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Mosc. Math. J., 2002, Volume 2, Number 3, Pages 533–553 (Mi mmj62)  

This article is cited in 4 scientific papers (total in 4 papers)

Normalized intertwining operators and nilpotent elements in the Langlands dual group

A. Braverman, D. A. Kazhdan

Department of Mathematics, Harvard University

Abstract: Let $F$ be a local non-archimedean field and $\mathbf G$ be a split reductive group over $F$ whose derived group is simply connected. Set $G=\mathbf G(F)$. Let also $\psi\colon F\to\mathbb C^\times$ be a nontrivial additive character of $F$. For two parabolic subgroups $P$$Q$ in $G$ with the same Levi component $M$, we construct an explicit unitary isomorphism $\mathcal F_{P,Q,\psi}\colon L^2(G/[P,P])\overset\sim\to L^2(G/[Q,Q])$ commuting with the natural actions of the group $G\times M/[M,M]$ on both sides. In some special cases, $\mathcal F_{P,Q,\psi}$ is the standard Fourier transform. The crucial ingredient in the definition is the action of the principal $\mathfrak{sl}_2$-subalgebra in the Langlands dual Lie algebra $\mathfrak m^\vee$ on the nilpotent radical a $\mathfrak u_\mathfrak p^\vee$ of the Langlands dual parabolic.
For $M$ as above, we use the operators $\mathcal F_{P,Q,\psi}$ to define a Schwartz space $S(G,M)$. This space contains the space $C_c{(G/[P,P])}$ of locally constant compactly supported functions on $G/[P,P]$ for every $P$ for which $M$ is a Levi component (but does not depend on $P$). We compute the space of spherical vectors in $S(G,M)$ and study its global analogue.
Finally, we apply the above results in order to give an alternative treatment of automorphic $L$-functions associated with standard representations of classical groups.

Key words and phrases: Intertwining operators, principal nilpotent, automorphic $L$-functions.

DOI: https://doi.org/10.17323/1609-4514-2002-2-3-533-553

Full text: http://www.ams.org/.../abst2-3-2002.html
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MSC: 22E50, 22E55
Received: May 18, 2002
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Citation: A. Braverman, D. A. Kazhdan, “Normalized intertwining operators and nilpotent elements in the Langlands dual group”, Mosc. Math. J., 2:3 (2002), 533–553

Citation in format AMSBIB
\Bibitem{BraKaz02}
\by A.~Braverman, D.~A.~Kazhdan
\paper Normalized intertwining operators and nilpotent elements in the Langlands dual group
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 3
\pages 533--553
\mathnet{http://mi.mathnet.ru/mmj62}
\crossref{https://doi.org/10.17323/1609-4514-2002-2-3-533-553}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1988971}
\zmath{https://zbmath.org/?q=an:1022.22015}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208593500003}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Sakellaridis Y., “Spherical Varieties and Integral Representations of l-Functions”, Algebr. Number Theory, 6:4 (2012), 611–667  crossref  mathscinet  zmath  isi  elib
    2. Sakellaridis Y., “Spherical Functions on Spherical Varieties”, Am. J. Math., 135:5 (2013), 1291–1381  crossref  mathscinet  zmath  isi
    3. Pollack A., “Unramified Godement-Jacquet Theory For the Spin Similitude Group”, J. Ramanujan Math. Soc., 33:3 (2018), 249–282  mathscinet  isi
    4. Li W.-W., “Zeta Integrals, Schwartz Spaces and Local Functional Equations Preface”: Li, WW, Zeta Integrals, Schwartz Spaces and Local Functional Equations, Lect. Notes Math., Lecture Notes in Mathematics, 2228, Springer International Publishing Ag, 2018, V+  mathscinet  isi
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