Parabolic Simple $\mathcal{L}$-invariants and Local-global Compatibility -

Parabolic Simple $L$ -invariants and Local-global Compatibility

Time: 2022-11-11 Published By: He Liu

Speaker(s): Yiqin He (PKU)

Time: 11:00-12:00 November 14, 2022

Venue: Room 77201, Jingchunyuan 78, BICMR

Let

L

be a finite extension of

Q_{p}

and

ρ_{L}

be a potentially semistable noncrystalline

p

-adic representation of

{Gal}_{L}

such that the associated

F

-semisimple Weil-Deligne representation is absolutely indecomposable. Via a study of Breuil's parabolic simple

L

-invariants, we attach to

ρ_{L}

a locally

Q_{p}

-analytic representation

Π (ρ_{L})

{GL}_{n} (L)

, which carries the exact information of the Fontaine-Mazur parabolic simple

L

-invariants of

ρ_{L}

. When

ρ_{L}

comes from a patched automorphic representation of

G (A_{F^{+}})

(for a unitary group

G

over a totally real field

F^{+}

which is compact at infinite places and

{GL}_{n}

p

-adic places), we prove under mild hypothesis that

Π (ρ_{L})

is a subrepresentation of the associated Hecke-isotypic subspace of the Banach spaces of (patched)

p

-adic automophic forms on

G (A_{F^{+}})