θ10
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In mathematics, θ10 is a q(q – 1)2/2-dimensional complex representation of the symplectic group Sp4(Fq) over a finite field Fq of order q found by Srinivasan (1968). It is also the name of a similar unitary representation of the symplectic group over a local field, and an automorphic form on the symplectic group over a global field.
θ10 is the only cuspidal unipotent representation of Sp4(Fq). It is the simplest example of a cuspidal unipotent representation of a reductive group, and also the simplest example of a degenerate representation (one without a Whittaker model). General linear groups have no cuspidal unipotent representations and no degenerate representations, so θ10 shows that the representation theory of more general reductive groups shows phenomena that cannot be seen for general linear groups.
The name θ10 is a historical accident that has stuck. Srinivasan (1968) arbitrarily named some of the characters of Sp4(Fq) as θ1, θ2, ..., θ13, and the tenth one in the list just happens to be the cuspidal unipotent character.
Howe & Piatetski-Shapiro (1979) used representations of type θ10 in their construction of counterexamples to the generalized Ramanujan conjecture for the symplectic group. Adams (2004) described the representation θ10 of the Lie group Sp4(R) in detail.
References
- Adams, Jeffrey (2004), "Theta-10", in Hida, Haruzo; Ramakrishnan, Dinakar; Shahidi, Freydoon (eds.), Contributions to automorphic forms, geometry, and number theory: a volume in honor of Joseph A. Shalika, Baltimore, MD: Johns Hopkins Univ. Press, pp. 39–56, ISBN 978-0-8018-7860-2, MR2058602
{{citation}}: External link in|chapterurl=|chapterurl=ignored (|chapter-url=suggested) (help) - Deshpande, Tanmay (2008), An exceptional representation of Sp(4,Fq), arXiv:0804.2722
- Gol'fand, Ya. Yu. (1978), "An exceptional representation of Sp(4,Fq)", Functional Analysis and its Applications, 12 (4), Institute of Problems in Management, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya: 83–84, doi:10.1007/BF01076387, MR 0515634.
- Howe, Roger; Piatetski-Shapiro, I. I. (1979), "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 315–322, ISBN 978-0-8218-1435-2, MR546605
- Kim, Ju-Lee; Piatetski-Shapiro, Ilya I. (2001), "Quadratic base change of θ10", Israel Journal of Mathematics, 123: 317–340, doi:10.1007/BF02784134, MR 1835303
- Srinivasan, Bhama (1968), "The characters of the finite symplectic group Sp(4,q)", Transactions of the American Mathematical Society, 131: 488–525, ISSN 0002-9947, MR0220845