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1. CJM Online first
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$ Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the
representations $\pi_1,\dots,\pi_k$. This is
the global analogue of the metaplectic tensor product
defined by P. Mezo in the sense that locally at each place $v$,
$\pi_v$ is equivalent to the local metaplectic tensor product of
$\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all
of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then
the metaplectic tensor product is cuspidal (resp. square-integrable
modulo center). We also show that (both
locally and globally) the metaplectic tensor product behaves in the
expected way under the action of a Weyl group element, and show the
compatibility with parabolic inductions.
Keywords:automorphic forms, representations of covering groups Category:11F70 |
2. CJM 2011 (vol 65 pp. 22)
Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters We prove a non-vanishing result for families of
$\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip,
as one factor runs over twists by Hecke characters. As an
application, we simplify the proof, due to Luo, Rudnick, and Sarnak,
of the best known bounds towards the Generalized Ramanujan Conjecture
at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is
the regularization of the units in residue classes by the use of an
Arakelov ray class group.
Keywords:non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture Categories:11F70, 11M41 |
3. CJM 1999 (vol 51 pp. 952)
On Limit Multiplicities for Spaces of Automorphic Forms Let $\Gamma$ be a rank-one arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$-Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
square-integrable $\Gamma$-automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorge-Wallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 |