representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, a branch of mathematics, θ₁₀ is a cuspidal unipotent complex irreducible representation of the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...

Sp₄ over a

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...

, or

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

. introduced θ₁₀ for the

Sp₄(F_''q'') over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

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 In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between rep ...

of Sp₄(F_''q''). It is the simplest example of a cuspidal unipotent representation of a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

, and also the simplest example of a degenerate cuspidal representation (one without a

Whittaker model In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named aft ...

). 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. used the representations θ₁₀ over local and global fields in their construction of counterexamples to the

generalized Ramanujan conjecture A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common character ...

for the symplectic group. described the representation θ₁₀ of the Lie group Sp₄(R) over the local field R in detail.

References

* * *. * * * {{DEFAULTSORT:Theta10 Representation theory Automorphic forms