We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$ -function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$ -function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of  $\unicode[STIX]{x1D70B}$ .

Given a compact Lie group $G$, in this paper we establish $L^{p}$-bounds for pseudo-differential operators in $L^{p}(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space $G\times \widehat{G}$, where $\widehat{G}$ is the unitary dual of $G$. We obtain two different types of $L^{p}$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$ classes which are a suitable extension of the well-known $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ ones on the Euclidean space. The results herein extend classical $L^{p}$ bounds established by C. Fefferman on $\mathbb{R}^{n}$. While Fefferman’s results have immediate consequences on general manifolds for $\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$, our results do not require the condition $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$. Moreover, one of our results also does not require $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$. Examples are given for the case of $\text{SU}(2)\cong \mathbb{S}^{3}$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^{m}$ and $\mathscr{S}_{\frac{1}{2},0}^{m}$ naturally appear, and where conditions $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$ fail, respectively.

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$ , whose characteristic is positive and does not divide the order of the Weyl group of $G$ , and let $\breve{G}$ be its Langlands dual group over $k$ . Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$ -bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$ -local systems on $C$ . Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$ . In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$ . This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$ . Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$ , see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of $\text{GL}(n,\mathbb{R})$ and $\text{GL}(n,\mathbb{C})$ , Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

Let $k$ be a finite extension of $\mathbb{Q}_{p}$ , let ${\mathcal{G}}$ be an absolutely simple split reductive group over  $k$ , and let $K$ be a maximal unramified extension of $k$ . To each point in the Bruhat–Tits building of ${\mathcal{G}}_{K}$ , Moy and Prasad have attached a filtration of ${\mathcal{G}}(K)$ by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when $p$ is sufficiently large. By passing to a finite unramified extension of $k$ if necessary, we obtain new supercuspidal representations of ${\mathcal{G}}(k)$ .

In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of $\mathbb{Q}_{p}$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using the work of M. Emerton, we then describe admissible representations of $\text{GL}_{2}(L)$ in terms of sheaves on the projective limit of these formal schemes. As an application, we show that representations coming from certain equivariant line bundles on Drinfeld’s first étale covering of the $p$-adic upper half plane are admissible.

We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.

In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. Associated with these elements are hyperelliptic curves defined over the residue field, and the numbers we obtain can be expressed in terms of Frobenius eigenvalues on the first $\ell$ -adic cohomology of the curves, generalizing previous result of Hales on stable subregular Shalika germs. These Shalika germ formulas imply new results on stability and endoscopic transfer of nilpotent orbital integrals of ramified unitary groups. We also describe how the same numbers appear in the local character expansions of specific supercuspidal representations and consequently dimensions of degenerate Whittaker models.

Let $\widetilde{\text{Sp}}(2n)$ be the metaplectic covering of $\text{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\text{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

An important result of Arkhipov–Bezrukavnikov–Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of ‘mixed modular sheaves’ recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov’s ‘exotic t-structure’ on the Springer resolution.

In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

Let $G=\mathbb{H}^{n}\rtimes K$ be the Heisenberg motion group, where $K=U(n)$ acts on the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$ by automorphisms. We formulate and prove two analogues of Hardy’s theorem on $G$ . An analogue of Miyachi’s theorem for $G$ is also formulated and proved. This allows us to generalize and prove an analogue of the Cowling–Price uncertainty principle and prove the sharpness of the constant $1/4$ in all the settings.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

We study induced representations of the form $\unicode[STIX]{x1D6FF}_{1}\times \unicode[STIX]{x1D6FF}_{2}\rtimes \unicode[STIX]{x1D70E}$ , where $\unicode[STIX]{x1D6FF}_{1},\unicode[STIX]{x1D6FF}_{2}$ are irreducible essentially square-integrable representations of general linear group and $\unicode[STIX]{x1D70E}$ is a strongly positive discrete series of classical $p$ -adic group, which naturally appear in the nonunitary dual. For $\unicode[STIX]{x1D6FF}_{1}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{a}\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D708}^{b}\unicode[STIX]{x1D70C}_{1}])$ and $\unicode[STIX]{x1D6FF}_{2}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{c}\unicode[STIX]{x1D70C}_{2},\unicode[STIX]{x1D708}^{d}\unicode[STIX]{x1D70C}_{2}])$ with $a\geqslant 1$ and $c\geqslant 1$ , we determine composition factors of such induced representation.

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$ . Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$ . The primary example in our mind is when $G=\text{Sp}(2n)$ , whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$ . As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of  $G$ .

We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$ -cliques, then $G$ admits a ${\it\mu}$ -measurable $d$ -coloring with respect to any Borel probability measure ${\it\mu}$ on $X$ , and a Baire measurable $d$ -coloring with respect to any compatible Polish topology on $X$ . The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$ -colorings of Cayley graphs of degree $d$ , except in two exceptional cases.

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.

A locally compact group G is compact if and only if its convolution algebras contain non-zero (weakly) completely continuous elements. Dually, G is discrete if its function algebras contain non-zero completely continuous elements. We prove non-commutative versions of these results in the case of locally compact quantum groups.