# θ10

In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field.

Srinivasan (1968) introduced θ10 for the symplectic group Sp4(Fq) over a finite field Fq of order q, and showed that in this case it is q(q  1)2/2-dimensional. The subscript 10 in θ10 is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp4(Fq) as θ1, θ2, ..., θ13, and the tenth one in her list happens to be the cuspidal unipotent character.

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 cuspidal representation (one without a Whittaker model). General linear groups have no cuspidal unipotent representations and no degenerate cuspidal representations, so θ10 exhibits properties of general reductive groups that do not occur for general linear groups.

Howe & Piatetski-Shapiro (1979) used the representations θ10 over local and global fields 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) over the local field R in detail.

## References

• Adams, Jeffrey (2004), Hida, Haruzo; Ramakrishnan, Dinakar; Shahidi, Freydoon (eds.), "Theta-10", Contributions to automorphic forms, geometry, and number theory: a volume in honor of Joseph A. Shalika, American Journal of Mathematics, Supplement, Baltimore, MD: Johns Hopkins Univ. Press: 39–56, ISBN 978-0-8018-7860-2, MR 2058602
• 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, Institute of Problems in Management, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, 12 (4): 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, MR 0546605
• 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, doi:10.2307/1994960, ISSN 0002-9947, JSTOR 1994960, MR 0220845