2022/10/04 by Zicheng Qian, Qian, Zicheng
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2210.01381
openalex publication_date 2022/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be prime number and K be a p-adic field. We systematically compute the higher Ext-groups between locally analytic generalized Steinberg representations (LAGS for short) of GLn(K) via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each Ext-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their Ext-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an Ext-group, and then describe the cup product maps between such Ext-groups using these bases. As an application, we generalize Breuil's \mathscrL-invariants for GL2(ℚp) and Schraen's higher \mathscrL-invariants for GL3(ℚp) to GLn(K). Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for GL3(ℚp).