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Duflo-Serganova functors and Brundan-Goodwin's parabolic inductions

Shunsuke Hirota

Abstract

Duflo--Serganova functors play a central role in the representation theory of Lie superalgebras and have been studied extensively, especially in the finite-dimensional setting. On the other hand, in the general linear case, Brundan--Goodwin observed that certain distinguished parabolic induction functors are particularly compatible with the principal Whittaker coinvariants functor $H_0$, and these functors are crucial in their theory. In this paper we show that, for certain rank-one DS functors, the image under DS can be computed explicitly for a class of infinite-dimensional modules which can be realized as images of these parabolic induction functors. In particular, for $\mathfrak b$-Verma supermodules attached to a suitable class of Borel subalgebras $\mathfrak b$ and a family of modules whose $H_0$-images are tensor products of one-dimensional evaluation modules for the super Yangian, we obtain an elegant explicit formula.

Duflo-Serganova functors and Brundan-Goodwin's parabolic inductions

Abstract

Duflo--Serganova functors play a central role in the representation theory of Lie superalgebras and have been studied extensively, especially in the finite-dimensional setting. On the other hand, in the general linear case, Brundan--Goodwin observed that certain distinguished parabolic induction functors are particularly compatible with the principal Whittaker coinvariants functor , and these functors are crucial in their theory. In this paper we show that, for certain rank-one DS functors, the image under DS can be computed explicitly for a class of infinite-dimensional modules which can be realized as images of these parabolic induction functors. In particular, for -Verma supermodules attached to a suitable class of Borel subalgebras and a family of modules whose -images are tensor products of one-dimensional evaluation modules for the super Yangian, we obtain an elegant explicit formula.
Paper Structure (7 sections, 20 theorems, 148 equations)

This paper contains 7 sections, 20 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. basic facts about $\mathfrak{gl}(n|n)$
  4. Brundan-Goodwin functors
  5. Duflo-Serganova functors
  6. Main results
  7. $\mathfrak{gl}(2|2)$-examles

Key Result

Theorem 1.1

Let $\mathfrak g=\mathfrak{gl}(n|n)$ and let $\operatorname{BG}_{n|n}(\lambda)$ be the module obtained by applying the Brundan--Goodwin parabolic induction functor to an irreducible $\mathfrak{gl}(1|1)^{\oplus n}$-module with highest weight $\lambda$. If $H_0\operatorname{BG}_{n|n}(\lambda)$ is one- and moreover $H_0\operatorname{BG}_{n-1|n-1}\bigl(\operatorname{pr}_\alpha(\lambda)\bigr)$ is also

Theorems & Definitions (51)

  • Conjecture 1
  • Theorem 1.1: \ref{['thm:DS_on_maBG']}
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1: See also brundan2014representations, Lemma 2.2
  • Definition 4
  • Definition 5: integral Weyl vectors brundan2014representations
  • Definition 6
  • Lemma 2
  • ...and 41 more