1. CJM 2011 (vol 63 pp. 1083)
|Decomposition of Splitting Invariants in Split Real Groups|
For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor
Categories:11F70, 22E47, 11S37, 11F72, 17B22
2. CJM 2007 (vol 59 pp. 1323)
|On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases |
We prove two spectral identities. The first one relates the relative trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a weighted trace formula for $\GL_2$. The second relates a spherical variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67
3. CJM 2007 (vol 59 pp. 1121)
|Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series |
Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.
Categories:11F72, 32N10, 32D15
4. CJM 2002 (vol 54 pp. 92)
|Comparisons of General Linear Groups and their Metaplectic Coverings I |
We prepare for a comparison of global trace formulas of general linear groups and their metaplectic coverings. In particular, we generalize the local metaplectic correspondence of Flicker and Kazhdan and describe the terms expected to appear in the invariant trace formulas of the above covering groups. The conjectural trace formulas are then placed into a form suitable for comparison.
Categories:11F70, 11F72, 22E50
5. CJM 2001 (vol 53 pp. 122)
|A Truncated Integral of the Poisson Summation Formula |
Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V(\bA)$. The Poisson summation formula then yields an identity of distributions on $V(\bA)$. The truncation used is due to Arthur.
6. CJM 2000 (vol 52 pp. 172)
|Cubic Base Change for $\GL(2)$ |
We prove a relative trace formula that establishes the cubic base change for $\GL(2)$. One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.
7. 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
8. CJM 1999 (vol 51 pp. 771)
|Stable Bi-Period Summation Formula and Transfer Factors |
This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to ``periodic'' automorphic forms, are related to analogous bi-period distributions associated to ``periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the ``fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a ``transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$.
Categories:11F72, 11F70, 14G27, 14L35
9. CJM 1999 (vol 51 pp. 266)
|Spectral Estimates for Towers of Noncompact Quotients |
We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$.
Categories:11F72, 58G25, 22E40