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 introduced θ10 for the symplectic group Sp4 over a finite field Fq of order q, showed that in this case it is q2/2-dimensional; the subscript 10 in θ10 is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp4 as θ1, θ2... θ13, the tenth one in her list happens to be the cuspidal unipotent character. Θ10 is the only cuspidal unipotent representation of Sp4. It is the simplest example of a cuspidal unipotent representation of a reductive group, the simplest example of a degenerate cuspidal representation. 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 used the representations θ10 over local and global fields in their construction of counterexamples to the generalized Ramanujan conjecture for the symplectic group.

Adams described the representation θ10 of the Lie group Sp4 over the local field R in detail. Adams, Hida, Haruzo. Press: 39–56, ISBN 978-0-8018-7860-2, MR 2058602 Deshpande, Tanmay. "An exceptional representation of Sp". ArXiv:0804.2722. Gol'fand, Ya. Yu. "An exceptional representation of Sp", 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: 83–84, doi:10.1007/BF01076387, MR 0515634. Howe, Roger. "A counterexample to the "generalized Ramanujan conjecture" for split groups", in Borel, Armand. Sympos. Pure Math. XXXIII, Providence, R. I.: American Mathematical Society, pp. 315–322, ISBN 978-0-8218-1435-2, MR 0546605 Kim, Ju-Lee. "Quadratic base change of θ10", Israel Journal of Mathematics, 123: 317–340, doi:10.1007/BF02784134, MR 1835303 Srinivasan, Bhama, "The characters of the finite symplectic group Sp", Transactions of the American Mathematical Society, 131: 488–525, doi:10.2307/1994960, ISSN 0002-9947, JSTOR 1994960, MR 0220845

Florida Gulf Coast Eagles

The Florida Gulf Coast Eagles refer to the fifteen sports teams representing Florida Gulf Coast University in Fort Myers, Florida in intercollegiate athletics, including men and women's basketball, cross country, golf and tennis. The Eagles are members of the Atlantic Sun Conference. FGCU is notable as the youngest institution competing in NCAA Division I, having been founded in 1991 and started classes in 1997. FGCU athletics began in the NAIA. In 2002, Florida Gulf Coast became an independent member of NCAA Division II. In 2006, Florida Gulf Coast applied for NCAA Division I status and became a transitory Division I effective in the 2007–08 season. Florida Gulf Coast became a full Division I member on August 11, 2011. In the 2013 NCAA Men's Division I Basketball Tournament, FGCU became the first number 15 seed to advance to the Sweet 16. FGCU Athletics earned $4.5 million in revenue for the 2015 fiscal year, with revenue from ticket sales earning over $800,000 for FY 2015. Men’s basketball is the athletic department’s "cash cow."

Men's and women's basketball account for over $600,000, baseball draws $28,000 in ticket revenue. The FGCU baseball team became eligible to compete in the NCAA post season in the 2010 season after the two-year NCAA reclassifying period. FGCU has several former baseball players playing in the minor leagues, has three MLB pitchers: Chris Sale, Richard Bleier, Jacob Barnes. In 2017, the Green and Blue appeared in the NCAA Tournament for the first time in school history. FGCU defeated Michigan in the first game 10-6 before falling in the next two contests of the double-elimination regional to Davidson and UNC, respectively. In 2012, FGCU's first year of full Division I post-season eligibility, the men's team played in the Atlantic Sun Championship final, losing to Belmont. In 2013, the team won the Atlantic Sun championship, beating top-seeded Mercer in the conference tournament final; that earned the team an automatic invitation to the 2013 NCAA Men's Division I Basketball Tournament. In its first-ever tournament appearance, the No. 15 seeded Eagles upset the No. 2 seeded Georgetown Hoyas in the first round of the tournament, the No. 7 seeded San Diego State Aztecs in the second round.

The Eagles became the seventh No. 15 seed to advance to the round of 32 and the first to advance to the Sweet 16. On March 24, 2007, FGCU women's basketball team ended their 35–1 season with a loss in the Division II National Championship to Southern Connecticut State University. One year in 2008, the Women's Basketball team qualified for the WNIT, becoming the first team to qualify for the WNIT in its first season of Division I sports. During the 2008 WNIT Tourney, the team became the first Atlantic Sun conference team to win a post-season game since 1998, when Florida International University won in the Women's NCAA tourney; the Women's team won the Atlantic Sun Conference regular season championship with a conference record of 17–3 but was ineligible to compete in the conference tourney, as the team was in transition from Division II to Division I athletics. In the 2010–11 season, the FGCU woman's basketball team won the Atlantic Sun Conference again. In the 2011–12 Season, the Women's team won both the Atlantic Sun Conference's regular season and conference tourney titles, earning its first berth into the NCAA Women's Basketball Championship.

The Eagles lost to St. Bonaventure University in the first round of the tournament 72-65 in Tallahassee, FL; the Florida Gulf Coast Eagles men's soccer team represents Florida Gulf Coast University in all NCAA Division I men's soccer competitions. Founded in 2007, the Eagles are one of the youngest men's soccer programs in the country; the team made the NCAA Tournament in the first year of eligibility, in 2011, returned in 2012. They most returned to the NCAA Tournament in 2014 where they lost to Coastal Carolina 1–0; the men's soccer team plays its home games at the FGCU Soccer Complex. The Florida Gulf Coast Eagles women's soccer team represents Florida Gulf Coast University in all NCAA Division I women's soccer competitions; the Green and Blue have competed in the NCAA Tournament 6 times, with a win against South Florida during the 2015 tournament. The women's team played their first home NCAA tournament game in 2014, before a record crowd of 2,014 at the FGCU Soccer Complex. Official website