#
θ_{10}

In representation theory, a branch of mathematics, **θ _{10}** is a cuspidal unipotent complex irreducible representation of the symplectic group Sp

_{4}over a finite, local, or global field.

Srinivasan (1968) introduced θ_{10} for the symplectic group Sp_{4}(**F**_{q}) over a finite field **F**_{q} of order *q*, and showed that in this case it is
*q*(*q* – 1)^{2}/2-dimensional. The subscript 10 in θ_{10} is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp_{4}(**F**_{q}) as θ_{1}, θ_{2}, ..., θ_{13}, and the tenth one in her list happens to be the cuspidal unipotent character.

θ_{10} is the only cuspidal unipotent representation of Sp_{4}(**F**_{q}). 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 θ_{10} exhibits properties of general reductive groups that do not occur for general linear groups.

Howe & Piatetski-Shapiro (1979) used the representations θ_{10} over local and global fields in their construction of counterexamples to the generalized Ramanujan conjecture for the symplectic group. Adams (2004) described the representation θ_{10} of the Lie group Sp_{4}(**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,**F**_{q})". arXiv:0804.2722. - Gol'fand, Ya. Yu. (1978), "An exceptional representation of Sp(4,F
_{q})",*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 θ
_{10}",*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