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Search: MSC category 11F70 ( Representation-theoretic methods; automorphic representations over local and global fields )

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1. CMB Online first

Matringe, Nadir
A specialisation of the Bump-Friedberg $L$-function
We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha,\pi)L(2s,\Lambda^2,\pi)$ which we denote by $L^{lin}(s,\pi,\alpha)$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,\mathbb{A})$ for $\mathbb{A}$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi,\alpha)$ has a pole at $1/2$, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha+1/2,\pi)\neq 0$ and $L(1,\Lambda^2,\pi)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is $\geq 0$.

Keywords:automorphic L functions
Categories:11F70, 11F66

2. CMB 2006 (vol 49 pp. 578)

Muić, Goran
On the Structure of the Full Lift for the Howe Correspondence of $(Sp(n), O(V))$ for Rank-One Reducibilities
In this paper we determine the structure of the full lift for the Howe correspondence of $(Sp(n),O(V))$ for rank-one reducibilities.

Categories:22E35, 22E50, 11F70

3. CMB 2002 (vol 45 pp. 364)

Deitmar, Anton
Mellin Transforms of Whittaker Functions
In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

Categories:11F30, 22E30, 11F70, 22E45

4. CMB 2002 (vol 45 pp. 220)

Hakim, Jeffrey; Murnaghan, Fiona
Globalization of Distinguished Supercuspidal Representations of $\GL(n)$
An irreducible supercuspidal representation $\pi$ of $G= \GL(n,F)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be ``distinguished'' by a subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if $\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue of this notion for and irreducible, automorphic, cuspidal representation associated to $\GL(n)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

Categories:22E50, 22E35, 11F70

5. CMB 2001 (vol 44 pp. 160)

Langlands, Robert P.
The Trace Formula and Its Applications: An Introduction to the Work of James Arthur
James Arthur was awarded the Canada Gold Medal of the National Science and Engineering Research Council in 1999. This introduction to his work is an attempt to explain his methods and his goals to the mathematical community at large.

Categories:11F70, 11F72, 58G25

6. CMB 2000 (vol 43 pp. 380)

Shahidi, Freydoon
Twists of a General Class of $L$-Functions by Highly Ramified Characters
It is shown that given a local $L$-function defined by Langlands-Shahidi method, there exists a highly ramified character of the group which when is twisted with the original representation leads to a trivial $L$-function.

Categories:11F70, 22E35, 22E50

7. CMB 1999 (vol 42 pp. 393)

Savin, Gordan
A Class of Supercuspidal Representations of $G_2(k)$
Let $H$ be an exceptional, adjoint group of type $E_6$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of $G_2(k)$.

Categories:22E35, 22E50, 11F70

8. CMB 1997 (vol 40 pp. 376)

Gross, Benedict H.; Savin, Gordan
The dual pair $PGL_3 \times G_2$
Let $H$ be the split, adjoint group of type $E_6$ over a $p$-adic field. In this paper we study the restriction of the minimal representation of $H$ to the closed subgroup $PGL_3 \times G_2$.

Categories:22E35, and, 50, 11F70

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