Bernstein center - Knowlpedia

Definition

For a reductive $p$ -adic group $G(F)$ , the Bernstein center is the commutative algebra

\mathfrak Z(G)= \operatorname{End}(\operatorname{Id}_{\operatorname{Rep}(G(F))})

of natural endomorphisms of the identity functor on the category of smooth complex representations. Thus an element $z$ assigns to every smooth representation $V$ an endomorphism $z_V$ , functorially in $V$ .

On an irreducible representation, Schur's lemma makes $z_V$ a scalar. The scalar varies algebraically in unramified families.

Bernstein variety

Under the Bernstein decomposition, the center is the product of the centers of the individual blocks. It identifies with the ring of regular functions on the Bernstein variety, whose components are quotients of unramified-character tori attached to inertial supercuspidal data.

Spectral comparison

The ordinary Bernstein center acts on representations. The spectral Bernstein center is instead the ring of functions on the stack of $L$ -parameters. Fargues–Scholze construct a map from the spectral center to this ordinary center; the two terms should not be treated as synonyms. The action of perfect complexes is the stronger categorical spectral action, not the definition of the spectral center itself.

References

Joseph Bernstein, “Le ‘centre’ de Bernstein,” in Représentations des groupes réductifs sur un corps local, Travaux en Cours, Hermann, 1984, 1–32.
David Helm, “The Bernstein center of the category of smooth $W(k)[\mathrm{GL}_n(F)]$ -modules,” Forum of Mathematics, Sigma 4 (2016), e11. DOI.