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Search: MSC category 22E35 ( Analysis on $p$-adic Lie groups )

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1. CJM 2013 (vol 66 pp. 566)

Choiy, Kwangho
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms
Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.

Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence
Categories:22E50, 11F70, 22E55, 22E35

2. CJM 2011 (vol 63 pp. 1137)

Moy, Allen
Distribution Algebras on p-adic Groups and Lie Algebras
When $F$ is a $p$-adic field, and $G={\mathbb G}(F)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space ${\mathcal H}(G)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a ${\mathbb C}$-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$-adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of an identity element, and the process $G \mapsto \mathcal{H}(G)$ is not a functor. Bernstein introduced an enlargement $\mathcal{H}\,\hat{\,}(G)$ of $\mathcal{H}(G)$. The algebra $\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left essentially compact. We show that the process $G \mapsto \mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow H$ is a morphism of $p$-adic groups, let $F(\tau) \colon \mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the natural projection, we show that $F(\tau)$ maps $G$-invariant distributions on $\mathcal{G}$ to $N_G (\mathfrak{m})$-invariant distributions on $\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$-invariant non-degenerate symmetric bilinear form on ${\mathfrak g}$ and in the case of $SL(2)$ show how certain members of the family can be moved to the group.

Keywords:distribution algebra, p-adic group
Categories:22E50, 22E35

3. CJM 2010 (vol 62 pp. 1340)

Mœglin, C.
Holomorphie des opérateurs d'entrelacement normalisés à l'aide des paramètres d'Arthur
In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series. To do that we need to normalize using the Langlands--Shahidi's normalization arising from the twisted endoscopy and the associated representations of the general linear group.

Categories:22E50, 22E35

4. CJM 2009 (vol 62 pp. 94)

Kuo, Wentang
The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series
Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects.

Keywords:Langlands correspondence, nilpotent orbits, principal series
Categories:22E50, 22E35

5. CJM 2009 (vol 61 pp. 427)

Tadi\'c, Marko
On Reducibility and Unitarizability for Classical $p$-Adic Groups, Some General Results
The aim of this paper is to prove two general results on parabolic induction of classical $p$-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

Categories:22E50, 22E35

6. CJM 2008 (vol 60 pp. 1306)

Mui\'c, Goran
Theta Lifts of Tempered Representations for Dual Pairs $(\Sp_{2n}, O(V))$
This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $(\Sp_{2n}, O(V))$ over a non-archimedean field $F$ of characteristic different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

Categories:22E35, 22E50, 11F70

7. CJM 2007 (vol 59 pp. 1050)

Raghuram, A.
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

8. CJM 2007 (vol 59 pp. 148)

Muić, Goran
On Certain Classes of Unitary Representations for Split Classical Groups
In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups

Categories:22E35, 22E50, 11F70

9. CJM 2006 (vol 58 pp. 897)

Courtès, François
Distributions invariantes sur les groupes réductifs quasi-déployés
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis.

Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions
Categories:22E35, 20G40

10. CJM 2006 (vol 58 pp. 344)

Goldberg, David
Reducibility for $SU_n$ and Generic Elliptic Representations
We study reducibility of representations parabolically induced from discrete series representations of $SU_n(F)$ for $F$ a $p$-adic field of characteristic zero. We use the approach of studying the relation between $R$-groups when a reductive subgroup of a quasi-split group and the full group have the same derived group. We use restriction to show the quotient of $R$-groups is in natural bijection with a group of characters. Applying this to $SU_n(F)\subset U_n(F)$ we show the $R$ group for $SU_n$ is the semidirect product of an $R$-group for $U_n(F)$ and this group of characters. We derive results on non-abelian $R$-groups and generic elliptic representations as well.

Categories:22E50, 22E35

11. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types
Categories:20G25, 22E35, 20H25

12. CJM 2005 (vol 57 pp. 616)

Muić, Goran
Reducibility of Generalized Principal Series
In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to M\oe glin and Tadi\'c.

Categories:22E35, and, 50, 11F70

13. CJM 2002 (vol 54 pp. 263)

Chaudouard, Pierre-Henri
Intégrales orbitales pondérées sur les algèbres de Lie : le cas $p$-adique
Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\ggo$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\ggo(F)$ sont des distributions $J_M(X,f)$---$f$ est une fonction test---indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d'Arthur. Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont bornées sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un résultat bien connu de Harish-Chandra. Si $M \subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_M^b(X,f)$, égales à $J_M(X,f)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, {\it etc.}) restent bornées quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

Categories:22E35, 11F70

14. CJM 2000 (vol 52 pp. 1101)

Zhang, Yuanli
Discrete Series of Classical Groups
Let $G_n$ be the split classical groups $\Sp(2n)$, $\SO(2n+1)$ and $\SO(2n)$ defined over a $p$-adic field F or the quasi-split classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a quadratic extension $E/F$. We prove the self-duality of unitary supercuspidal data of standard Levi subgroups of $G_n(F)$ which give discrete series representations of $G_n(F)$.

Category:22E35

15. CJM 2000 (vol 52 pp. 306)

Cunningham, Clifton
Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group
This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals---an expression which is ideally suited for the study of the stability of those characters. Building on work of F.~Murnaghan, our proof involves Lusztig's Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of $p$-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

Categories:22E50, 22E35

16. CJM 1999 (vol 51 pp. 130)

Savin, Gordan; Gan, Wee Teck
The Dual Pair $G_2 \times \PU_3 (D)$ ($p$-Adic Case)
We study the correspondence of representations arising by restricting the minimal representation of the linear group of type $E_7$ and relative rank $4$. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.

Categories:22E35, 22E50, 11F70

17. CJM 1998 (vol 50 pp. 74)

Flicker, Yuval Z.
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

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