A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908468746649600 |
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| author | Khare, Apoorva Teja, G. Krishna |
| author_facet | Khare, Apoorva Teja, G. Krishna |
| contents | Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $λ\in\mathfrak{h}^*$. For $M(λ)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula for the weights of $V$. This is novel even in finite type, and is obtained from $λ$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$.
Our proofs use and reveal a finite family (for each $λ$) of "higher order Verma modules" $\mathbb{M}(λ,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(λ,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(λ)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(λ)$ of Verma and $M(λ,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(λ,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$.
We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(λ,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(λ,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_05515 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$ Khare, Apoorva Teja, G. Krishna Representation Theory 22E47 (primary), 17B10, 17B67, 20C15 (secondary) Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $λ\in\mathfrak{h}^*$. For $M(λ)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula for the weights of $V$. This is novel even in finite type, and is obtained from $λ$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$. Our proofs use and reveal a finite family (for each $λ$) of "higher order Verma modules" $\mathbb{M}(λ,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(λ,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(λ)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(λ)$ of Verma and $M(λ,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(λ,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$. We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(λ,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(λ,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups. |
| title | A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$ |
| topic | Representation Theory 22E47 (primary), 17B10, 17B67, 20C15 (secondary) |
| url | https://arxiv.org/abs/2203.05515 |