Nilpotent Orbits of Orthogonal Groups over p -adic Fields, and the DeBacker Parametrization
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Nilpotent Orbits of Orthogonal Groups over p -adic Fields, and the DeBacker Parametrization Tobias Bernstein1 · Jia-Jun Ma2 · Monica Nevins3
· Jit Wu Yap4
Received: 30 January 2019 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract For local non-archimedean fields k of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups GLn (D), SLn (D) (where D is a central division algebra over k) or classical groups, give a new characterisation of the “building set” (defined by DeBacker) of an sl2 (k)-triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker’s parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building. Keywords p-adic groups · Nilpotent orbits · DeBacker classification · Quadratic forms · Bruhat-Tits buildings Mathematics Subject Classification (2010) 20G25 (17B08, 17B45)
Presented by: Michela Varagnolo Monica Nevins’s research is supported by a Discovery Grant from NSERC Canada. Monica Nevins
[email protected] Tobias Bernstein [email protected] Jia-Jun Ma [email protected] Jit Wu Yap [email protected] 1
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
2
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Rd, Shanghai, 200240, China
3
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada
4
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
T. Bernstein et al.
1 Introduction Rational, or arithmetic, nilpotent adjoint orbits of algebraic groups over a local field k arise in representation theory in several contexts. For example, the Harish-Chandra–Howe character formula locally expresses a character of a representation as a linear combination of (Fourier transforms of) nilpotent orbital integrals. As another example, the orbit method would parametrize representations by admissible coadjoint orbits, with the admissible nilpotent orbits corresponding to core singular cases. Algebraic, or geometric, nilpotent adjoint orbits can be thought of as those under the algebraic group over the algebraic closure of the local field. These orbits can be parameterized in multiple ways, including the Bala-Carter classification (extended to low characteristic by McNinch and others), weighted Dynkin diagrams, and partition-type classifications (for classical groups). The rational points of an algebraic orbit form zero or more rational orbits, and these can in principle be counted using Galois cohomology; yet it remained an open combinatorial problem to count these orbits for orthogonal groups. Solving this is the first goal of this paper, in Section 4. Our second goal is to presen
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