Bernstein decomposition

Statement

Let $G$ be a connected reductive group over a nonarchimedean local field $F$ . A cuspidal pair is a pair $(M,\sigma)$ consisting of a Levi subgroup and an irreducible supercuspidal representation of $M(F)$ . Two such pairs are inertially equivalent if they become $G(F)$ -conjugate after twisting $\sigma$ by an unramified character of $M(F)$ .

The Bernstein decomposition is the product decomposition

\operatorname{Rep}(G(F)) \simeq\prod_{\mathfrak s}\operatorname{Rep}^{\mathfrak s}(G(F))

of the category of smooth complex representations into full subcategories indexed by inertial classes $\mathfrak s=[M,\sigma]_G$ . An irreducible representation belongs to the block $\mathfrak s$ exactly when its supercuspidal support has inertial class $\mathfrak s$ .

Blocks and geometry

Each block is an indecomposable summand cut out by a central idempotent in an appropriate categorical sense. Unramified characters of $M(F)$ form a complex torus, and a finite stabilizer acts on it. The resulting quotient is the Bernstein variety attached to $\mathfrak s$ .

Regular functions on these components form the Bernstein center. Types and Hecke algebras often give explicit algebraic models of individual blocks.

References

Joseph Bernstein, “Le ‘centre’ de Bernstein,” in Représentations des groupes réductifs sur un corps local, Travaux en Cours, Hermann, 1984, 1–32.
Colin J. Bushnell and Philip C. Kutzko, The Admissible Dual of $\mathrm{GL}(N)$ via Compact Open Subgroups, Annals of Mathematics Studies 129, Princeton University Press, 1993.