Θ10

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In representation theory, a branch of mathematics, θ10 is a particular unitary representation of the symplectic group Sp4, which has some exceptional properties. The symplectic group Sp4 is the set of 4×4 matrices that preserve a skew-symmetric non-degenerate matrix, but its unitary representations – that is, the ways of realizing that set as a group of linear operators preserving the inner product in a Hilbert space – are often subtle.

The notation θ10 was first used for a complex irreducible representation of the finite group Sp4(Fq), where Fq is a finite field with q elements. In this finite-field case, for q odd, θ10 has dimension q(q – 1)2/2. It is notable because it is simultaneously cuspidal, unipotent, and degenerate, meaning that it has no Whittaker model. Related representations, also denoted θ10, occur for Sp4 over local and global fields.

Finite-field case

Srinivasan (1968) introduced θ10 in her determination of the irreducible characters of the finite symplectic group Sp4(Fq) for q odd. She showed that θ10 has degree q(q – 1)2/2.

The subscript 10 is historical. Srinivasan labelled certain characters of Sp4(Fq) as θ1, θ2, ..., θ13; the tenth character in this list is the cuspidal unipotent character now known as θ10.

In the finite-field setting, θ10 is the only cuspidal unipotent representation of Sp4(Fq). It is also the smallest example of a cuspidal unipotent representation of a reductive group and the smallest example of a degenerate cuspidal representation. General linear groups do not have cuspidal unipotent representations, and they do not have degenerate cuspidal representations. Thus θ10 exhibits features of the representation theory of general reductive groups that are absent for general linear groups.

Construction

One construction of θ10 uses the Weil representation and Howe duality. Let V be a four-dimensional symplectic vector space over Fq, and let E be a two-dimensional anisotropic quadratic space over Fq. Then V ⊗ E is an eight-dimensional symplectic vector space.

The Weil representation of Sp(V ⊗ E) restricts to an action of Sp(V) × O(E). Since O(E) contains the special orthogonal group SO(E) as a subgroup of index two, the representation can be decomposed using the characters of SO(E) and the involution coming from the nontrivial coset of O(E). In this decomposition, one of the resulting irreducible Sp(V)-representations, usually denoted W−1, has dimension q(q – 1)2/2. Its character is the representation θ10.[1]

Properties

The representation θ10 is irreducible, so it has no nonzero proper invariant subspaces. It is cuspidal, meaning that it does not arise from a proper parabolic subgroup by parabolic induction. Equivalently, in the finite-field setting, its restriction to the unipotent radical of a proper parabolic subgroup does not contain the trivial representation.

It is also degenerate: it does not admit a Whittaker model. For representations of reductive groups, the existence of a Whittaker model is often called genericity. Thus θ10 is an example of a cuspidal but non-generic representation.

Finally, θ10 is unipotent in the sense of Deligne–Lusztig theory: it occurs in the part of the character theory attached to the trivial character of a maximal torus. The simultaneous occurrence of all three properties – cuspidal, unipotent, and degenerate – is what makes θ10 exceptional.

Local and global analogues

Representations also denoted θ10 occur for Sp4 over local and global fields. Howe & Piatetski-Shapiro (1979) used such representations in their construction of counterexamples to the generalized Ramanujan conjecture for quasi-split groups. These examples showed that a naive generalization of the Ramanujan conjecture to all cuspidal automorphic representations of reductive groups could not hold without additional hypotheses, such as genericity.

Adams (2004) described the real local representation θ10 for the Lie group Sp4(R). Kim & Piatetski-Shapiro (2001) studied quadratic base change for θ10.

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