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- W3039968654 abstract "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." @default.
- W3039968654 created "2020-07-10" @default.
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- W3039968654 date "2020-07-05" @default.
- W3039968654 modified "2023-10-06" @default.
- W3039968654 title "The Bernstein projector determined by a weak associate class of good cosets" @default.
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- W3039968654 doi "https://doi.org/10.48550/arxiv.2007.02396" @default.
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