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(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 θ₁₀ is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp₄(F_q) 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(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 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,F_q)". arXiv:0804.2722 [math.RT].
Gol'fand, Ya. Yu. (1978), "An exceptional representation of Sp(4,F_q)", 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, S2CID 122223668.
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, S2CID 121587192
Srinivasan, Bhama (1968), "The characters of the finite symplectic group Sp(4,q)", Transactions of the American Mathematical Society, 131 (2): 488–525, doi:10.2307/1994960, ISSN 0002-9947, JSTOR 1994960, MR 0220845