Isobaric automorphic representation

Let $F$ be a global field and let $n=n_1+\cdots+n_r$ . Given unitary cuspidal automorphic representations $\pi_i$ of $\operatorname{GL}_{n_i}(\mathbb A_F)$ , where $\mathbb A_F$ is the adele ring, their isobaric sum

\pi_1\boxplus\cdots\boxplus\pi_r

is the irreducible automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$ obtained as the Langlands quotient of the normalized parabolic induction of $\pi_1\otimes\cdots\otimes\pi_r$ from the standard Levi subgroup $\prod_i\operatorname{GL}_{n_i}$ .

An automorphic representation obtained this way is isobaric. More general Langlands data may include real powers of determinant; the unitary isobaric normalization places those exponents into the cuspidal constituents in the standard way.

Local factors

At every place, the local component is the corresponding local Langlands quotient at nonarchimedean places. Standard $L$ -functions multiply:

L(s,\pi_1\boxplus\cdots\boxplus\pi_r) =\prod_i L(s,\pi_i).

The multiset of cuspidal summands is unique. This follows from strong multiplicity one and the classification of the discrete spectrum of general linear groups.

References

Hervé Jacquet and Joseph Shalika, “On Euler products and the classification of automorphic representations I,” American Journal of Mathematics 103 (1981), 499–558. JSTOR.
C. Mœglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics 113, Cambridge University Press, 1995.