Contragredient representation

For a finite-dimensional representation $(\pi,V)$ of a group $G$ , the contragredient representation on $V^*=\operatorname{Hom}_{\mathbb C}(V,\mathbb C)$ is

(\pi^\vee(g)\ell)(v)=\ell(\pi(g^{-1})v).

In a basis, $\pi^\vee(g)$ is the transpose of $\pi(g)^{-1}$ .

Smooth representations

If $G$ is locally profinite and $V$ is a smooth representation, the correct contragredient space is the smooth dual

V^\vee = \{\ell\in V^*:\ell\text{ is fixed by some compact open subgroup}\},

not generally the entire algebraic dual. If $V$ is admissible, then $V^\vee$ is admissible and the natural map $V\to(V^\vee)^\vee$ is an isomorphism.

Unitary distinction

For a unitary Hilbert representation, the Hilbert-space contragredient is naturally related to the conjugate Hilbert space. Passing among the algebraic, smooth, and Hilbert categories requires specifying which dual is being used.

Langlands compatibility

The local Langlands correspondence is expected, and known in established cases, to carry contragredients to the Chevalley-dual parameter. This compatibility is listed separately in local Langlands compatibilities.

Relation to the letter

The letter denotes this operation by $\pi^e$ . Its finite-dimensional dual-group representations use the ordinary algebraic dual; modern automorphic representations require the smooth local formulation above.

References

Joseph Bernstein and Andrei Zelevinsky, “Induced representations of reductive $p$ -adic groups I,” Annales scientifiques de l'ÉNS 10 (1977), 441–472. Numdam.