Higher order Verma modules, and a positive formula for all highest weight modules - talk 2

Apoorva Khare (Indian Institute of Science)

Thu Jun 2, 06:00-07:00 (4 weeks ago)

Abstract: We study weights of highest weight modules $V$ over a Kac-Moody algebra $\mathfrak{g}$ (one may assume this to be $\mathfrak{sl}_n$ throughout the talk, without sacrificing novelty). We begin by recalling the notation, "higher order holes", and "higher order Verma modules" (along with their universal property, via examples). We then recall our result from last time: the weights of any highest weight module equal the weights of its "higher order" Verma cover.

Next, we define the higher order category $\mathcal{O}^\mathcal{H}$, and recall some properties in the zeroth and first order cases (work of Bernstein–Gelfand–Gelfand and Rocha-Caridi), and end by explaining that in the higher order cases, (a) the category $\mathcal{O}^\mathcal{H}$ still has enough projectives and injectives; (b) BGG reciprocity does not always hold "on the nose", yet (c) the "Cartan matrix" (of simple multiplicities in projective covers) is still symmetric over $\mathfrak{g} = \mathfrak{sl}_2^{\oplus n}$.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

( video )

Series comments: Timings may vary depending on the time zone of the speakers.

 Organizers: Amritanshu Prasad*, Apoorva Khare*, Pooja Singla*, R. Venkatesh* *contact for this listing

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