Expand all Collapse all | Results 1 - 5 of 5 |
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 2011 (vol 63 pp. 1107)
Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$,
where $F$ is a non-Archimedean local field
of characteristic
$0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$-case of a
conjecture of
Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$-function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.
Keywords:generic representations, local Langlands parameters Categories:22E50, 11S37 |
3. CJM 2007 (vol 59 pp. 1050)
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$ Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicity-free by proving an appropriate theorem on invariant
distributions; this then proves a multiplicity-one theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.
Categories:22E50, 22E35, 11S37 |
4. CJM 2000 (vol 52 pp. 804)
The Distributions in the Invariant Trace Formula Are Supported on Characters J.~Arthur put the trace formula in invariant form for all connected
reductive groups and certain disconnected ones. However his work was
written so as to apply to the general disconnected case, modulo two
missing ingredients. This paper supplies one of those missing
ingredients, namely an argument in Galois cohomology of a kind first
used by D.~Kazhdan in the connected case.
Categories:22E50, 11S37, 10D40 |
5. CJM 1998 (vol 50 pp. 74)
Elementary proof of the fundamental lemma for a unitary group The fundamental lemma in the theory of automorphic forms is proven
for the (quasi-split) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$-adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
Categories:22E35, 11F70, 11F85, 11S37 |