170 research outputs found
Global analysis by hidden symmetry
Hidden symmetry of a G'-space X is defined by an extension of the G'-action
on X to that of a group G containing G' as a subgroup. In this setting, we
study the relationship between the three objects:
(A) global analysis on X by using representations of G (hidden symmetry);
(B) global analysis on X by using representations of G';
(C) branching laws of representations of G when restricted to the subgroup
G'.
We explain a trick which transfers results for finite-dimensional
representations in the compact setting to those for infinite-dimensional
representations in the noncompact setting when is -spherical.
Applications to branching problems of unitary representations, and to spectral
analysis on pseudo-Riemannian locally symmetric spaces are also discussed.Comment: Special volume in honor of Roger Howe on the occasion of his 70th
birthda
Transfer of K-types on local theta lifts of characters and unitary lowest weight modules
In this paper we study representations of the indefinite orthogonal group
O(n,m) which are local theta lifts of one dimensional characters or unitary
lowest weight modules of the double covers of the symplectic groups. We apply
the transfer of K-types on these representations of O(n,m), and we study their
effects on the dual pair correspondences. These results provide examples that
the theta lifting is compatible with the transfer of K-types. Finally we will
use these results to study subquotients of some cohomologically induced
modules
Dirac cohomology, elliptic representations and endoscopy
The first part (Sections 1-6) of this paper is a survey of some of the recent
developments in the theory of Dirac cohomology, especially the relationship of
Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology;
the second part (Sections 7-12) is devoted to understanding the unitary
elliptic representations and endoscopic transfer by using the techniques in
Dirac cohomology. A few problems and conjectures are proposed for further
investigations.Comment: This paper will appear in `Representations of Reductive Groups, in
Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa,
published by Springe
Mathematicians’ Central Role in Educating the STEM Workforce
In the recent report Engage to Excel,1 President Obama’s Council of Advisors on Science and Technology (PCAST) identifies mathematics as a bottleneck in undergraduate Science, Technology, Engineering, and Mathematics (STEM) education. Among PCAST’s recommendations are ones
calling for the development and teaching of college-level mathematics courses “by faculty from mathematics-intensive disciplines other than mathematics” and for “a new pathway for producing K–12 mathematics teachers…in programs in mathematics-intensive fields other than mathematics.”2 While we are in sharp disagreement with these specific recommendations, we do share PCAST’s concern for the state of STEM education. We encourage the mathematics
community to focus constructively on the broad view the report sketches. We appeal to the community to amplify its communications with other STEM disciplines, to publicize its teaching innovations, and to redouble its efforts to meet the challenges discussed by PCAST
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
This is the second of a series of four articles studying various
generalisations of Khovanov's diagram algebra. In this article we develop the
general theory of Khovanov's diagrammatically defined "projective functors" in
our setting. As an application, we give a direct proof of the fact that the
quasi-hereditary covers of generalised Khovanov algebras are Koszul.Comment: Minor changes, extra sections on Kostant modules and rigidity of cell
modules adde
Irreducible Characters of General Linear Superalgebra and Super Duality
We develop a new method to solve the irreducible character problem for a wide
class of modules over the general linear superalgebra, including all the
finite-dimensional modules, by directly relating the problem to the classical
Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture
of Brundan on the irreducible characters in the BGG category \mc{O} of the
general linear superalgebra. We also prove the super duality conjecture
Notes on Stein-Sahi representations and some problems of non harmonic analysis
We discuss one natural class of kernels on pseudo-Riemannian symmetric
spaces.Comment: 40p
On unitarizability in the case of classical p-adic groups
In the introduction of this paper we discuss a possible approach to the
unitarizability problem for classical p-adic groups. In this paper we give some
very limited support that such approach is not without chance. In a forthcoming
paper we shall give additional evidence in generalized cuspidal rank (up to)
three.Comment: This paper is a merged and revised version of ealier preprints
arXiv:1701.07658 and arXiv:1701.07662. The paper is going to appear in the
Proceedings of the Simons Symposium on Geometric Aspects of the Trace Formul
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