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1. CJM Online first

Pan, Shu-Yen
$L$-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams/Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality
Categories:22E50, 11F27, 20C33

2. CJM Online first

Varma, Sandeep
On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group $\operatorname{G}$ over a $p$-adic field $F$. Assume that there exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$ to an opposite parabolic subgroup. When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$-orbit, F. Shahidi and X. Yu describe a certain distribution $D$ on $\operatorname{M}(F)$ such that, for irreducible unitary supercuspidal representations $\pi$ of $\operatorname{M}(F)$ with $\pi \cong \pi \circ \operatorname{Int} w_0$, $\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is irreducible if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of $\pi$. Since this irreducibility is conjecturally related to $\pi$ arising via transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is of interest to realize $D$ as endoscopic transfer from a simpler distribution on a twisted endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations where $\operatorname{N}$ is abelian. Here, we handle the `standard examples' in cases where $\operatorname{N}$ is nonabelian but admits a Zariski dense $\operatorname{Int} \operatorname{M}$-orbit.

Keywords:induced representation, intertwining operator, endoscopy
Categories:22E50, 11F70

3. CJM Online first

Crann, Jason
Amenability and covariant injectivity of locally compact quantum groups II
Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\infty}(\widehat{\mathbb{G}})$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers--Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Keywords:locally compact quantum group, amenability, injective module
Categories:22D35, 46M10, 46L89

4. CJM Online first

Xu, Bin
On Moeglin's parametrization of Arthur packets for p-adic quasisplit $Sp(N)$ and $SO(N)$
We give a survey on Moeglin's construction of representations in the Arthur packets for $p$-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing Moeglin's parametrization of elements in the Arthur packets with that of Arthur.

Keywords:symplectic and orthogonal group, Arthur packet, endoscopy
Categories:22E50, 11F70

5. CJM Online first

Ganguly, Arijit; Ghosh, Anish
Dirichlet's theorem in function fields
We study metric Diophantine approximation for function fields specifically the problem of improving Dirichlet's theorem in Diophantine approximation.

Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristic
Categories:11J83, 11K60, 37D40, 37A17, 22E40

6. CJM Online first

Kamgarpour, Masoud
On the notion of conductor in the local geometric Langlands correspondence
Under the local Langlands correspondence, the conductor of an irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representations
Categories:17B67, 17B69, 22E50, 20G25

7. CJM 2016 (vol 68 pp. 1067)

Runde, Volker; Viselter, Ami
On Positive Definiteness over Locally Compact Quantum Groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

8. CJM 2016 (vol 68 pp. 309)

Daws, Matthew
Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups
We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the ``maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser
Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25

9. CJM 2014 (vol 66 pp. 1201)

Adler, Jeffrey D.; Lansky, Joshua M.
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes
Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},\Gamma)$, and consider any group $G$ satisfying the axioms. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in $G^*(k)$ to the analogous set for $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

Keywords:reductive group, lifting, conjugacy class, representation, Lusztig series
Categories:20G15, 20G40, 20C33, 22E35

10. CJM 2014 (vol 66 pp. 993)

Beuzart-Plessis, Raphaël
Expression d'un facteur epsilon de paire par une formule intégrale
Let $E/F$ be a quadratic extension of $p$-adic fields and let $d$, $m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $\pi$ and $\sigma$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and $\sigma$ are conjugate-dual. That is to say $\pi\simeq \pi^{\vee,c}$ and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $\pi$ to an unitary representation $\tilde{\pi}$ of a nonconnected group $GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same way. We state and prove an integral formula for $\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of $\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the local Gan-Gross-Prasad conjecture for unitary groups.

Keywords:epsilon factor, twisted groups
Categories:22E50, 11F85

11. CJM 2013 (vol 66 pp. 1250)

Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module for the group $G^a$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$, we call the closure $\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$ of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where even for fundamental weights $\omega$ the varieties $\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula.

Keywords:Lie algebras, flag varieties, symplectic groups, representations
Categories:14M15, 22E46

12. CJM 2013 (vol 66 pp. 1287)

Henniart, Guy; Sécherre, Vincent
Types et contragrédientes
Soit $\mathrm{G}$ un groupe réductif $p$-adique, et soit $\mathrm{R}$ un corps algébriquement clos. Soit $\pi$ une représentation lisse de $\mathrm{G}$ dans un espace vectoriel $\mathrm{V}$ sur $\mathrm{R}$. Fixons un sous-groupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une représentation lisse irréductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel $\mathrm{W}$ de dimension finie sur $\mathrm{R}$. Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algèbre d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$. Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes $\mathrm{V}^ėe$ et $\mathrm{W}^ėe$, et donnons en particulier des conditions sur $\mathrm{W}$ ou sur la caractéristique de $\mathrm{R}$ pour que le comportement soit semblable au cas des représentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriétés générales de $\mathrm{G}$. Nous terminons par une application à la théorie des types pour le groupe $\mathrm{GL}_n$ et ses formes intérieures sur un corps local non archimédien.

