Highest Weights for Categorical Representations

Ben‐Zvi, Dani; Gunningham, Sam; Orem, Hendrik

doi:10.1093/imrn/rny258

Cited by 8 publications

(5 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Informally, the theorem allows one to reconstruct a piece of the entire module from its K invariants, which are often easier to analyze directly. The result and its proof are similar to work of Ben-Zvi-Gunningham-Orem [BZGO18]; for a generalization see also the interesting paper of Yang [Yan21].…”

Section: Hecke Algebras and Invariant Vectorssupporting

confidence: 73%

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

Campbell,

Dhillon

2021

Preprint

View full text Add to dashboard Cite

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the category of Lie algebra representations with a generalized infinitesimal character as the parabolic induction, in the sense of categorical representation theory, of a Steinberg module. Contents 1. Introduction 1 2. Preliminaries and notation 8 3. Hecke algebras and invariant vectors 15 4. A category of Lie algebra representations 20 5. Localization I: The sign representation and the Steinberg module 22 6. parabolic induction for groups and Hecke categories 26 7. Localization II: the parabolic induction of the Steinberg module and Lie algebra representations 28 8. The linkage principle for positive energy representations 32 9. Some applications 38 10. Affine Harish-Chandra bimodules 40 References 43

show abstract

Section: Hecke Algebras and Invariant Vectorssupporting

confidence: 73%

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

Campbell,

Dhillon

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We do not formulate the result in these terms. The assertion stated here follows from our Theorem 3.0.1 using the methods of[BZGO18].…”

mentioning

confidence: 67%

Localization for affine $W$-algebras

Dhillon¹,

Raskin²

2020

Preprint

View full text Add to dashboard Cite

We prove a localization theorem for affine W-algebras in the spirit of Beilinson-Bernstein and Kashiwara-Tanisaki. More precisely, for any non-critical regular weight λ, we identify λ-monodromic Whittaker D-modules on the enhanced affine flag variety with a full subcategory of Category O for the W-algebra.To identify the essential image of our functor, we provide a new realization of Category O for affine Walgebras using Iwahori-Whittaker modules for the corresponding Kac-Moody algebra. Using these methods, we also obtain a new proof of Arakawa's character formulae for simple positive energy representations of the W-algebra.

show abstract

“…4.5.4. We would also like to mention, for a reductive group G, a nice analogue of highest weight theory in the present setting due to Ben-Zvi-Gunningham-Orem [BZGO18]. Namely, they showed that, for any maximal unipotent subgroup N , an arbitrary D-mod(G) representation C can be reconstructed from its N -invariants, or even the weak Cartan invariants of the latter…”

Section: Let Us Mention a Few Features And Examplesmentioning

confidence: 81%

“…Remark 4.5.5. The argument of [BZGO18] yields the following useful variation. Given any subgroup H of a group K such that the quotient of K by the normalizer of H is proper, the tautological map…”

Section: Let Us Mention a Few Features And Examplesmentioning

confidence: 99%

An informal introduction to categorical representation theory and the local geometric Langlands program

Dhillon¹

2022

Preprint

View full text Add to dashboard Cite

We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way we emphasize applications, old and new, to the usual representation theory of reductive and affine Lie algebras. Contents 1. Introduction 1 2. Representations of group algebras and convolution algebras 3 3. Beilinson-Bernstein localization and hidden symmetries 6 4. Categorical representations of groups 19 5. Categorical representations of loop groups 32 6. Local geometric Langlands and affine Beilinson-Bernstein localization 36 Appendix A. From functions on X(F q ) to D-modules on X(C) 67 References 78

show abstract

Highest Weights for Categorical Representations

Cited by 8 publications

References 4 publications

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

Localization for affine $W$-algebras

An informal introduction to categorical representation theory and the local geometric Langlands program

Contact Info

Product

Resources

About