Θ10
Encyclopedia
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
.
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: 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 representation (one without a Whittaker model
).
General linear groups have no cuspidal unipotent representations and no degenerate 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 for the symplectic group. described the representation θ10 of the Lie group Sp4(R) over the local field R in detail.
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 studiesmodules over these abstract algebraic structures...
, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
Sp4 over a finite
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, local
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
, or global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
.
introduced θ10 for the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
Sp4(Fq) over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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: 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
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
, and also the simplest example of a degenerate representation (one without a Whittaker model
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 GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T...
).
General linear groups have no cuspidal unipotent representations and no degenerate 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 for the symplectic group. described the representation θ10 of the Lie group Sp4(R) over the local field R in detail.