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SUMMARY:Dubravka Ban (Southern Illinois University)
DTSTART:20201017T143000Z
DTEND:20201017T150000Z
DTSTAMP:20260418T110209Z
UID:MRTC2020/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MRTC2020/6/"
 >From $\\mathrm{GL}(2\,Q_p)$ to $\\mathrm{SL}(2\,Q_p)$</a>\nby Dubravka Ba
 n (Southern Illinois University) as part of The 2020 Paul J. Sally\, Jr. M
 idwest Representation Theory Conference\n\n\nAbstract\nLet $E$ be a finite
  extension of $\\mathbb{Q}_p$. The $p$-adic Langlands correspondence for $
 G=GL_2(\\mathbb{Q}_p)$ is a bijection between the set of absolutely irredu
 cible 2-dimensional $E$-representations of \n${\\rm Gal} \\overline{\\math
 bb{Q}}_p/\\mathbb{Q}_p)$ and the set of absolutely irreducible admissible 
 non-ordinary $E$-Banach space representations of $G$. We study the corresp
 onding objects for $H=SL_2(\\mathbb{Q}_p)$. Let $\\Pi$ be an irreducible a
 dmissible Banach space representation of $G$. Then $\\Pi|_H$ decomposes as
  a direct sum of inequivalent representations.\n\nLet $\\psi: {\\rm Gal}(\
 \overline{\\mathbb{Q}}_p/\\mathbb{Q}_p) \\to GL_2(E)$ be the associated Ga
 lois representation.\nAssume that $\\psi$ is de Rham with Hodge-Tate weigh
 ts 0 and 1. We compute  the centralizer in $PGL_2(\\overline{E})$ of the i
 mage of the corresponding projective Galois representation $\\overline{\\p
 si}: {\\rm Gal}(\\overline{\\mathbb{Q}}_p/\\mathbb{Q}_p) \\to PGL_2(E)$. W
 e show that the order of the centralizer is equal to the number of compone
 nts of $\\Pi|_H$.\n\nEncapsulated in the $p$-adic Langlands correspondence
  for $G$ is the classical smooth Langlands correspondence.\nThe representa
 tion $\\Pi$ contains a smooth representation $\\pi$. In the case when $\\p
 si$ is non-trianguline\, $\\pi$ is supercuspidal.\nWe investigate the conn
 ection between $\\Pi|_H$ and $\\pi|_H$. This is a joint work with Matthias
  Strauch.\n
LOCATION:https://researchseminars.org/talk/MRTC2020/6/
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