Keywords:modular representations of p-adic reductive groups, types, contragredient, intertwining

13. CJM 2013 (vol 66 pp. 241)

Broussous, P.
Transfert du pseudo-coefficient de Kottwitz et formules de caractère pour la série discrète de $\mathrm{GL}(N)$ sur un corps local
Soit $G$ le groupe $\mathrm{GL}(N,F)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish-Chandra de certaines telles représentations en certains éléments elliptiques réguliers.

Keywords:reductive p-adic groups , discrete series, Harish-Chandra character, pseudo-coefficient

14. 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

15. CJM 2013 (vol 66 pp. 354)

Kellerhals, Ruth; Kolpakov, Alexander
The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space
Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

Keywords:hyperbolic Coxeter group, growth rate, Salem number
Categories:20F55, 22E40, 51F15

16. CJM 2013 (vol 66 pp. 303)

Elekes, Márton; Steprāns, Juris
Haar Null Sets and the Consistent Reflection of Non-meagreness
A subset $X$ of a Polish group $G$ is called Haar null if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb R$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszyński and Burke-Miller.

Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real
Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

17. CJM 2012 (vol 66 pp. 102)

Birth, Lidia; Glöckner, Helge
Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map $C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$, $(\gamma,\eta)\mapsto \gamma*\eta$ taking a pair of test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact. More generally, consider $r,s,t \in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$ and a continuous bilinear map $b\colon E_1\times E_2\to F$ to a complete locally convex space $F$. Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$, $(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map. The main result is a characterization of those $(G,r,s,t,b)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed, as well as convolution of compactly supported $L^1$-functions and convolution of compactly supported Radon measures.

Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates
Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

18. CJM 2012 (vol 65 pp. 1043)

Hu, Zhiguo; Neufang, Matthias; Ruan, Zhong-Jin
Convolution of Trace Class Operators over Locally Compact Quantum Groups
We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$. We prove that the reduced quantum group $C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution $\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module $\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is equal to $C_0(\mathbb{G})$. On the other hand, we show that the left $T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product $C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. We establish a natural isomorphism between the completely bounded right multiplier algebras of $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove that if $\mathbb{G}$ is semi-regular, then the space $\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right $\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. In the representation framework formulated by Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B(L_2(\mathbb{G}))$. We also characterize some commutation relations of completely bounded multipliers of $(T(L_2(\mathbb{G})), \triangleright)$ over $B(L_2(\mathbb{G}))$.

Keywords:locally compact quantum groups and associated Banach algebras
Categories:22D15, 43A30, 46H05

19. CJM 2012 (vol 65 pp. 222)

Sauer, N. W.
Distance Sets of Urysohn Metric Spaces
A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With $\operatorname{dist}(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous, universal, separable and complete. (It is not difficult to deduce that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds every separable metric space $\mathrm{M}$ with $\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.) The main results are: (1) A characterization of the sets $\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\mathrm{M}$ is homogeneous.

Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability
Categories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99

20. CJM 2012 (vol 64 pp. 721)

Achab, Dehbia; Faraut, Jacques
Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations
We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-function
Categories:17C36, 22E46, 32M15, 33C80

21. CJM 2012 (vol 65 pp. 66)

Deng, Shaoqiang; Hu, Zhiguang
On Flag Curvature of Homogeneous Randers Spaces
In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groups
Categories:22E46, 53C30

22. CJM 2012 (vol 64 pp. 497)

Li, Wen-Wei
Le lemme fondamental pondéré pour le groupe métaplectique
Dans cet article, on énonce une variante du lemme fondamental pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l'aide de la méthode de descente, qui est conditionnelle: on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s'attend à ce que cette condition soit ultérieurement vérifiée.

Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula
Categories:11F70, 11F27, 22E50

23. CJM 2011 (vol 64 pp. 669)

Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana A.
The Genuine Omega-regular Unitary Dual of the Metaplectic Group
We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal series

24. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
Gosset Polytopes in Picard Groups of del Pezzo Surfaces
In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively.

Categories:51M20, 14J26, 22E99

25. CJM 2011 (vol 64 pp. 481)

Chamorro, Diego
Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition.

Keywords:Sobolev inequalities, polynomial volume growth Lie groups
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