Restricted tensor-product factorization of an automorphic representation

Let $G$ be a connected reductive group over a global field $F$ . An irreducible admissible automorphic representation has a factorization

\pi \simeq \bigotimes_v' \pi_v

into irreducible admissible representations $\pi_v$ of the groups over the completions $F_v$ . The restricted tensor product is taken with respect to distinguished spherical vectors at almost all finite places.

Construction

Choose compact open subgroups $K_v\subset G(F_v)$ such that, outside a finite set $S$ , the space $\pi_v^{K_v}$ is one-dimensional. Choose a nonzero vector $e_v\in\pi_v^{K_v}$ . Algebraically,

\bigotimes_v'(\pi_v,e_v) = \varinjlim_{S'} \left( \bigotimes_{v\in S'}\pi_v \otimes \bigotimes_{v\notin S'} \mathbb C e_v \right),

where $S'$ ranges over finite sets containing $S$ . Rescaling finitely many $e_v$ does not change the isomorphism class.

At archimedean places one uses the appropriate completed topological tensor product or, more commonly in algebraic statements, the tensor product of Harish–Chandra modules together with the finite-adelic restricted product.

The theorem and its converse

Flath's tensor-product theorem gives the factorization for irreducible admissible representations of the adelic group under the standard hypotheses. Conversely, a suitable restricted tensor product of irreducible admissible local representations is an irreducible admissible representation of $G(\mathbb A_F)$ ; being automorphic is an additional global condition.

Arithmetic role

The factorization makes local-to-global constructions possible. For almost all $v$ , the unramified $\pi_v$ has a Satake parameter. Applying a representation of the $L$ -group gives the local Euler factor. The resulting Euler product depends on the global representation, not merely on an arbitrary collection of local factors.

References

D. Flath, “Decomposition of representations into tensor products,” in Automorphic Forms, Representations and $L$ -Functions, Proc. Sympos. Pure Math. 33, part 1, 1979, pp. 179–183.
A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” ibid., pp. 189–207.