F72, 17B22 |
2. CJM 2009 (vol 61 pp. 1375)
Stable Discrete Series Characters at Singular Elements Write $\Theta^E$ for the stable discrete series character associated
with an irreducible finite-dimensional representation $E$ of a connected
real reductive group $G$. Let $M$ be the centralizer of the split
component of a maximal torus $T$, and denote by $\Phi_M(\gm,\Theta^E)$
Arthur's extension of $ |D_M^G(\gm)|^{\lfrac 12}
\Theta^E(\gm)$ to $T(\R)$. In this paper we give a simple
explicit expression for
$\Phi_M(\gm,\Theta^E)$ when $\gm$ is elliptic in $G$. We do not assume $\gm$ is regular.
Category:22E47 |
3. CJM 2004 (vol 56 pp. 293)
Structure of modules induced from simple modules with minimal annihilator We study the structure of generalized Verma modules over a
semi-simple complex finite-dimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
Bernstein-Gelfand-Gelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of Harish-Chandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.
Keywords:parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categories Categories:17B10, 22E47 |
4. CJM 2003 (vol 55 pp. 1155)
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ |
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{-1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$-orbit (or $K^0$-orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$-orbit (resp.\ $K^0$-orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$-orbits (resp.\ $K^0$-orbits) of smaller dimensions. The
description of the closure of nilpotent $K$-orbits has been known for
some time, but not so for the nilpotent $K^0$-orbits. A conjecture
describing the closure of nilpotent $K^0$-orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$-diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 |
5. 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 |
6. CJM 1997 (vol 49 pp. 417)
Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the Kazhdan-Lusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 |