2011/10/10 by Igor Frenkel, Frenkel, Igor, Matvei Libine +1
Mathematics · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1110.2106
openalex publication_date 2011/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the series of papers [FL,FL2] we approach quaternionic analysis from the\npoint of view of representation theory of the conformal group SL(4,C) and its\nreal forms. This approach has proven very fruitful and pushed further the\nparallel with complex analysis and develop a rich theory. In [FL2] we study the\ncounterparts of Cauchy-Fueter and Poisson formulas on the spaces of split\nquaternions HR and Minkowski space M and show that they solve the problem of\nseparation of the discrete and continuous series on SL(2,R) and the imaginary\nLobachevski space SL(2,C)/SL(2,R). In particular, we introduce an operator\nPlR, compute its effect on the discrete and continuous series components of\nthe space of functions H(HR) and obtain a surprising formula for the\nPlancherel measure of SL(2,R). The proof is based on a transition to the\nMinkowski space M and some pretty lengthy computations. In this paper we\nintroduce an operator d/dR PlR on H(HR) and show that its effect on the\ndiscrete and continuous series components can be easily computed using the\nSchrodinger model for the minimal representation of O(p,q) (with p=q=3) and the\nresults of Kobayashi-Mano from [KM], particularly their computation of the\nintegral expression for the operator FC. This provides an independent\nverification of the coefficients involved in the formula for PlR. This paper\nonce again demonstrates a close connection between quaternionic analysis and\nrepresentation theory of various O(p,q)'s.\n