Kirillov's character formula for reductive Lie groups

1978 ◽  
Vol 48 (3) ◽  
pp. 207-220 ◽  
Author(s):  
Wulf Rossmann
Keyword(s):  
Lie Groups ◽  
Character Formula ◽  
Reductive Lie Groups

scholarly journals The elliptic Weyl character formula

2014 ◽  
Vol 150 (7) ◽  
pp. 1196-1234 ◽  
Author(s):  
Nora Ganter
Keyword(s):  
Line Bundle ◽  
Lie Groups ◽  
Theoretical Framework ◽  
Flag Variety ◽  
Schubert Calculus ◽  
Character Formula ◽  
Elliptic Cohomology ◽  
Image Position ◽  
Weyl Character Formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

scholarly journals Characteristic Cycles and Wave Front Cycles of Representations of Reductive Lie Groups

2000 ◽  
Vol 151 (3) ◽  
pp. 1071 ◽  
Author(s):  
Wilfried Schmid ◽  
Kari Vilonen
Keyword(s):  
Wave Front ◽  
Lie Groups ◽  
Reductive Lie Groups

Parabolic Higgs Bundles for Real Reductive Lie Groups

2018 ◽  
pp. 653-680 ◽  
Author(s):  
Ignasi Mundet i Riera
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Lie Group ◽  
Vector Bundles ◽  
Structure Group ◽  
Higgs Bundles ◽  
Local Systems ◽  
Parabolic Structure ◽  
Reductive Lie Groups ◽  
Reductive Lie Group

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

scholarly journals Survey on real forms of the complex A2(2)-Toda equation and surface theory

2019 ◽  
Vol 6 (1) ◽  
pp. 194-227 ◽  
Author(s):  
Josef F. Dorfmeister ◽  
Walter Freyn ◽  
Shimpei Kobayashi ◽  
Erxiao Wang
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Loop Group ◽  
Additional Parameter ◽  
Lagrangian Surfaces ◽  
Affine Spheres ◽  
Reductive Lie Groups ◽  
Real Forms ◽  
Minimal Lagrangian Surfaces

AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

Harmonic analysis on reductive Lie groups

1981 ◽  
Vol 15 (4) ◽  
pp. 490-529 ◽  
Author(s):  
D. P. Zhelobenko
Keyword(s):  
Harmonic Analysis ◽  
Lie Groups ◽  
Reductive Lie Groups
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