TY - JOUR

T1 - Categorical actions and multiplicities in the Deligne category Rep_(GLt)

AU - Entova-Aizenbud, Inna

N1 - Funding Information:
The problem of studying categorical actions on Deligne categories was suggested by I. Losev. The author would like to thank I. Losev for his suggestion and for sharing his ideas on the subject, and P. Etingof and V. Serganova for their helpful comments and suggestions. The author is extremely grateful to J. Brundan for his thorough explanations on categorical actions and pointing out an error in the earlier version of the article. The author was supported by ERC grant no. 030-8915 (PI: David Kazhdan).
Funding Information:
The problem of studying categorical actions on Deligne categories was suggested by I. Losev. The author would like to thank I. Losev for his suggestion and for sharing his ideas on the subject, and P. Etingof and V. Serganova for their helpful comments and suggestions. The author is extremely grateful to J. Brundan for his thorough explanations on categorical actions and pointing out an error in the earlier version of the article. The author was supported by ERC grant no. 030-8915 (PI: David Kazhdan).
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/6/15

Y1 - 2018/6/15

N2 - We study the categorical type A action on the Deligne category Dt=Rep_(GLt) (t∈C) and its “abelian envelope” Vt constructed in [13]. For t∈Z, this action categorifies an action of the Lie algebra slZ on the tensor product of the Fock space F with Ft ∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category Vt the tensor product (in the sense of Losev and Webster, [20]) of categorical slZ-modules Pol and Polt ∨. The latter categorify F and Ft ∨ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18]. When t∉Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of slZ×slZ. This action categorifies the slZ×slZ-module F⊠F∨, making Dt the exterior tensor product of the categorical slZ-modules Pol, Pol∨. Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt.

AB - We study the categorical type A action on the Deligne category Dt=Rep_(GLt) (t∈C) and its “abelian envelope” Vt constructed in [13]. For t∈Z, this action categorifies an action of the Lie algebra slZ on the tensor product of the Fock space F with Ft ∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category Vt the tensor product (in the sense of Losev and Webster, [20]) of categorical slZ-modules Pol and Polt ∨. The latter categorify F and Ft ∨ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18]. When t∉Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of slZ×slZ. This action categorifies the slZ×slZ-module F⊠F∨, making Dt the exterior tensor product of the categorical slZ-modules Pol, Pol∨. Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt.

KW - Categorical actions

KW - Deligne categories

UR - http://www.scopus.com/inward/record.url?scp=85042940161&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2018.02.016

DO - 10.1016/j.jalgebra.2018.02.016

M3 - Article

AN - SCOPUS:85042940161

VL - 504

SP - 391

EP - 431

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -