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This article is cited in 12 scientific papers (total in 12 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.
Received: May 18, 2002
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
Linking options:
https://www.mathnet.ru/eng/mmj62 https://www.mathnet.ru/eng/mmj/v2/i3/p533
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