Explicit fundamental solutions of some second order differential\n operators on Heisenberg groups

2012/05/24 by Isolda Cardoso, Cardoso, Isolda, Linda Saal +1
Mathematics · Physics and Astronomy · #43A80 (Primary) 35A08 (Secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1205.5489

openalex publication_date 2012/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p,q,n be natural numbers such that p+q=n. Let FF be either CC,\nthe complex numbers field, or HH, the quaternionic division algebra. We\nconsider the Heisenberg group N(p,q, FF) defined as N(p,q, FF)= FFⁿ\×\n mathfrakIm FF, with group law given by (v,
zeta)(v',
zeta')=(v+v',\n
zeta+
zeta'-1/2
mathfrakIm B(v,v')), where B(v,w)=\∑_j=1^p\nv_j\w_j - \∑_j=p+1ⁿ v_j\w_j. Let U(p,q, FF) be the\ngroup of n\× n matrices with coefficients in FF that leave invariant\nthe form B. In this work we compute explicit fundamental solutions of some\nsecond order differential operators on N(p,q, FF) which are canonically\nassociated to the action of U(p,q, FF).\n

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