The Bernstein projector determined by a weak associate class of good cosets
| Autor: | Sandeep Varma, Loren Spice, Yeansu Kim |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
Characteristic function (probability theory) General Mathematics Zero (complex analysis) Field (mathematics) Reductive group Combinatorics Lie algebra FOS: Mathematics Coset Equivalence relation 22E35 Representation Theory (math.RT) Mathematics::Representation Theory Mathematics - Representation Theory Mathematics |
| Popis: | Let $G$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$. In [Kim04], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for $G$ in the sense of A. Moy and G. Prasad. Following [Kim04], we attach to the set \(\overline{\mathfrak s}\) of good \(K\)-types in a weak associate class of positive-depth unrefined minimal $K$-types a $G(F)$-invariant open and closed subset $\mathfrak g(F)_{\overline{\mathfrak s}}$ of the Lie algebra $\mathfrak g(F)$ of $G(F)$, and a subset $\tilde G_{\overline{\mathfrak s}}$ of the admissible dual \(\tilde G\) of \(G(F)\) consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak s}$. Then \(\tilde G_{\overline{\mathfrak s}}\) is the union of finitely many Bernstein components for $G$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak s}}$ that it determines. We show that $E_{\overline{\mathfrak s}}$ vanishes outside the Moy--Prasad $G(F)$-domain $G(F)_r \subset G(F)$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak s}}$ to $G(F)_r$, pushed forward via the logarithm to the Moy--Prasad $G(F)$-domain $\mathfrak g(F)_r \subset \mathfrak g(F)$, agrees on $\mathfrak g(F)_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak g(F)_{\overline{\mathfrak s}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan and Y. Varshavsky in arXiv:1504.01353 for the depth-$r$ Bernstein projector. 16 pages |
| Databáze: | OpenAIRE |
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